On the geometry and cohomology of some simple Shimura varieties

Michael Harris ^{1} Department of Mathematics,

University of Paris 7, Paris, France.

Richard Taylor ^{2} Department of Mathematics, Harvard University, Cambridge, MA 02138, U.S.A.

August 30, 1999

1Institut de Mathématiques de Jussieu, U.M.R. 7586 du CNRS 2Partially supported by NSF Grant DMS-9702885 and by the Miller Institute at

the University of California at Berkeley

Introduction

This paper has twin aims. On the one hand we prove the local Langlands conjecture for GL_{n} over a p-adic field. On the other hand in many cases we are able to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the “simple” Shimura varieties studied by Kottwitz in [Ko4]. These two problems go hand in hand.

The local Langlands conjecture is one of those hydra like conjectures which seems to grow as it gets proved. However the generally accepted formulation seems to be the following (see [He2]). Let K be a finite extension of ℚ_{p}. Fix a non-trivial additive character ψ : K → ℂ^{×}. We will denote the absolute value on K which takes uniformisers to the reciprocal of the number of elements in the residue field by | |_{K}. We will let W_{K} denote its Weil group. Recall that local class field theory gives us a canonical isomorphism

Art_{K} : K^{×} → W_{K}^{ab}.

(Normalised so that geometric Frobenius elements correspond to uniformisers.) The local Langlands conjecture provides some sort of description of the whole of W_{K} in the same spirit.

We will let Irr(GL_{n}(K)) denote the set of isomorphism classes or irreducible admissible representations of GL_{n}(K) over ℂ (or what comes to the same thing: irreducible smooth representations). If [π_{1}] ∈ Irr(GL_{n_{1}}(K)) and [π_{2}] ∈ Irr(GL_{n_{2}}(K)) then there is an L-factor L(π_{1} × π_{2}, s) and an epsilon factor ϵ(π_{1} × π_{2}, s, ψ) associated to the pair π_{1}, π_{2} (see for instance [JPSS]).

On the other hand let WDRep_{n}(W_{K}) denote the set of isomorphism classes of n-dimensional Frobenius semi-simple Weil-Deligne representations of the Weil group, W_{K}, of K over ℂ. By a Frobenius semi-simple Weil-Deligne representation of W_{K} over ℂ we mean a pair (r, N) where r is a semi-simple representation of W_{K} on a finite dimensional complex vector space , V, which is trivial on an open subgroup and an element N ∈ End_{ℂ}(V) such that

r(σ)Nr(σ)^{−1} = |Art _{K}^{−1}(σ)|_{K}N

for all σ ∈ W_{K}. Again if [(r, N)] ∈ WDRep_{n_{1}}(W_{K}) then there is an L-factor L((r, N), s) and an epsilon factor ϵ((r, N), s, ψ) associated to (r, N) (see for instance [Tat2] and section 12 of this paper for the precise normalisations we are using).

By a local Langlands correspondence for K we shall mean a collection of bijections

rec_{K} : Irr(GL_{n}(K)) ⟶ WDRep_{n}(W_{K})

for every n ≥ 1 satisfying the following properties.

1. If π ∈ Irr(GL_{1}(K)) then rec_{K}(π) = π ∘ Art _{K}^{−1}.

2. If [π_{1}] ∈ Irr(GL_{n_{1}}(K)) and [π_{2}] ∈ Irr(GL_{n_{2}}(K)) then

L(π_{1} × π_{2}, s) = L(rec_{K}(π_{1}) ⊗ rec_{K}(π_{2}), s)

and

ϵ(π_{1} × π_{2}, s, ψ) = ϵ(rec_{K}(π_{1}) ⊗ rec_{K}(π_{2}), s, ψ).

3. If [π] ∈ Irr(GL_{n}(K)) and χ ∈ Irr(GL_{1}(K)) then rec_{K}(π ⊗ (χ ∘ det)) = rec_{K}(π) ⊗ rec_{K}(χ).

4. If [π] ∈ Irr(GL_{n}(K)) and π has central character χ then det rec_{K}(π) = rec_{K}(χ).

5. If [π] ∈ Irr(GL_{n}(K)) then rec_{K}(π^{∨}) = rec_{K}(π)^{∨} (where ∨ denotes contragredient).

Henniart showed (see [He5]) that there is at most one set of bijections rec_{K} with these properties. The commonest formulation of the local Langlands conjecture for GL_{n} is the following theorem.

Theorem A A local Langlands correspondence rec_{K} exists for any finite extension K/ℚ_{p}.

However it seems to us that one would really like more than this simple existence theorem. On the one hand it would be very useful if one had some sort of explicit description of this map rec_{K}. Our methods shed no light on this. One might well hope that the methods of Bushnell, Henniart and Kutzko will lead to an explicit version of this theorem. On the other hand one would also like to know that the local reciprocity map rec_{K} is compatible with global reciprocity maps whenever the global map is known to exist. Our methods do not resolve this latter question but they do shed considerable light on it. For instance in the cases considered by Clozel in [Cl1] we settle this question affirmatively up to semisimplification (in particular we do not identify the two N’s).

Maybe a remark on the history of this problem is in order. The existence of rec_{K}|_{Irr(GL_{1}(K))} with the desired properties follows from local class field theory (due originally to Hasse [Has]), but this preceded the general conjecture. The key generalisation to n > 1 is due to Langlands (see [Lan]), who formulated some much more wide ranging, if less precise, conjectures. The formulation in the form described here, with its emphasis on epsilon factors of pairs, seems to be due to Henniart (see [He2]). Henniart’s formulation has the advantage that there is a most one such correspondence, but as remarked above it limits somewhat the scope of Langlands’ original desirata. The existence of rec_{K}|_{Irr(GL_{2}(K))} with the desired properties was established by Kutzko ([Ku]), following earlier partial work by a number of people. The existence of rec_{K}|_{Irr(GL_{3}(K))} with almost all the desired properties was established by Henniart ([He1]). In particular, his correspondence had enough of these properties to characterise it uniquely. Both the work of Kutzko and Henniart relied on a detailed classification of all elements of Irr(GL_{n}(K)). These methods have since been pushed much further, but to date have not provided a construction of rec_{K} which demonstrably has the desired properties on Irr(GL_{n}(K)) for any n > 3. In the case of completions of functions fields of transcendence degree 1 over finite fields, the corresponding theorem was proved by Laumon, Rapoport and Stuhler ([LRS]).

We will let Cusp (GL_{n}(K)) denote the subset of Irr(GL_{n}(K)) consisting of equivalence classes of supercuspidal representations. Let Rep_{n}(W_{K}) denote the subset of WDRep_{n}(W_{K}) consisting of equivalence classes of pairs (r, N) with N = 0. Also let Irr_{n}(W_{K}) denote the subset of Rep_{n}(W_{K}) consisting of equivalence classes of pairs (r, 0) with r irreducible. It follows from important work of Zelevinsky [Z] that it suffices to construct bijections

rec_{K} : Cusp (GL_{n}(K)) ⟶ Irr_{n}(W_{K})

with the properties listed above (see [He2].) In a key breakthrough, Henniart [He4] showed that there did exist bijections

rec_{K} : Cusp (GL_{n}(K)) ⟶ Irr_{n}(W_{K}),

which preserved conductors and were compatible with twists by unramified characters. He was however unable to show that these bijections had enough of the other desired properties to characterise them uniquely. The usefulness of this result is that it allows one to use counting arguments, for instance any injection Cusp (GL_{n}(K)) ↪ Irr_{n}(W_{K}) satisfying the desired properties must be a bijection. (This result is usually referred to as the numerical local Langlands theorem.)

We will give a natural construction of a map

rec_{K} : Cusp (GL_{n}(K)) ⟶ Irr_{n}(W_{K}),

which we will show is compatible with the association of l-adic representations to many automorphic forms on certain unitary groups. Using this global compatibility and some instances of non-Galois automorphic induction discovered by one of us (M.H., see [Har2]) we will see that there is a subset Cusp (GL_{n}(K))′ ⊂ Cusp (GL_{n}(K)) such that

rec_{K} : Cusp (GL_{n}(K))′ ⟶^{∼} Irr_{n}(W_{K})

and such that rec_{K}|_{Cusp (GL_{n}(K))′} has all the desired properties. The subset Cusp (GL_{n}(K))′ may be described as those elements of Cusp (GL_{n}(K)) which become unramified after some series of cyclic base changes. Appealing to Henniart’s numerical local Langlands theorem ([He4]) we can conclude that Cusp (GL_{n}(K)) = Cusp (GL_{n}(K))′ and so deduce theorem A.

One of us (M.H. see [Har1]) had previously given a different construction of a map

rec′_{K} : Cusp (GL_{n}(K)) ⟶ Irr_{n}(W_{K}).

In some cases he was able to show its compatibility with the association of ladic representations to certain classes of automorphic forms on unitary groups. As a result he could deduce the local Langlands conjecture only for p > n (see [BHK] and [Har2]). A posteriori we can show that rec′_{K} = rec_{K}. Since the distribution of a preliminary version of this paper, but before the distribution of the final version, Henniart [He6] has given a much simpler proof of theorem A by making much cleverer use of the non-Galois automorphic induction of [Har2] and of his own numerical local Langlands theorem [He4]. He does not need the a priori construction of a map rec_{K} compatible with some instances of the global correspondence, and thus he is able to by-pass all the main results in this paper. For the reader interested only in theorem A his is clearly the better proof. None the less we believe the results of this paper are still important as they establish many instances of compatibility between the global and local correspondences.

Let us now explain our construction of maps

rec_{K} : Cusp (GL_{n}(K)) ⟶ Rep_{n}(W_{K}).

To this end choose a prime l ≠ p and fix an isomorphism ℂ ≅ ℚ_{l}^{ac}. Let k denote the residue field of K. For any g ≥ 1 there is, up to isomorphism, a unique one-dimensional formal O_{K}-module Σ_{K,g}/k^{ac} of O_{K}-height g. Then End_{O_{K}}(Σ_{K,g}) ⊗_{ℤ} ℚ ≅ D_{K,g}, the division algebra with centre K and Hasse invariant 1/g. Drinfeld showed that the functor which associates to any Artinian local O_{K}-algebra A with residue field k^{ac} the set of isomorphism classes of deformations of Σ_{K,g} to A is prorepresented by a complete noetherian local O_{K} algebra R_{K,g} with residue field k^{ac}. (In fact he showed that R_{K,g} is a formal power series ring in g − 1 variables over the ring of integers of the completion of the maximal unramified extension of K.) We will let Σ̃_{K,g} denote the universal deformation of Σ_{K,g} over R_{K,g}. (In the case g = 1 one just obtains the base change to the ring of integers of the completion of the maximal unramified extension of K of any Lubin-Tate formal O_{K} module over K.) Drinfeld further showed that for any integer m ≥ 0 there is a finite flat R_{K,g}- algebra R_{K,g,m} over which Σ̃_{K,g} has a universal Drinfeld level p^{m}-structure. We will consider the direct limit over m of the formal vanishing cycle sheaves of Spf R_{K,g,m} with coefficients in ℚ_{l}^{ac}. This gives a collection {Ψ_{K,l,g}^{i}} of infinitedimensional ℚ_{l}^{ac} vector spaces with natural admissible actions of the subgroup of GL_{g}(K) × D_{K,g}^{×} × W_{K} consisting of elements (γ, δ, σ) such that

| det δ|| det γ|^{−1}|Art _{K}^{−1}σ| = 1.

For any irreducible representation ρ of D_{K,g}^{×} we set

Ψ_{K,l,g}^{i}(ρ) = Hom_{O_{D_{K,g}}^{×}} (ρ, Ψ_{K,l,g}^{i}).

This becomes an admissible GL_{g}(K) × W_{K}-module. In the case g = 1 we have Ψ_{K,l,1}^{i} = (0) for i > 0, while it follows from the theory of Lubin-Tate formal groups (see [LT]) that Ψ_{K,l,1}^{0}(ρ) = ℚ_{l}^{ac} with an action of K^{×} × W_{K} via ρ^{−1} × (ρ ∘ Art _{K}^{−1}) (see section 3.4 of [Car3]).

To describe Ψ_{K,l,g}^{i}(ρ) in greater generality we must recall that Deligne, Kazhdan and Vigneras (see [DKV]) and Rogawski ([Rog2]) have given a bijection between irreducible representations of D_{K,g}^{×} and (quasi-)square integrable irreducible admissible representations of GL_{g}(K) characterised by a natural character identity (see appendix IV). This generalises work of Jacquet and Langlands in the case g = 2 so we will denote the correspondence ρ ↦ JL (ρ). Carayol essentially conjectured ([Car3]) that if JL (ρ) is supercuspidal then

Ψ_{K,l,g}^{g−1}(ρ) ≅ JL (ρ)^{∨} × rec_{K}(JL (ρ) ⊗ | det |^{(1−g)/2}).

We do not quite prove this (though it may be possible by our methods to do so). However motivated by Carayol’s conjecture our first main theorem is the following. To state it let [Ψ_{K,g}(ρ)] denote the virtual representation (−1)^{g−1} ∑_{i=0}^{g−1}(−1)^{i}[Ψ_{K,l,g}^{i}(ρ)].

Theorem B If π is an irreducible supercuspidal representation of GL_{g}(K) then there is a (true) representation

r_{l}(π) : W_{K} → GL_{g}(ℚ_{l}^{ac}),

such that in the Grothendieck group

[Ψ_{K,l,g}(JL (π)^{∨}] = [π ⊗ r_{l}(π)].

In the case n = 1 Lubin-Tate theory allows one to identify

r_{l}(π) = π^{−1} ∘ Art _{K}^{−1}.

We use this theorem to define

rec_{K} : Cusp (GL_{g}(K)) ⟶ Rep_{g}(W_{K})

by the formula

rec_{K}(π) = r_{l}(π^{∨} ⊗ (| | ∘ det)^{(1−g)/2}).

It will also be convenient for us to extend r_{l} to all irreducible admissible representations of GL_{g}(K) as follows. If π is an irreducible admissible representation of GL_{g}(K), then we can find positive integers g_{1}, ..., g_{t} which sum to g and irreducible supercuspidal representations π_{i} of GL_{g_{i}}(K) such that π is a subquotient of n-Ind (π_{1} × ... × π_{t}), where we are using the usual normalised induction (see appendix I). Then we set

r_{l}(π) = ⨁_{i=1}^{t} r_{l}(π_{i}) ⊗ |Art _{K}^{−1}|^{(g_{i}−g)/2}.

This is well defined and

rec_{K}(π) = (r_{l}(π^{∨} ⊗ (| | ∘ det)^{(1−g)/2}), N),

for some N.

Our second key result is that r_{l} is compatible with many instances of the global Langlands correspondence. The following theorem strengthens a theorem of Clozel [Cl1] (in which he only identifies [R(Π)|_{W_{Fy}}] for all but finitely many places y, and specifically for none of the bad places).

Theorem C Suppose that L is a CM field and that Π is a cuspidal automorphic representation of GL_{g}(𝔸_{L}) satisfying the following conditions:

• Π^{∨} ≅ Π^{c},

• Π_{∞} has the same infinitesimal character as some algebraic representation over ℂ of the restriction of scalars from L to ℚ of GL_{g},

• and for some finite place x of L the representation Π_{x} is square integrable.

Then there is a non-zero integer a(Π) and a continuous representation R(Π) of Gal (L^{ac}/L) over ℚ_{l}^{ac} such that for any finite place y of L not dividing l we have

[R(Π)|_{W_{Ly}}] = a(Π)[r_{l}(Π_{y})].

In the case n = 2 and K = ℚ_{p} and F^{+} = ℚ both theorems B and C were essentially proved by Deligne in his beautiful letter [De2]. (The argument was completed by Brylinski [Bry].) Carayol [Car2] generalised Deligne’s method to essentially prove both theorems B and C in the general n = 2 case. We will simply generalise Deligne’s approach to n > 2. The combination of theorems B and C, Henniart’s numerical local Langlands theorem [He4] and the non-Galois automorphic induction of [Har2] suffice to prove theorem A.

Both theorems B and C follow without great difficulty from an analysis of the bad reduction of certain Shimura varieties. We will next explain this analysis. Unfortunately we must first establish some notation. Let E denote an imaginary quadratic field in which p splits: p = uu^{c}. Let F^{+} denote a totally real field of degree d and set F = EF^{+}. Fix a place w of F above u. Let B be a division algebra with centre F such that

• the opposite algebra B^{op} is isomorphic to B ⊗_{E,c} E;

• B is split at w;

• at any place x of F which is not split over F^{+}, B_{x} is split;

• at any place x of F which is split over F^{+} either B_{x} is split or B_{x} is a division algebra,

• if n is even then 1 + dn/2 is congruent modulo 2 to the number of places of F^{+} above which B is ramified.

Let n denote [B : F]^{1/2}. We can pick a positive involution of the second kind ∗ on B (i.e. ∗|_{F} = c and tr_{B/ℚ}(xx^{∗}) > 0 for all nonzero x ∈ B). If β ∈ B^{∗=−1} then we will let

• G denote the algebraic group with G(ℚ) the subgroup of elements x ∈ (B^{op})^{×} so that x^{∗}βx = ν(x)β for some ν(x) ∈ ℚ^{×},

• ν : G → 𝔾_{m} the corresponding character,

• G_{1} the kernel of ν, • and ( , ) the pairing on B defined by

(x, y) = (tr_{F/ℚ} ∘ tr_{B/F})(xβy^{∗}).

We can and will choose β such that

• G is quasi-split at all rational primes x which do not split in E

• and G_{1}(ℝ) ≅ U(n − 1, 1) × U(n)^{[F^{+}:ℚ]−1}.

If U ⊂ G(𝔸^{∞}) is an open compact subgroup we will consider the following moduli problem. If S is a connected F -scheme and s is a closed geometric point of S then we consider equivalence classes of quadruples (A, λ, i, \overline{η}) where

• A is an abelian scheme of dimension [F^{+} : ℚ]n^{2};

• λ : A → A^{∨} is a polarisation;

• i : B ↪ End (A) ⊗_{ℤ} ℚ such that λ ∘ i(b) = i(b^{∗})^{∨} ∘ λ for all b ∈ B and such that

tr (b|_{Lie(A)}) = (c ∘ tr_{F/E} ∘ tr_{B/F})(nb) + tr_{B/F} (b) − (c ∘ tr_{B/F})(b)

for all b ∈ B;

• \overline{η} is a π_{1}(S, s)-invariant U-orbit of isomorphisms of B ⊗_{ℚ} 𝔸^{∞}-modules η : V ⊗_{ℚ} 𝔸^{∞} → V A_{s} which take the standard pairing ( , ) on V to a scalar multiple of the λ-Weil pairing on V A_{s}.

We consider two such quadruples equivalent if the abelian varieties are isogenous in a way that preserves the rest of the structure (but only need preserve the polarisation up to ℚ^{×} multiples). The set of equivalence classes is canonically independent of the choice of s. If U is sufficiently small this moduli problem is represented by a smooth proper scheme of finite type X_{U}/F.

If ξ is a representation of the algebraic group G over ℚ_{l}^{ac} then we can define a lisse ℚ_{l}^{ac} sheaf L_{ξ} on X_{U}. Then we will consider the ℚ_{l}^{ac}-vector spaces

H^{i}(X, L_{ξ}) = lim_{→U} H_{et}^{i}(X_{U} × F^{ac}, L_{ξ}).

This is naturally an admissible G(𝔸^{∞}) × Gal (F^{ac}/F)-module. In fact we can write

H^{i}(X, L_{ξ}) = ⨁_{π} π ⊗ R_{ξ}^{i}(π),

where π runs over irreducible admissible representations of G(𝔸^{∞}) and R_{ξ}^{i}(π) is a finite dimensional continuous representation of Gal (ℚ^{ac}/ℚ). We will focus on the virtual representation

[R_{ξ}(π)] = (−1)^{n−1} ∑_{i}(−1)^{i}[R_{ξ}^{i}(π)].

Kottwitz (see [Ko4]) determined tr [R_{ξ}(π)](Frob_{x}) in terms of ρ for all but finitely many places x of F. He thus completely determined the virtual representation [R_{ξ}(π)]. We will extend Kottwitz’s description to all places x |̸l of F (see corollary 11.12).

We have an isomorphism

G(ℚ_{p}) ≅ E_{u^{c}}^{×} × ∏_{x|u}(B_{x}^{op})^{×}

and hence a decomposition

G(𝔸^{∞}) ≅ G(𝔸^{∞,p}) × E_{u^{c}}^{×} × ∏_{x|u}(B_{x}^{op})^{×}.

Thus we may decompose an irreducible admissible representation π of G(𝔸^{∞}) as

π ≅ π^{p} ⊗ π_{p,0} ⊗ ⨂_{x|u} π_{x}.

For h = 1, ..., n we will let P_{h}^{op} denote the parabolic subgroup of GL_{n}(F_{w}) consisting of block lower triangular matrices with an (n − h) × (n − h)-block in the top left and an h × h block in the bottom right. It has Levi component GL_{n−h}(F_{w}) × GL_{h}(F_{w}). Let N_{h}^{op} denote its unipotent radical. Suppose that ρ is an irreducible representation of D_{F_{w},n−h}^{×}. We will let φ_{JL (ρ)^{∨}} ∈ C^{∞}(GL_{n−h}(F_{w})) denote a pseudo-coefficient for JL (ρ)^{∨} (so that φ_{JL (ρ)^{∨}} is compactly supported mod centre, and for any irreducible tempered admissible representation α of GL_{n−h}(F_{w}) with the same central character as JL (ρ)^{∨} we have tr α(φ_{JL (ρ)^{∨}}) = vol (D_{F_{w},n−h}^{×}/F_{w}^{×}) if α ≅ JL (ρ)^{∨} and = 0 otherwise). Then we define a homomorphism

n-red_{ρ}^{(h)} : Groth (GL_{n}(F_{w})) ⟶ Groth (GL_{h}(F_{w}))

as a composite

Groth (GL_{n}(F_{w})) → Groth (GL_{n−h}(F_{w}) × GL_{h}(F_{w})) → Groth (GL_{h}(F_{w})),

where • the first map takes the class, [π], of an irreducible admissible represen-

tation, π, to the class, [J_{N_{h}^{op}}(π)], of its normalised Jacquet module (see appendix I),

• and the second map takes the class, [α ⊗ β], to

– vol (D_{F_{w},n−h}^{×}/F_{w}^{×})^{−1}tr α(φ_{JL (ρ)^{∨}}) times [β] if the central characters of α and JL (ρ)^{∨} are equal,

– and to 0 otherwise.

Our key technical result is the following theorem relating R_{ξ}(π) and [Ψ_{F_{w},l,g}] for 1 ≤ g ≤ n. From it both theorem B and theorem C follow without undue difficulty. (As does corollary 11.12.)

Theorem D Suppose that π is an irreducible admissible representation of G(𝔸^{∞}) such that π_{p,0}|_{ℤ_{p}^{×}} = 1. Then

n[π_{w}][R_{ξ}(π)|_{W_{Fw}}] = (dim[R_{ξ}(π)]) ∑_{h=0}^{n−1} ∑_{ρ} n-Ind _{P_{h}^{op}(F_{w})}^{GL_{n}(F_{w})}((n-red_{ρ}^{(h)}[π_{w}])
[Ψ_{F_{w},l,n−h}(ρ) ⊗ ((π_{p,0}^{−1} ⊗ | |_{p}^{−h/2}) ∘ Art _{ℚ_{p}}^{−1})|_{W_{Fw}}])

where ρ runs over irreducible admissible representations of D_{F_{w},n−h}^{×}.

Almost all of this paper is devoted to proving this theorem. In the rest of this introduction we will give a very brief sketch of the strategy. We caution the reader that in the rest of this introduction we will not make precise mathematical statements, but rather comments that we hope will convey an idea of our methods. We refer the reader to the body of our article for the accurate formulation of these ideas.

We compute the cohomology groups H_{et}^{i}(X_{U} × F_{w}^{ac}, L_{ξ}) via vanishing cycle sheaves on the special fibre \overline{X}_{U} of X_{U}. Thus we are led to try to compute the cohomology groups

H_{et}^{i}(\overline{X}_{U} × k(w)^{ac}, R^{j}Ψ_{η}(ℚ_{l}^{ac}) ⊗ L_{ξ}),

where k(w) denotes the residue field of w. The first key idea is to introduce a certain stratification on \overline{X}_{U} and compute stratum by stratum. Consider the w^{∞} torsion points on the universal abelian variety over X_{U}. It has an action of B_{w} ≅ M_{n}(F_{w}). Applying the idempotent (e_{ij}) ∈ M_{n}(F_{w}) with e_{11} = 1 and e_{ij} = 0 otherwise, we obtain a divisible O_{F_{w}}-module G/X_{U} of O_{F_{w}}-height n and of dimension 1. For h = 0, ..., n−1 we will let \overline{X}_{U}^{(h)} denote the (h-dimensional) locally closed reduced subscheme of \overline{X}_{U} where the maximal etale quotient of G has O_{F_{w}}-height h. (Then the closure of \overline{X}_{U}^{(h)} is the union of the \overline{X}_{U}^{(h′)} for h′ ≤ h.) It will suffice to compute

H_{c}^{i}(\overline{X}_{U}^{(h)} × k(w)^{ac}, R^{j}Ψ_{η}(ℚ_{l}^{ac}) ⊗ L_{ξ}).

Next we restrict to U of the form U^{w} ×U_{w} where O_{E,u^{c}}^{×} ⊂ U^{w} ⊂ G(𝔸^{∞,p})× E_{u^{c}}^{×} × ∏_{x|u,x≠w}(B_{x}^{op})^{×} and U_{w} ⊂ GL_{n}(F_{w}). If U_{w} is the group of matrices in GL_{n}(O_{F,w}) congruent to 1 modulo w^{m} we will write U = U^{w}(m). To analyse \overline{X}_{U(m)}^{(h)} we introduce an analogue of Igusa curves in this setting: we call them Igusa varieties of the first kind. More precisely I_{U^{w},m}^{(h)} will denote the etale

cover of \overline{X}_{U(0)}^{(h)} which parametrises isomorphisms

(O_{F_{w}}/w^{m})^{h} →^{∼} G^{et}[w^{m}].

One can define this Igusa variety of the first kind not only as a variety in characteristic p but as a formal scheme. Thus we obtain formal schemes (I_{U^{w},m}^{(h)})^{∧}(t) with special fibre I_{U^{w},m}^{(h)} over which there is a universal deformation of the formal O_{F_{w}}-module G^{0} together with its Drinfeld level w^{t}-structure.

One can show that \overline{X}_{U(m)}^{(h)} is a disjoint union of copies of I_{U^{w},m}^{(h)} except that

the structure map down to \overline{X}_{U(0)}^{(h)} is twisted by a power of Frobenius. If P_{h} denotes the opposite parabolic to P_{h}^{op} then one can obtain an isomorphism

lim_{→m} H_{c}^{i}(\overline{X}_{U(m)}^{(h)} × k(w)^{ac}, R^{j}Ψ_{η}(ℚ_{l}^{ac}) ⊗ L_{ξ}) ≅
Ind _{P_{h}}^{GL_{n}(F_{w})} lim_{→m,t} H_{c}^{i}(I_{U^{w},m}^{(h)} × k(w)^{ac}, R^{j}Ψ_{η}(ℚ_{l}^{ac})_{(I_{U^{w},m}^{(h)})^{∧}(t)_{η}} ⊗ L_{ξ}).

Thus it will suffice to compute

H_{c}^{i}(I_{U^{w},m}^{(h)} × k(w)^{ac}, R^{j}Ψ_{η}(ℚ_{l}^{ac})_{(I_{U^{w},m}^{(h)})^{∧}(t)_{η}} ⊗ L_{ξ}).

The next step is to understand the vanishing cycle sheaves

R^{j}Ψ_{η}(ℚ_{l}^{ac})_{(I_{U^{w},m}^{(h)})^{∧}(t)_{η}}.

To do so we introduce a second generalisation of Igusa varieties, which we will call Igusa varieties of the second kind. More specifically we let J_{U^{w},m,s}^{(h)}/I_{U^{w},m}^{(h)} × k(w)^{ac} denote the moduli space for isomorphisms

α : Σ_{F_{w},n−h}[w^{s}] →^{∼} G^{0}[w^{s}],

which for every s′ > s lift etale locally to isomorphisms of the w^{s′}-division schemes. As s varies we get a system of finite etale Galois covers and the system has Galois group O_{D_{Fw,n−h}}^{×}. It is perhaps worth noting three things. One is that we need now the technical condition that the isomorphism must lift locally - this is because Aut (Σ_{F_{w},n−h}) → Aut (Σ_{F_{w},n−h}[w^{s}]) is not usually surjective. Secondly we remark that if one looked at a similar construction in the case of the ordinary locus of a modular curve one would just obtain the familiar Igusa curves. This is because in that case there is a duality between the connected and etale part of the analogue of G (i.e. the p-divisible group of the universal elliptic curve). To the best of our knowledge, for n − h > 1 the varieties J_{U^{w},m,s}^{(h)} do not occur in the reduction of any Shimura variety. They seem to naturally exist only in characteristic p.

The idea is now that over the “pro-object” lim_{←s}J_{U^{w},m,s}^{(h)} we have an isomorphism G^{0} ≅ Σ_{F_{w},n−h} and R^{j}Ψ_{η}(ℚ_{l}^{ac})_{(I_{U^{w},m}^{(h)})^{∧}(e_{Fw/ℚp}t)_{η}} becomes the constant

sheaf R^{j}Ψ_{η}(ℚ_{l}^{ac})_{(Spf R_{Fw,n−h,t})_{η}}. If one descends this isomorphism back down to I_{U^{w},s}^{(h)} one obtains an isomorphism

lim_{→t} R^{j}Ψ_{η}(ℚ_{l}^{ac})_{(I_{U^{w},m}^{(h)})^{∧}(t)_{η}} ≅ ⨁_{ρ} F(Ψ_{F_{w},l,n−h}^{j}[ρ]),

where ρ runs over irreducible representations of D_{F_{w},n−h}^{×} (up to unramified twist). If F_{ρ} is the lisse ℚ_{l}^{ac}-sheaf on I_{U^{w},m}^{(h)} associated to the representation ρ of Gal (J_{U^{w},m,∞}^{(h)}/I_{U^{w},m}^{(h)}) then the sheaf F(Ψ_{F_{w},l,n−h}^{j}[ρ]) is closely related to

F_{ρ} ⊗ Ψ_{F_{w},l,n−h}^{j}(ρ),

where now the rather mysterious action of

GL_{n−h}(O_{F,w}) × I_{F_{w}}

is concentrated on the constant sheaf Ψ_{F_{w},l,n−h}^{j}(π). At least in our unskilled hands it took some effort to make sense of the non-mathematical ideas of this paragraph. We are very grateful to Berkovich for providing a key step in the argument.

In this way we obtain an isomorphism

lim_{→m} H_{c}^{i}(\overline{X}_{U(m)}^{(h)} × k(w)^{ac}, R^{j}Ψ(ℚ_{l}^{ac}) ⊗ L_{ξ})^{⊕(n−h)} ≅ ⨁_{ρ} Ind _{P_{h}}^{GL_{n}(F_{w})}(lim_{→m} H_{c}^{i}(I_{U^{w},m}^{(h)} × k(w)^{ac}, L_{ξ} ⊗ F_{ρ})) ⊗ Ψ_{F_{w},l,n−h}^{j}(ρ)^{⊕(n−h)/e[ρ]}

for some explicit integers e[ρ]|(n − h) (see section 3). (Here we are using unnormalised induction.) To complete the proof of theorem D it remains to compute

lim_{→} H_{c}^{i}(I_{U^{w},m}^{(h)} × k(w)^{ac}, L_{ξ} ⊗ F_{π})

as a G(𝔸^{∞,p}) × E_{u^{c}}^{×} × ℤ × GL_{h}(F_{w}) × ∏_{x|u,x≠w}(B_{x}^{op})^{×}-module.

At this point we return to Langlands’ idea of using the Lefschetz trace formula to calculate the trace of the action of correspondences on Shimura varieties in characteristic p. In our case we use Fujiwara’s “Deligne conjecture” to compute the trace of a Hecke operator acting on

lim_{→} H_{c}^{i}(I_{U^{w},m}^{(h)} × k(w)^{ac}, L_{ξ} ⊗ F_{π})

in terms of data at fixed points. For this to be applicable there is a condition on the Hecke operator which corresponds to it being sufficiently twisted by Frobenius. Following Kottwitz we combine the results of Honda and Tate with some group theory (which we need in order to understand polarisations) to describe the points of I_{U^{w},m}^{(h)} × k(w)^{ac}. We find an expression for the sum of the terms at fixed points in terms of orbital integrals in the group

G(𝔸^{∞,p}) × E_{u^{c}}^{×} × D_{F_{w},n−h}^{×} × GL_{h}(F_{w}) × ∏_{x|u,x≠w}(B_{x}^{op})^{×}.

Unlike Kottwitz’s work there is no distinguished Frobenius element and we use a classification of points over k(w)^{ac} rather than over finite extensions of k(w). We are able to manipulate this expression so that it becomes a sum of orbital integrals in G(𝔸). In doing so we use again the condition that the Hecke operator “was sufficiently twisted by Frobenius” (cf [Cas]). Next we apply the trace formula on X_{U} × F_{u}^{ac} to relate traces of Hecke operators on

∑_{i}(−1)^{i}[lim_{→} H_{c}^{i}(I_{U^{w},m}^{(h)} × k(w)^{ac}, L_{ξ} ⊗ F_{π})]

with traces of related Hecke operators on

∑_{i}(−1)^{i}[lim_{→} H_{et}^{i}(X_{U} × F_{w}^{ac}, L_{ξ})].

From this comparison it is not hard to deduce theorem D.

We remark that we recover in this way some of Kottwitz’s results from [Ko4]. Although we have borrowed many of Kottwitz’s ideas our argument in the case of overlap does seem to be somewhat different. For instance we make no appeal to the fundamental lemma for stable base change (at this point in the argument).

Acknowledgements

One of us (R.T.) is very grateful to the Miller Institute at the University of California at Berkeley for its hospitality and support during the very lengthy revision of this paper.

We are very grateful for the help (conscious and unconscious) of many mathematicians. We would like to thank Bultel, Clozel, Coleman,B.Conrad, Gabber, Henniart, R.Huber, de Jong, Katz, Kazhdan, Kottwitz, Labesse, Laumon, Mazur, Rapoport and Strauch for helpful conversations. We came at this work from different starting points. One of us (M.H.) was inspired by Boyer’s Paris XI thesis [Bo]; he would like to thank Laumon for explaining Boyer’s work. The other (R.T.) was inspired by teaching a class on Deligne’s letter [De2] at Harvard; he would like to thank the students of math 253x for their help with understanding this letter.

Our debt to the work of Kottwitz (see eg [Ko3] and [Ko4]) on the l-adic cohomology of these same Shimura varieties will be clear to the reader. We have borrowed many of his ideas.

Our work would have been impossible without Berkovich’s vanishing cycles for formal schemes. We are extremely grateful to him for a number of things. Firstly for the help he has given us in understanding his results and in writing appendix II. Secondly because he found a proof of a key result we needed, which he kindly wrote it up in an appendix to this paper [Berk4].

Finally we would like to acknowledge our debt to the work of Carayol, Deligne and Drinfeld; on whose ideas the present paper is based. In particular we owe a very great debt to Deligne: our paper is simply the natural generalisation of the arguments of his beautiful letter [De2] from modular curves to the unitary group Shimura varieties we consider.

1 Notation

In this section we will introduce some notation which we will use throughout this paper. The reader should also consult appendix I for some basic group theory notation.

We will let p and l denote distinct rational primes, ℤ_{(p)} ring of elements of ℚ with denominator coprime to p, val_{p} the p-adic valuation (so that val_{p}(p) = 1 and | |_{p} the p-adic absolute value (so that |p|_{p} = 1/p).

If X is a scheme and x is a point of X we will let k(x) denote the residue field at x. We will let O_{X,x} denote the local ring of X at x and we will let O_{X,x}^{∧} denote its completion at its maximal ideal. If Y ⊂ X is a locally closed subscheme we will let X_{Y}^{∧} denote the completion of X along Y. For instance X_{x}^{∧} = Spf O_{X,x}^{∧}. If X is a locally noetherian formal scheme then X has a unique largest ideal of definition I. The formal scheme with the same underlying topological space as X and with structure sheaf O_{X} /I is in fact a scheme which we will refer to as the reduced subscheme of X, and will denote X^{red}. (See section 10.5 of [EGAI].)

If k is a field and A/k is an abelian variety we will let T A denote the Tate module of A, i.e.

T A = lim_{←N} A[N](k^{ac}),

where the limit is over all positive integers N. We will also introduce the characteristic zero version of the Tate module

V A = T A ⊗_{ℤ} ℚ.

If S is a finite set of rational primes we will let T^{S}A and V^{S}A denote the “away from S” Tate modules, i.e.

T^{S}A = lim_{←N} A[N](k^{ac})

where the limit is over all positive integers coprime to S, and V^{S}A = T^{S}A⊗_{ℤ}ℚ. We similarly define the “at S” Tate modules T_{S}A and V_{S}A.

If A and A′/S are abelian schemes then by an isogeny α : A → A′ we shall mean an invertible element of Hom (A, A′)⊗_{ℤ}ℚ. We will denote Hom (A, A)⊗_{ℤ} ℚ by End^{0}(A). By a polarisation λ of A we shall mean a homomorphism λ : A → A^{∨} such that for each geometric point s of S the homomorphism λ_{s} is a polarisation in the usual sense. If p is a rational prime then by a primeto-p-isogeny we shall mean an invertible element of Hom (A, A′) ⊗_{ℤ} ℤ_{(p)}. By a prime-to-p-polarisation of A we shall mean a polarisation λ : A → A^{∨} which is also a prime-to-p-isogeny.

If L′/L is a finite field extension we will let N_{L′/L} denote the norm from L′ to L.

If R is an 𝔽_{p}-algebra we will let Fr : R → R denote the Frobenius morphism which takes x ∈ R to x^{p} ∈ R. If X/𝔽_{p} is a scheme we will let Fr^{∗} : X → X denote the Frobenius morphism induced by Fr on structure sheaves. If Y → X is a morphism of schemes over 𝔽_{p} then we will let Y_{/X}^{(p)} (or simply Y^{(p)}, if no confusion seems likely to arise) denote the pull back of Y by Fr^{∗} : X → X. We will also let F_{Y/X} : Y → Y_{/X}^{(p)} (or simply F : Y → Y^{(p)} when no confusion seems likely to arise) denote the relative Frobenius, i.e. the morphism that arises from Fr : Y → Y and the universal property of the pull back Y_{/X}^{(p)}. If Y/X is a finite flat group scheme then we will let V : Y^{(p)} → Y denote the dual of F : Y^{∨} → Y^{∨,(p)} = Y^{(p),∨}, where Y^{∨} is the Cartier dual of Y. This definition then extends to p-divisible groups Y/X. The morphism V : Y^{(p)} → Y induces a morphism of quasi-coherent sheaves of O_{X}-modules

V_{∗} : (Fr^{∗})^{∗}Lie Y ≅ Lie Y^{(p)} → Lie Y.

Combining this with the natural map Fr : Lie Y → (Fr^{∗})^{∗}Lie Y we get a map, which we will also denote V_{∗}, from Lie Y to itself over X, which satisfies

V_{∗}(xy) = x^{p}V_{∗}(y)

for x a section of O_{X} and for y a section of Lie Y.

If k/𝔽_{p} is a finite extension we will let Frob_{k} ∈ Gal (k^{ac}/k) denote Fr^{−[k:𝔽_{p}]}, i.e. it will denote a geometric Frobenius element.

We will let K denote a p-adic field, i.e. a finite extension of ℚ_{p}. We will let v_{K} : K^{×} → → ℤ denote its unique valuation which is normalised to send uniformisers to 1. We will let O_{K} denote its ring of integers, ℘_{K} the unique maximal ideal of O_{K} and k(v_{K}) = k(℘_{K}) = O_{K}/℘_{K} its residue field. We will often use ϖ_{K} to denote a uniformiser in O_{K}. We will define an absolute value | |_{K} = | |_{v_{K}} on K by

|x|_{K} = (#k(v_{K}))^{−v_{K}(x)},

for x ∈ K^{×}. We will let ℂ_{v_{K}} = ℂ_{℘_{K}} denote the completion of the algebraic closure of K. We will let K^{nr} denote the maximal unramified extension of K and we will let K̂^{nr} denote the completion of K^{nr}.

We will let I_{K} ⊂ Gal (K^{ac}/K) denote the inertia subgroup, so that

Gal (K^{ac}/K)/I_{K} →^{∼} Gal (K^{nr}/K) →^{∼} Gal (k(v_{K})^{ac}/k(v_{K})).

We will let W_{K} ⊂ Gal (K^{ac}/K) denote the Weil group, i.e. the inverse image in Gal (K^{ac}/K) of Frob_{k(v_{K})}^{ℤ} ⊂ Gal (k(v_{K})^{ac}/k(v_{K})). We will write Frob_{v_{K}} for Frob_{k(v_{K})}, and will without comment think of it as an element of W_{K}/I_{K}. If σ ∈ W_{K} then we define v_{K}(σ) by σI_{K} = Frob_{v_{K}}^{v_{K}(σ)}. We will let f_{K} = f_{v_{K}} = [k(v_{K}) : 𝔽_{p}] and e_{K} = e_{v_{K}} = [K : ℚ_{p}]/f_{K}. If g is a positive integer we will let D_{K,g} denote the division algebra with centre K and Hasse invariant 1/g. The algebra D_{K,g} has a unique maximal order, which we will denote O_{D_{K,g}}. We will also let det denote the reduced norm from D_{K,g} to K and Π_{K,g} a uniformiser in O_{D_{K,g}}.

Local class field theory gives us a canonical isomorphism

Art_{K} : K^{×} ⟶^{∼} W_{K}^{ab}.

There is a choice of sign in the definition of Art_{K}. We will choose a normalisation which makes uniformisers and geometric Frobenius elements correspond. If σ ∈ W_{K} we will define

|σ|_{K} = |Art _{K}^{−1}σ|_{K} = p^{−f_{K}v_{K}(σ)}.

We will let c denote the non-trivial element of Gal (ℂ/ℝ). We take Art_{ℂ} : ℂ^{×} → Gal (ℂ/ℂ) to be the trivial homomorphism and we take

Art_{ℝ} : ℝ^{×}/ℝ_{>0}^{×} ⟶^{∼} Gal (ℂ/ℝ)

to be the unique isomorphism. We take | |_{ℝ} to be the usual absolute value and | |_{ℂ} to be the square of the usual absolute value, i.e. |z|_{ℂ} = |zz^{c}|_{ℝ}.

If L is a number field we will let 𝔸_{L} denote the adeles of L. If S is a finite set of places of L we decompose 𝔸_{L} = 𝔸_{L}^{S} × L_{S} where 𝔸_{L}^{S} denotes the adeles away from S and where L_{S} = ∏_{x∈S} L_{x}. Also let \overline{𝔸}_{L}^{S} denote

lim →𝔸_{L′}^{S(L′)},

where L′ runs over finite extensions of L and where S(L′) is the set of places of L′ above S. The product of the normalised absolute values gives a homomorphism

| | = ∏_{x} | |_{L_{x}} : L^{×}\𝔸_{L}^{×} ⟶ ℝ_{>0}^{×}.

Global class field theory tells us that the product of the local Artin maps gives an isomorphism

Art_{L} : L^{×}\𝔸_{L}^{×}/(L_{∞}^{×})^{0} ⟶^{∼} Gal (L^{ac}/L)^{ab},

where (L_{∞}^{×})^{0} denotes the connected component of the identity in L_{∞}^{×}. Suppose that ı : ℚ_{l}^{ac} →^{∼} ℂ. If

ψ : 𝔸_{L}^{×}/L^{×} ⟶ ℂ^{×}

is a continuous character, we will call ψ algebraic if we can find integers n_{σ} for each embedding σ : L ↪ ℂ such that

ψ|_{L⊗_{ℚ}ℝ} = ∏_{σ}(σ ⊗ 1)^{n_{σ}}.

In this case there is a unique continuous character

rec_{l,ı}(ψ) = rec(ψ) : Gal (L^{ac}/L) ⟶ (ℚ_{l}^{ac})^{×}

such that for any finite place x |̸l of L we have

ı ∘ rec_{l,ı}(ψ) ∘ Art_{L_{x}} = ψ|_{L_{x}^{×}}.

More explicitly rec_{l,ı}(ψ) = ψ′ ∘ Art _{L}^{−1} where

ψ′ : L^{×}\𝔸_{L}^{×}/(L_{∞}^{×})^{0} ⟶ (ℚ_{l}^{ac})^{×}

is defined by

ψ′(x) = ı(ψ(x) ∏_{σ}(σ ⊗ 1)(x_{∞})^{−n_{σ}}) ∏_{σ}((ı ∘ σ) ⊗ 1)(x_{l})^{n_{σ}}.

If M is any Gal (L^{ac}/L)-module we will let

ker^{1}(L, M)

denote the subset of H^{1}(L, M) of elements that become trivial in H^{1}(L_{x}, M) for every place x of L.

If L is a CM field we will let c denote complex conjugation on L, i.e. the unique automorphism of L which coincides with complex conjugation on ℂ for any embedding L ↪ ℂ.

Let ε ∈ M_{n}(ℤ) denote the idempotent

( \begin{matrix}1 & 0 & 0 & & 0 \\ 0 & 0 & 0 & … & 0 \\ 0 & 0 & 0 & & 0 \\ & ⋮ & & ... & ⋮ \\ 0 & 0 & 0 & … & 0 \end{matrix} ) .

Thus for any ring R we have isomorphisms

• M_{n}(R)ε ≅ R^{n} via the map sending x to its first column;

• εM_{n}(R) ≅ (R^{n})^{∨} via the map sending x to its first row;

• the map M_{n}(R)ε ⊗_{R} εM_{n}(R) → M_{n}(R) sending xε ⊗ εy to xεy is an isomorphism.

Suppose that L is a field of characteristic 0 and that C is a finite dimensional semi-simple L-algebra. Suppose that ∗ is an involution on C (i.e.

∗ : C → C satisfies (x + y)^{∗} = x^{∗} + y^{∗}, (xy)^{∗} = y^{∗}x^{∗} and ∗^{2} = 1) such that ∗|_{L} ≠ 1. Set L^{+} = L^{∗=1}. Suppose that W is a C-module which is finite dimensional over L. We will call a non-degenerate L^{+}-alternating pairing

⟨ , ⟩ : W × W ⟶ L^{+}

∗-Hermitian if

⟨γx, y⟩ = ⟨x, γ^{∗}y⟩

for all x, y ∈ W and all γ ∈ C. We will call two such pairings ⟨ , ⟩ and ⟨ , ⟩_{1} equivalent if we can find δ ∈ End_{C}(W)^{×} and μ ∈ (L^{+})^{×} such that

⟨x, y⟩_{1} = μ⟨δx, δy⟩

for all x, y ∈ W. We may classify equivalence classes of non-degenerate L^{+}- alternating ∗-Hermitian pairings W × W → L^{+} as follows.

Fix one such pairing ⟨ , ⟩_{0} (if one exists). We will denote by ∗_{0} the involution on End_{L^{+}}(W) such that

⟨δx, y⟩ = ⟨x, δ^{∗}y⟩

for all x, y ∈ W and all δ ∈ End_{L^{+}}(W). Note that ∗_{0}|_{C} = ∗ and that ∗_{0} preserves End_{C}(W). Define a reductive algebraic group H/L^{+} by setting, for any L^{+}-algebra R, H(R) equal to the set of δ ∈ (End_{C}(W) ⊗_{L^{+}} R)^{×} such that

δδ^{∗_{0}} ∈ R^{×}.

Note that if R is an L-algebra then

End_{C}(W) ⊗_{L^{+}} R ≅ (End_{C}(W) ⊗_{L^{+}} L) ⊗_{L} R ≅
End_{C}(W) ⊗_{L} R ⊕ End_{C}(W) ⊗_{L,∗|_{L}} R

and ∗_{0} interchanges the two factors. Then H(R) consists of the set of pairs (x, λx^{∗_{0}}), where x ∈ (End_{C}(W) ⊗_{L} R)^{×} and λ ∈ R^{×}, i.e.

H(R) ≅ (End_{C}(W) ⊗_{L} R)^{×} × R^{×}.

By Hilbert 90 we see that H^{1}(L, H) = (0) and so

H^{1}(Gal (L/L^{+}), H(L)) ⟶^{∼} H^{1}(L^{+}, H).

We can describe H^{1}(Gal (L/L^{+}), H(L)) as the set of equivalence classes of pairs (λ, γ) ∈ L^{×} × End_{C}(W)^{×} such that λ has norm 1 down to L^{+} and such that

γ^{∗_{0}} = λ^{∗}γ.

We consider (λ, γ) and (λ′, γ′) equivalent if there exists μ ∈ L^{×} and δ ∈ End_{C}(W)^{×} such that

(λ′, γ′) = (λμ/μ^{∗}, μ^{−∗}δγδ^{∗_{0}}).

Applying Hilbert 90 to L/L^{+} we see that H^{1}(Gal (L/L^{+}), H(L)) is also in bijection with equivalence classes of

γ ∈ End_{C}(W)^{×} ∩ End_{C}(W)^{∗_{0}=1},

where we consider γ and γ′ equivalent if there exists μ ∈ (L^{+})^{×} and δ ∈ End_{C}(W)^{×} such that

γ′ = μδgammaδ^{∗_{0}}.

Any non-degenerate, L^{+}-alternating, ∗-Hermitian form W × W → L^{+} is of the form

⟨x, y⟩_{δ} = ⟨δx, δy⟩_{0}

for some δ ∈ End_{L^{+}}(W)^{×} with

δ^{∗_{0}}δ ∈ End_{C}(W)^{∗_{0}=1}.

Moreover ⟨ , ⟩_{δ} and ⟨ , ⟩_{δ′} are equivalent if and only if there exists γ ∈ End_{C}(W)^{×} and λ ∈ (L^{+})^{×} such that

(δ′)^{∗_{0}}δ′ = λγ^{∗_{0}}(δ^{∗_{0}}δ)γ.

Note that any element γ ∈ End_{C}(W)^{∗_{0}=1} can be written δ^{∗_{0}}δ for some δ ∈ End_{L^{+}}(W). (If we choose an L^{+}-basis of W we get an isomorphism of End_{L^{+}}(W) with M_{2N}(L^{+}) for some integer N. Moreover ⟨ , ⟩ is represented by an anti-symmetric matrix J ∈ GL_{2N}(L^{+}) and if δ ∈ M_{2N}(L^{+}) then δ^{∗_{0}} = J^{−1}δ^{t}J (where t denotes the transpose). Thus if δ^{∗_{0}} = δ we see that Jδ is antisymmetric and hence that

Jδ = (δ′)^{t}Jδ′

for some δ′ ∈ GL_{2N}(L^{+}). Thus δ = (δ′)^{∗_{0}}δ′.) We deduce that the correspondence which associates ⟨ , ⟩_{δ} with δδ^{∗} sets up a bijection between

• equivalence classes of non-degenerate L^{+}-alternating ∗-Hermitian forms on W

• and H^{1}(L^{+}, H).

Suppose now that L is a number field. Suppose that two classes ψ_{1}, ψ_{2} ∈ H^{1}(L^{+}, H) correspond to non-degenerate L^{+}-alternating ∗-Hermitian forms ⟨ , ⟩_{1} and ⟨ , ⟩_{2}. Then the same arguments show that ⟨ , ⟩_{1} and ⟨ , ⟩_{2} become equivalent over 𝔸_{L^{+}}^{S} (with the same definition of equivalence as over a field) if and only if ψ_{1} and ψ_{2} have the same image in H^{1}(L^{+}, \overline{𝔸}_{L^{+}}^{S}).

We will call an L^{+}-bilinear pairing

( , ) : W × W ⟶ L

∗-symmetric if ( , ) is L-linear in the first variable and satisfies

(y, x) = (x, y)^{∗}

for all x, y ∈ W. We will call this pairing non-degenerate if (x, y) = 0 for all y ∈ W implies that x = 0. We will call it 8-Hermitian if

(γx, y) = (x, γ^{∗}y)

for all x, y ∈ W and all γ ∈ C. We call two ∗-Hermitian ∗-symmetric pairings ( , )_{1} and ( , )_{2} equivalent if there exists δ ∈ End_{C}(W) and λ ∈ (L^{+})^{×} such that

(x, y)_{2} = λ(γx, γy)

for all x, y ∈ W.

Suppose that L = L^{+}(\sqrt{a}) where \sqrt{a}^{2} = a ∈ L^{+}. Then there is a bijection between equivalence classes of non-degenerate, ∗-Hermitian, ∗-symmetric pairings W × W → L and equivalence classes of non-degenerate, ∗-Hermitian, L^{+}-alternating pairings W × W → L^{+} given as follows. If ⟨ , ⟩ is a nondegenerate, ∗-Hermitian, L^{+}-alternating pairing W × W → L^{+} then associate to the equivalence class of ⟨ , ⟩ the equivalence class of the non-degenerate, ∗-Hermitian, ∗-symmetric pairing given by

(x, y) = ⟨\sqrt{a}x, y⟩ + \sqrt{a}⟨x, y⟩.

conversely if ( , ) is a non-degenerate, ∗-Hermitian, ∗-symmetric pairing W × W → L^{+} then associate to the equivalence class of ( , ) the equivalence class of the non-degenerate, ∗-Hermitian, L^{+}-alternating pairing given by

⟨x, y⟩ = tr_{L/L^{+}} \sqrt{a}(x, y).

This bijection is independent of the choice of \sqrt{a} ∈ L.

Suppose that L = ℂ, L^{+} = ℝ, C = ℂ^{I}, ∗ = c^{I} and W = (ℂ^{n})^{I}. Then any non-degenerate ∗-Hermitian, ∗-symmetric pairing W × W → ℂ is of the form

(x⃗_{i})_{i∈I} × (y⃗_{i})_{i∈I} ⟼ ∑_{i∈I} x_{i}^{t}J_{i}y⃗_{i},

where J_{i} is the diagonal n×n-matrix with 1 on the diagonal a_{i} times and −1 on the diagonal b_{i} = n−a_{i} times. This establishes a bijection between equivalence classes of non-degenerate, ∗-Hermitian, ∗-symmetric pairings W timesW → ℂ and equivalence classes of I-tuples ((a_{i}, b_{i}))_{i∈I} of pairs of non-negative integers (a_{i}, b_{i}) with a_{i} + b_{i} = n for all i ∈ I. We call the I-tuples ((a_{i}, b_{i}))_{i∈I} and ((b_{i}, a_{i}))_{i∈I} equivalent, but consider no other pairs of I-tuples equivalent. We deduce that equivalence classes of non-degenerate, ∗-Hermitian, ℝ-alternating pairings W × W → ℝ are also parametrised by such equivalence classes of I-tuples.

Now suppose that L = ℂ, L^{+} = ℝ, C = M_{n}(ℂ)^{I}, (γ_{i})^{∗} = (γ_{i}^{c,t}) and W = C. Then equivalence classes of non-degenerate, ∗-Hermitian, ℝ-alternating pairings W × W → ℝ are still parametrised by equivalence classes of I-tuples ((a_{i}, b_{i}))_{i∈I} of pairs of non-negative integers (a_{i}, b_{i}) with a_{i} + b_{i} = n for all i ∈ I. again we call the I-tuples ((a_{i}, b_{i}))_{i∈I} and ((b_{i}, a_{i}))_{i∈I} equivalent, but consider no other pairs of I-tuples equivalent. To see this one can note that to give a non-degenerate, ∗-Hermitian, ℝ-alternating form W × W → ℝ is the same as giving a non-degenerate, ∗-Hermitian for ℂ^{I}, ℝ-alternating form εW × εW → ℝ. The equivalence sends ⟨ , ⟩ to the I-tuple parametrising

⟨ , ⟩|_{εW ×εW} .

Now suppose that L is an imaginary quadratic field, that M is a totally real field, that C is a central simple LM-algebra with dim_{LM} C = n^{2}, that ∗ is an involution of the second kind on C (i.e. ∗|_{LM} = c), that ∗ is positive (i.e. tr_{LM/ℚ}tr_{C/LM}(γγ^{∗}) > 0 for all non-zero γ ∈ C) and that W = C. Then the 5-tuple (L_{∞}, L_{∞}^{+}, C_{∞}, ∗, W_{∞}) is isomorphic to the 5-tuple

(ℂ, ℝ, M_{n}(ℂ)^{Hom (M,ℝ)}, (γ_{τ}) ↦ (γ_{τ}^{c,t}), M_{n}(ℂ)^{Hom (M,ℝ)}).

In particular we see that equivalence classes of non-degenerate, ∗-Hermitian, ℝ-alternating pairings W_{∞} ×W_{∞} → ℝ are parametrised by equivalence classes of Hom (M, ℝ)-tuples as above.

We will let E denote an imaginary quadratic field in which p splits. We will let c denote complex conjugation in Gal (E/ℚ). We will choose a prime u of E above p. We will also let F^{+}/ℚ denote a totally real field of degree d. We will set F = E.F^{+} so that F is a CM-field with maximal totally real subfield F^{+}. Let w = w_{1}, w_{2}, ..., w_{r} denote the places of F above u and let v = v_{1}, ..., v_{r} denote their restrictions to F^{+}. We will denote [k(w_{i}) : 𝔽_{p}] by f_{i}. We will let B/F denote a division algebra of dimension n^{2} such that

• F is the centre of B;

• the opposite algebra B^{op} is isomorphic to B ⊗_{E,c} E;

• B is split at w;

• at any place x of F which is not split over F^{+}, B_{x} is split;

• at any place x of F which is split over F^{+} either B_{x} is split or B_{x} is a division algebra,

• if n is even then 1 + dn/2 is congruent modulo 2 to the number of places of F^{+} above which B is ramified.

We will let tr_{B/F} denote the reduced trace and det_{B/F} the reduced norm for B/F. When no confusion seems likely to arise we may drop the subscripts. Define n ∈ ℤ_{>0} by [B : F] = n^{2}.

We may pick an involution of the second kind ∗ on B. (That we may chose such an involution follows from the second and fourth of the above assumptions on B. More precisely lemma 8.1 of [Sc] defines homomorphism Br (F)^{op=c} → (F^{+})^{×}/N(F^{×}) and shows that B has an involution of the second kind if and only if [B] is in the kernel of this homomorphism. But [B] is in the kernel if and only if [B_{x}] is in the kernel for all places x of F which are non split over F^{+}.) We may and will further assume that ∗ is positive, i.e. for all nonzero x ∈ B we have (tr_{F/ℚ} ∘ tr_{B/F})(xx^{∗}) > 0. (To see that we may suppose that ∗ is positive one may argue as follows. The involutions of the second kind on B are exactly the maps of the form

x ⟼ bx^{∗}b^{−1},

where b ∈ B^{×} and b^{∗}b^{−1} ∈ F. By Hilbert’s theorem 90 we may alter any such b by an element of F^{×} so that b^{∗} = b. Thus we may suppose that b^{∗} = b. By lemma 2.8 of [Ko3] the set of invertible b ∈ (B^{∗=1} ⊗_{ℚ}ℝ) such that x ↦ bx^{∗}b^{−1} is positive is a non-empty open set. Thus we can find an invertible b ∈ B^{∗=1} such that x ↦ bx^{∗}b^{−1} is positive.)

We will let V denote the B ⊗_{F} B^{op} module B. We will be interested in alternating pairings V × V → ℚ which are ∗-Hermitian for the action of B on V. Any such pairing is of the form

(x_{1}, x_{2})_{β} = (tr_{F/ℚ} ∘ tr_{B/F})(x_{1}βx_{2}^{∗}),

for some β ∈ B^{∗=−1}. Define an involution of the second kind #_{β} on B by x^{#_{β}} = βx^{∗}β^{−1}. Then we have that

((b_{1} ⊗ b_{2})x_{1}, x_{2}) = (x_{1}, (b_{1}^{∗} ⊗ b_{2}^{#_{β}})x_{2})

for all x_{1}, x_{2} ∈ V, b_{1} ∈ B and b_{2} ∈ B^{op}. Also let G_{β}/ℚ be the algebraic group whose R-points (for any ℚ-algebra R) are the set of pairs

(λ, g) ∈ R^{×} × (B^{op} ⊗_{ℚ} R)^{×}

such that

gg^{#_{β}} = λ.

This comes with a homomorphism ν : G_{β} → 𝔾_{m} which sends (λ, g) to λ. We will let G_{β,1} denote the kernel of ν. Note that the structure map G_{β,1} → Spec ℚ factors through Spec F^{+} so we may also consider G_{β,1} as an algebraic group over F^{+}.

Choose a distinguished embedding τ_{0} : F ↪ ℂ.

Lemma 1.1 We can choose an element 0 ≠ β ∈ B^{∗=−1} such that

1. if x is a rational prime which is not split in E then G_{β} is quasisplit at x,

2. and if τ : F^{+} ↪ ℝ then G_{β,1} ×_{F^{+},τ} ℝ is isomorphic to U(1, n − 1) if τ = τ_{0} and U(n) otherwise.

Proof: Choose 0 ≠ β_{0} ∈ B^{∗=−1} and suppose that if x is an infinite place of F^{+} then G_{β_{0},1}(F_{x}^{+}) ≅ U(p_{0,x}, q_{0,x}). We will look for an element α ∈ B, such that the element β = αβ_{0} satisfies the conditions of the lemma. Firstly we require that

α^{#_{β_{0}}} = α.

Thus α defines a class in H^{1}(F/F^{+}, P G_{β_{0},1}), where P G_{β_{0},1} is the adjoint group of G_{β_{0},1}. Every class in H^{1}(F/F^{+}, P G_{β_{0},1}) arises in this way. (By definition such a class is represented by α ∈ (B^{op})^{×} such that αα^{−#_{β_{0}}} = λ ∈ F^{×}. Then λ has norm 1 in F^{+} and so by Hilbert’s theorem 90 can be written as μ^{c}/μ. Then μα represents the same class as α and μα = (μα)^{#_{β_{0}}}.) Moreover G_{αβ_{0},1} is the inner form of G_{β_{0},1} classified by

[α] ∈ H^{1}(F^{+}, P G_{β_{0},1}).

If x is a place of F^{+} which splits in F and if y is a place of F above x then we have natural maps

H^{1}(F_{x}^{+}, P G_{β_{0},1}) ≅ H^{1}(F_{y}, P G_{β_{0},1}) ≅ H^{2}(F_{y}, μ_{n}) ≅ ℤ/nℤ.

If x is a finite place of F^{+} which does not split in F then according to section 2 of [Cl1] we have

H^{1}(F_{x}^{+}, P G_{β_{0},1}) ≅ ℤ/2ℤ.

Moreover if x is an infinite place then H^{1}(F_{x}^{+}, P G_{β_{0},1}) is in bijection with the set of unordered pairs of non-negative integers {p_{x}, q_{x}} with p_{x} + q_{x} = n. In both these cases the map

H^{1}(F_{x}^{+}, P G_{β_{0},1}) ⟶ H^{1}(F_{x}, P G_{β_{0},1})

is trivial. Clozel also shows (lemma 2.1 of [Cl1]) that if n is odd then the map

H^{1}(F^{+}, P G_{β_{0},1}) ⟶ ⨁_{x} H^{1}(F_{x}^{+}, P G_{β_{0},1})

is surjective. If on the other hand n is even he shows there is a map

⨁_{x} H^{1}(F_{x}^{+}, P G_{β_{0},1}) ⟶ ℤ/2ℤ

whose kernel coincides with the image of H^{1}(F^{+}, P G_{β_{0},1}). Clozel describes this map as the sum of the natural maps

H^{1}(F_{x}^{+}, P G_{β_{0},1}) → → ℤ/2ℤ

if x is finite; and the map

H^{1}(F_{x}^{+}, P G_{β_{0},1}) → → ℤ/2ℤ

which sends {p_{x}, q_{x}} to p_{x,0} − p_{x} mod 2, if x is infinite. (See section 2 of [Cl1], particularly lemma 2.2.)

Suppose that B is ramified above s places of F^{+} which split in F. If x is a place of F^{+} which does not split in F let u_{x} ∈ H^{1}(F_{x}^{+}, P G_{β_{0},1}) denote the class of the quasi-split inner form of P G_{β_{0},1} over F_{x}^{+}. If n is even then we see that

s + nd/2 + ∑_{x|∞} p_{x,0} + ∑_{x |̸∞} u_{x} ≡ 0 mod 2.

Let A = ℤ/ℤ if n is odd and A = ℤ/2ℤ if n is even. Also suppose that B is ramified above s places of F^{+} which split in F. Then we see that we get maps

H^{1}(F/F^{+}, P G_{β_{0},1}) ⟶ ⨁_{x} H^{1}(F_{x}^{+}, P G_{β_{0},1}) ⟶ A,

where x runs over places of F^{+} which do not split in F, where the second map is as described above and where the sequence is exact in the middle. The lemma requires us to find a class in H^{1}(F/F^{+}, P G_{β_{0},1}) which maps to • u_{x} if x is a finite place of F^{+} which does not split in F,

• {1, n − 1} if x is an infinite place corresponding to τ_{0} : F^{+} ↪ ℝ,

• and {0, n} if x is any other infinite place of F^{+}.

If n is odd this will be possible. If n is even this is possible if

1 + ∑_{x |̸∞} u_{x} + ∑_{x|∞} p_{x,0} ≡ 0 mod 2,

i.e. if

1 ≡ s + nd/2 mod 2.

The lemma follows. □

Now fix β as in the lemma. We will drop the subscript β from #, G and G_{1}. Note that the corresponding alternating form on V,

(x_{1}, x_{2}) = (tr_{F/ℚ} ∘ tr_{B/F})(x_{1}βx_{2}^{∗}),

has parameters (n, 0) at any embedding τ ≠ τ_{0} and (n − 1, 1) at τ_{0}. If R is an E-algebra then G(R) can be identified with the set of pairs

(g_{1}, g_{2}) ∈ (B^{op} ⊗_{E} R) × (B^{op} ⊗_{E,c} R)

such that

(g_{1}g_{2}^{#}, g_{2}g_{1}^{#}) ∈ R^{×}.

Thus we have

G(R) ≅ (B^{op} ⊗_{E} R)^{×} × R^{×}

where

(g_{1}, g_{2}) ⟼ (g_{1}, g_{1}g_{2}^{#})

and inversely

(g, ν) ⟼ (g, νg^{−#}).

In particular we get an isomorphism

RS_{ℚ}^{E}(G ×_{ℚ} E) ≅ RS_{Q}^{E}(𝔾_{m}) × H_{B^{op}},

where RS denotes Weil’s restriction of scalars and where H_{B^{op}}/ℚ is the algebraic group defined by

H_{B^{op}}(R) = (B^{op} ⊗_{ℚ} R)^{×}.

Suppose that x is a place of ℚ which splits as x = yy^{c} in E. Then the choice of a place y|x allows us to consider ℚ_{x} →^{∼} E_{y} as an E-algebra and hence to identify

G(ℚ_{x}) ≅ (B_{y}^{op})^{×} × ℚ_{x}^{×}.

In particular, we get an isomorphism

G(ℚ_{p}) →^{∼} ℚ_{p}^{×} × ∏_{i=1}^{r}(B_{w_{i}}^{op})^{×},

which sends g to (ν(g), g_{1}, ..., g_{r}). We will often let

(g_{0}, g_{1}, ..., g_{r}) ∈ ℚ_{p}^{×} × ∏_{i=1}^{r}(B_{w_{i}}^{op})^{×}

denote a typical element of G(ℚ_{p}). Similarly we will decompose a typical element g ∈ G(𝔸) as (g_{x})_{x≠p} × (g_{p,0}, g_{w_{1}}, ..., g_{w_{r}}) with g_{x} ∈ G(ℚ_{x}), g_{p,0} ∈ ℚ_{p}^{×} and g_{w_{i}} ∈ B_{w_{i}}^{×}; or as g^{p} × g_{p,0} × g_{w} × g_{p}^{w}, where g^{p} = (g_{x})_{x≠p}, g_{w} = g_{w_{1}} and g_{p}^{w} = (g_{w_{2}}, ..., g_{w_{r}}). We will let G(𝔸^{w}) denote the subgroup of G(𝔸) consisting of elements with g_{p,0} = 1 and g_{w} = 1. Similarly if π is an irreducible admissible representation of G(𝔸) over an algebraically closed field of characteristic 0 we may decompose it π ≅ π^{p}⊗π_{p} ≅ π^{p}⊗π_{p,0}⊗π_{p,1}⊗...⊗π_{p,r} ≅ π^{p}⊗π_{p,0}⊗π_{w}⊗π_{p}^{w} ≅ π^{w} ⊗ π_{p,0} ⊗ π_{w}. Note that π_{p,0} = ψ_{π}|_{E_{u^{c}}^{×}}.

Fix a maximal order Λ_{i} = O_{B_{w_{i}}} in B_{w_{i}} for each i = 1, ..., r. Our pairing ( , ) gives a perfect duality between V_{w_{i}} and V_{w_{i}^{c}}. Let Λ_{i}^{∨} ⊂ V_{w_{i}^{c}} denote the dual of Λ_{i} ⊂ V_{w_{i}}. Then if

Λ = ⨁_{i=1}^{r} Λ_{i} ⊕ ⨁_{i=1}^{r} Λ_{i}^{∨} ⊂ V ⊗_{ℚ} ℚ_{p},

we see that Λ is a ℤ_{p}-lattice in V ⊗_{ℚ} ℚ_{p} and that the pairing ( , ) on V restricts to give a perfect pairing Λ × Λ → ℤ_{p}.

There is a unique maximal ℤ_{(p)}-order O_{B} ⊂ B such that O_{B}^{∗} = O_{B} and O_{B,w_{i}} = O_{B_{w_{i}}} for i = 1, ..., r. Then O_{B,p} equals the set of elements of B_{p} which carry Λ into itself. On the other hand the stabiliser of Λ in G(ℚ_{p}) is ℤ_{p}^{×} × ∏_{i=1}^{r} O_{B_{w_{i}}}^{×}.

Fix an isomorphism O_{B_{w}} ≅ M_{n}(O_{F,w}). Composing this with the transpose map t we also get an isomorphism O_{B_{w}}^{op} ≅ M_{n}(O_{F,w}). Moreover we get an isomorphism

εΛ_{1} ≅ (O_{F,w}^{n})^{∨}.

The action of g ∈ M_{n}(O_{F,w}) ≅ (O_{B_{w}}^{op}) on this module is via right multiplication by g^{t}. We will write Λ_{11} as an abbreviation for εΛ_{1}. We get an identification

Λ ≅ ((O_{F,w}^{n} ⊗ Λ_{11}) ⊕ (O_{F,w}^{n} ⊗ Λ_{11})^{∨}) ⊕ ⨁_{i=2}^{r}(Λ_{i} ⊕ Λ_{i}^{∨}).

Under this identification (g_{0}, g_{1}, ..., g_{r}) ∈ G(ℚ_{p}) acts as

((1 ⊗ g_{1}) ⊕ g_{0}(1 ⊗ g_{1}^{−1})^{∨}) ⊕ ⨁_{i=2}^{r}(g_{i} ⊕ g_{0}(g_{i}^{−1})^{∨}).

Fix a square root

| |_{K}^{1/2} : (K^{×} ⟶ ℚ_{l}^{ac})^{×}

of | |_{K} : K^{×} → ℚ_{l}^{×}, i.e. fix a square root of p^{f_{K}} in ℚ_{l}^{ac}. If f_{K} is even we assume that this square root is chosen to be p^{f_{K}/2}. Also choose ı : ℚ_{l}^{ac} ≅ ℂ, such that ı ∘ | |^{1/2} is valued in ℝ_{>0}^{×}. We apologise for making such an ugly choice. The reader will see that all our main results are independent of the choice of ı, but it would require a lot of extra notation to make the proofs free of such a choice. Some of our main results do involve the choice of | |^{1/2}, but in each case this choice is involved in more than one place and all that matters is that the same choice is made at each place.

We will let ξ denote an irreducible representation of the algebraic group G on a finite dimensional ℚ_{l}^{ac} vector space W_{ξ}.

2 Barsotti-Tate groups

For the definition of a Barsotti-Tate group over a scheme S we refer the reader to section 2 of chapter I of [Me]. Suppose that S is a O_{K} scheme, then by a Barsotti-Tate O_{K}-module H/S we shall mean a Barsotti-Tate group H/S together with an embedding O_{K} ↪ End (H) (ring morphisms are assumed to send the multiplicative identity to itself) such that the induced action of O_{K} on Lie H coincides with the action coming from the structural morphism H → S → Spec O_{K}. We call a Barsotti-Tate O_{K}-module H ind-etale if the underlying Barsotti-Tate group is ind-etale (see example 3.7 of chapter I of [Me]). There is an equivalence of categories between ind-etale Barsotti-Tate O_{K}-modules and finite, torsion free lisse etale O_{K}-sheaves on S (see example 3.7 of chapter I of [Me]). If S is connected we define the height of a Barsotti-Tate O_{K}-module H to be the unique integer h(H) such that H[℘_{K}^{n}] has rank q_{K}^{h(H)} for all n ≥ 1. The usual height of H as a Barsotti-Tate group is h(H)[K : ℚ_{p}]. In general we will let H^{∨} denote the unique Barsotti-Tate O_{K}- module such that H^{∨}[p^{r}] is the Cartier dual of H[p^{r}] for all p and such that the

inclusions H^{∨}[p^{r}] ↪ H^{∨}[p^{s}] for s ≥ r are the Cartier duals of H[p^{s}] ⟶^{p^{s−r}} H[p^{r}]. We will refer to H^{∨} as the Cartier dual of H.

Now suppose that p is locally nilpotent on S. We will call a Barsotti-Tate O_{K}-module H formal if the p-torsion H[p] in H is radicial. There is an equivalence of categories between the category of formal Barsotti-Tate O_{K}- module and the category of formal Lie groups Θ/S together with a morphism O_{K} → End (Θ) such that

• Θ[p] is finite and locally free;

• p : Θ → Θ is an epimorphism;

• the induced action of O_{K} on Lie Θ coincides with the action coming from the structural morphism Θ → S → Spec O_{K}.

(This follows from corollary 4.5 of chapter II of [Me].)

Let X be a locally noetherian formal scheme with ideal of definition I. We will let X_{n} denote the scheme with underlying topological space X and structure sheaf O_{X} /I^{n}. By a Barsotti-Tate O_{K}-module H over the locally noetherian formal scheme X we shall mean a system of Barsotti-Tate O_{K}- modules H_{n} over the schemes X_{n} together with compatible isomorphisms

H_{n} ×_{X_{n}} X_{m} ≅ X_{m}

whenever m ≤ n. This definition is easily checked to be canonically independent of the choice of ideal of definition I. We will call H ind-etale (resp. formal) if each H_{n} is ind-etale (resp. formal). Note that in fact H is indetale (resp. formal) if and only if H_{1} is ind-etale (resp. formal, see paragraph 3.2 of chapter II of [Me]). If A is a noetherian ring complete with respect to the I-adic topology for some ideal I, then there is a natural functor from Barsotti-Tate O_{K}-modules H/Spec A to Barsotti-Tate O_{K}-modules H/Spf A. It follows from lemma 4.16 of chapter II of [Me] that if I contains some power of p then this is in fact an equivalence of categories which preserves exact sequences. We remark that H/Spf A may be formal while the corresponding Barsotti-Tate O_{K}-module H/Spec A is not.

Lemma 2.1 Suppose that p is locally nilpotent on a locally noetherian scheme S and that H/S is a Barsotti-Tate O_{K}-module. Then for h ∈ Z_{≥0} we can find reduced closed subschemes S^{[h]} ⊂ S such that 1. S^{[h]} ⊃ S^{[h−1]};

2. the codimension of any component of S^{[h−1]} in any component of S^{[h]} which contains it is at most 1;

3. for any geometric point s of S we have that s lies in S^{[h]} if and only if #H[p](k(s)) ≤ p^{[K:ℚ_{p}]h};

4. on S^{(h)} = S^{[h]} − S^{[h−1]} there is a short exact sequence of Barsotti-Tate

O_{K}-module

(0) ⟶ H^{0} ⟶ H ⟶ H^{et} ⟶ (0)

where H^{0} is a formal Barsotti-Tate O_{K}-module and where H^{et} is an ind-etale Barsotti-Tate O_{K}-module of height h.

Proof: By proposition 4.9 of [Me] it suffices to show that for g ∈ ℤ^{≥0} we can find closed subschemes S′_{g} ⊂ S such that

1. S′_{g} ⊃ S′_{g−1};

2. if s is a geometric point of S then s lies in S′_{g} if and only if #H[p](k(s)) ≤ p^{g};

3. the codimension of any component of S′_{g−1} in any component of S′_{g} which contains it is at most 1.

The question is local on S so we may assume that S = Spec R for a noetherian ring R. We may further assume that S is reduced. By a simple inductive argument it suffices in fact to show that if for any geometric point s of S we have #H[p](k(s)) ≤ p^{g} then we can find a reduced closed subscheme S′ ⊂ S such that

1. a geometric point s of S lies in S′ if and only if #h[p](k(s)) < p^{g};

2. any irreducible component of S′ has codimension at most one in any irreducible component of S containing it.

Finally we may assume that S is in fact integral.

We now follow the arguments of page 97 of [O]. Let H/S denote the locally free sheaf Lie (H[p]^{∨}) = Lie H^{∨} (the equality here follows from remark 3.3.20 of chapter II of [Me] because p = 0 on S). For any geometric point s of S there is a canonical perfect pairing between H_{s}^{V_{∗}=1} and H_{s}[p](k(s)). (This seems to be well known, but we know of no reference for the statement in exactly this form, so we will sketch the proof. On page 138 of [Mu1] we see that we can identify H_{s} with Hom (H_{s}[p], 𝔾_{a}) and that V_{∗} then becomes identified with the map ϕ ↦ ϕ ∘ Fr^{∗}. We get a pairing

\begin{matrix}H_{s}[p](k(s)) & × & H_{s} & ⟶ & k(s) \\ x & × & ϕ & ⟼ & ϕ ∘ x, \end{matrix}

where ϕ ∈ Hom (H_{s}[p], 𝔾_{a}) and ϕ ∘ x ∈ 𝔾_{a}(k(s)) = k(s). We see that it restricts to a pairing

H_{s}[p](k(s)) × H_{s}^{V_{∗}=1} ⟶ 𝔽_{p}.

If ϕ ∘ x = 0 for all x ∈ H_{s}[p](k(s)) then ϕ factors through the local ring of 𝔾_{a} at 0. If moreover ϕ ∘ Fr^{∗} = ϕ then we see that ϕ = 0. Thus our pairing gives an injection

H_{s}^{V_{∗}=1} ↪ Hom (H_{s}[p](k(s)), 𝔽_{p}).

To show this is in fact an isomorphism one can count orders. Suppose that #H_{s}[p](k(s)) = p^{h}. Then we have an embedding μ_{p}^{h} ↪ H_{s}[p]^{∨} and so an embedding Lie μ_{p}^{h} ↪ H_{s}. But (Lie μ_{p})^{V_{∗}=1} has order p (as follows easily from the results on page 143 of [Mu1]), and so

#H_{s}^{V_{∗}=1} ≥ p^{h} = #H_{s}[p](k(s)).

The perfection of our pairing follows at once.)

Again shrinking S we may assume that in fact H is free. Choose a basis e_{1}, ..., e_{m} and suppose that V_{∗}e_{i} = ∑_{j} v_{i,j}e_{j}. Then

V_{∗} ∑_{i} x_{i}e_{i} = ∑_{i,j} x_{i}^{p}v_{i,j}e_{j}.

Let H^{V_{∗}=1} denote the subscheme of Aff_{R}^{m} defined by the equations

x_{j} = ∑_{i,j} v_{i}x_{i}^{p}

for j = 1, ..., m. Then H^{V_{∗}=1}/S is quasi-finite and etale (as the Jacobian is the identity matrix). Generically H^{V_{∗}=1}/S has degree ≤ p^{g}. We may suppose that in fact generically the degree equals p^{g}. The locus where the degree drops is closed (as the degree is locally constant). We must show that it has codimension 1. Let T denote the normalisation of S in a finite separable extension of the fraction field of R over which H^{V_{∗}=1} has p^{g} points. As T/S is finite, it suffices to prove the result for T. Let x_{1}, ..., x_{p^{g}} denote the sections of H^{V_{∗}=1} over the generic point of T. Then T′ is simply the locus where some x_{i} is not regular. But the locus where any given x_{i} is not regular has codimension 1 because T is normal. □

Corollary 2.2 Suppose that H/X is a Barsotti-Tate O_{K}-module over a locally noetherian formal scheme. Suppose also that p = 0 on X^{red} and that the function from geometric points of X^{red} to integers

s ↦ #H[p](k(s))

is constant. Then there is an exact sequence of Barsotti-Tate O_{K}-modules

(0) ⟶ H^{0} ⟶ H ⟶ H^{et} ⟶ (0)

over X with H^{0} formal and H^{et} ind-etale.

For any g ≥ 1 there is a unique 1-dimensional formal Barsotti-Tate O_{K}- module Σ_{K,g} over k(v_{K})^{ac} of height g. In fact if k is any separably closed field containing k(v_{K}) then any one-dimensional Barsotti-Tate O_{K}-module over k is of the form Σ_{K,g} × (K/O_{K})^{h} for some g and h. (If H/k(v_{K})^{ac} is a onedimensional Barsotti-Tate O_{K}-module over k we have an exact sequence

(0) ⟶ H^{0} ⟶ H ⟶ H^{et} ⟶ (0),

where H^{0} is formal and H^{et} is ind-etale. By proposition 1.7 of [Dr] we see that H^{0} ≅ Σ_{K,g}. It only remains to find a splitting H^{et} → H, but this is the same as finding a splitting H^{et}(k) → H(k). Finally note that H(k) → H^{et}(k) is an isomorphism.) Moreover End (Σ_{K,g}/k) = End (Σ_{K,g}/k(v_{K})^{ac}) ≅ O_{D_{K,g}} (proposition 1.7 of [Dr]). We can extend the (left)-action of O_{D_{K,g}}^{×} on Σ_{K,g}/k(v_{K})^{ac} to an action of D_{K,g}^{×} on Σ_{K,g}/k(v_{K}), such that for δ ∈ D_{K,g}^{×}

Σ_{K,g}  ⟶^{δ}  Σ_{K,g}
↓  ↓
Spec k(v_{K})^{ac} ⟶^{(Frob_{v_{K}}^{∗})^{v_{K} (det δ)}} Spec k(v_{K})^{ac}

commutes. To see this one need only consider the case v_{K}(det δ) ≥ 0: for v_{K}(det δ) < 0 we define the action of δ to be the inverse of the action of δ^{−1}. If v_{K}(det δ) ≥ 0 then the kernel of δ ∈ End (Σ_{K,g}) is the same as the kernel of F^{f_{K}v_{K}(det δ)} : Σ_{K,g} → Σ_{K,g}^{(f_{K}v_{K}(det δ))}. Thus δ induces a map Σ_{K,g}^{(f_{K}v_{K}(det δ))} → Σ_{K,g}. We define our semi-linear action of δ as the top row of the following diagram

Σ_{K,g}  ⟶  Σ_{K,g}^{(p^{f_{K} v_{K} (det δ)})} ⟶^{δ}  Σ_{K,g}
↓  ↓  ↓
Spec k(v_{K})^{ac} ⟶^{(Frob_{v_{K}}^{∗})^{v_{K} (det δ)}} Spec k(v_{K})^{ac} = Spec k(v_{K})^{ac}

where the left hand square is a pullback.

Consider the functor from Artinian local O_{K}-algebras with residue field k(v_{K})^{ac} to sets which sends A to the set of isomorphism classes of pairs (H, j) where H/Spec A is a Barsotti-Tate O_{K}-module and j : Σ_{K,g} →^{∼} H ×_{A} k(℘_{K})^{ac}. This functor is pro-represented by a complete noetherian local ring R_{K,g} with residue field k(℘_{K})^{ac} and in fact R_{K,g} ≅ OK̂^{nr}[[T_{2}, ..., T_{g}]] (proposition 4.2 of [Dr]). The universal deformation exists over Spec R_{K,g} (not just over Spf R_{K,g}, see lemma 4.16 of chapter II of [Me]). We will denote this universal deformation by (Σ̃_{K,g}, j̃)/Spec R_{K,g}. Note that R_{K,g} has a continuous left action of O_{D_{g}}^{×}. (If δ ∈ O_{D_{K,g}} then the push forward of (Σ̃_{K,g}, j̃) along δ : R_{K,g} → R_{K,g} is (Σ̃_{K,g}, j̃ ∘ δ).) We will let \overline{R}_{K,g} denote R_{K,g} ×_{W (k(℘_{K}))} k(℘_{K})^{ac}.

Set H_{0} = Σ_{k,g} × (K/O_{K})^{h} a Barsotti-Tate O_{K}-module over k(v_{K})^{ac}. Let T H_{0} denote its Tate module, i.e. T H_{0} = Hom_{O_{K}}(K/O_{K}, H_{0}(k(v_{K})^{ac})) ≅ O_{K}^{h}. Now consider the functor from Artinian local O_{K}-algebras with residue field k(v_{K})^{ac} to sets which sends A to the set of isomorphism classes of pairs (H, j) where H/Spec A is a Barsotti-Tate O_{K}-module and j : H_{0} →^{∼} H ×_{A} k(℘_{K})^{ac}. This functor is again pro-represented, this time by Hom (T H_{0}, Σ̃_{K,g}). By Hom (T H_{0}, Σ̃_{K,g}) we mean the R_{K,g}-formal scheme such that for any Artinian local R_{K,g} algebra S we have

Hom (T H_{0}, Σ̃_{K,g})(S) = Hom_{O_{K}}(T H_{0}, Σ̃_{K,g}(S)).

Noncanonically we have Hom (T H_{0}, Σ̃_{K,g}) ≅ Σ̃_{K,g}^{h}, where the fibre product is taken over Spf R_{K,g}. We also have, again noncanonically, Hom (T H_{0}, Σ̃_{K,g}) ≅ Spf OK̂^{nr}[[T_{2}, ..., T_{g+h}]]. The universal deformation of H_{0} over Hom (T H_{0}, Σ̃_{K,g}) is then the extension of Σ̃_{K,g} by (K/O_{K})^{h} classified by the tautological class in

Hom (T H_{0}, Σ̃_{K,g})(S) = Hom (T H_{0}, Σ̃_{K,g}(S)) ≅
Ext (T H_{0} ⊗_{O_{K}} (K/O_{K}), Σ̃_{K,g}(S)) ≅ Ext_{S}(T H_{0} ⊗_{O_{K}} (K/O_{K}), Σ̃_{K,g}),

where S = Hom (T H_{0}, Σ̃_{K,g}). (See proposition 4.5 of [Dr] and its proof.)

Lemma 2.3 Suppose that S/k(v_{K})^{ac} is reduced of finite type. Suppose also that H/S is a one-dimensional Barsotti-Tate O_{K}-module. Suppose moreover that over S there is an exact sequence of Barsotti-Tate O_{K}-modules

(0) ⟶ H^{0} ⟶ H ⟶ H^{et} ⟶ (0),

where H^{et} has constant height h and H^{0} has constant height g. Let s be a closed point of S and choose an isomorphism j : Σ_{K,g} →^{∼} H_{s}^{0}.

1. Then we get a morphism S_{s}^{∧} → Spf R_{K,g} which in fact factors through k(v_{K})^{ac}.

2. We also get a morphism S_{s}^{∧} → Hom (T H_{s}, Σ̃_{K,g}) which in fact factors through Hom (T H_{s}, Σ_{K,g}) ⊂ Hom (T H_{s}, Σ̃_{K,g}).

Proof: The statements are easily seen to be equivalent. We will prove the first one. Write R_{K,g} = OK̂^{nr}[[T_{2}, ..., T_{g}]], let P be a minimal prime of O_{S,s}^{∧} and let k denote the field of fractions of the image R_{K,g} → O_{S,s}^{∧}/P. As S is reduced, it suffices to show that T_{2}, ..., T_{g} map to 0 in k. Suppose not.

For the rest of this proof we will use without comment the notation of [Dr]. We can arrange that Σ̃_{K,g} corresponds to a morphism Λ_{O_{K}} = O_{K}[g_{1}, g_{2}, ...] → R_{K,g} which

• sends g_{p^{f_{K} i}−1} to T_{i+1} for 1 ≤ i ≤ g − 1;

• and sends g_{j} to zero for 1 ≤ j < p^{f_{K}g} − 1 and j ≠ p^{f_{K}i} − 1 for some i in the above range.

(See the proof of proposition 4.2 of [Dr].) Choose i minimal such that T_{i} does not map to zero in k. Then H^{0}×_{S}k corresponds to a morphism Λ_{O_{K}} → k which sends g_{j} to 0 for j = 1, 2, ..., p^{f_{K}(i−1)} − 2 and sends g_{p^{f_{K} (i−1)}−1} to something nonzero. Thus H^{0} ×_{S} k has height i − 1 < g (see the proof of proposition 1.6 of [Dr]). This contradicts the fact that H ×_{S} Spec k is a formal Barsotti-Tate O_{K}-module of height g (because H/S is a formal Barsotti-Tate O_{K}-module of height g). □

Corollary 2.4 Suppose that S/k(v_{K})^{ac} is a smooth scheme of finite type. Suppose that H/S is a one-dimensional Barsotti-Tate O_{K}-module of constant height g. Suppose moreover that for each closed point s of S the formal completion S_{s}^{∧} is isomorphic to the equidimensional universal formal deformation space of H_{s}. Then for h = 0, ..., g − 1 the locally closed subscheme S^{(h)} = S^{[h]} − S^{[h−1]} of S is either empty or smooth of dimension h. If s is a closed point of S^{(h)} and if j : Σ_{K,g−h} →^{∼} H_{s}^{0} then we get an identification S_{s}^{∧} ≅ Hom (T H_{s}, Σ̃_{K,g−h})×_{O_{K}} k(v_{K}) and under this identification (S^{(h)})_{s}^{∧} ⊂ S_{s}^{∧} corresponds to Hom (T H_{s}, Σ_{K,g−h}) ⊂ Hom (T H_{s}, Σ̃_{K,g−h}) ×_{O_{K}} k(v_{K}).

Proof: Because the formal completion of S at any closed point is isomorphic to k(v_{K})^{ac}[[T_{2}, ..., T_{g}]], every component of S has dimension g − 1. We must have S = S^{[g−1]}. Hence by lemma 2.1 every irreducible component of S^{[h]} has dimension at least h. Thus the same is true for S^{(h)}. On the other hand by the previous lemma if s is any closed point of S^{(h)} then the formal completion (S^{(h)})_{s}^{∧} corresponds to a sub-formal scheme of Hom (T H_{s}, Σ_{K,g−h}) ⊂ Hom (T H_{s}, Σ̃_{K,g−h}). Thus we must have (S^{(h)})_{s}^{∧} ≅ Hom (T H_{s}, Σ_{K,g−h}) and, assuming such a closed point exists, we have that S^{(h)} is smooth at s of dimension h. □

The functor on schemes S/k(v_{K})^{ac} which sends S to Aut (Σ_{K,g}[℘_{K}^{m}]/S) is represented by a scheme Aut (Σ_{K,g}[℘_{K}^{m}]) of finite type over k(v_{K})^{ac}. (To see this simply think of these automorphisms as maps on sheaves of Hopf algebras.) If m_{1} > m_{2} there is a natural morphism

Aut (Σ_{K,g}[℘_{K}^{m_{1}}]) ⟶ Aut (Σ_{K,g}[℘_{K}^{m_{2}}]).

We will let Aut^{1}(Σ_{K,g}[℘_{K}^{m}]) denote the intersection of the scheme theoretic images of Aut (Σ_{K,g}[℘_{K}^{m′}]) in Aut (Σ_{K,g}[℘_{K}^{m}]) as m′ varies over integers greater than or equal to m.

Lemma 2.5 Aut^{1}(Σ_{K,g}[℘_{K}^{m}]) is zero dimensional and

Aut^{1}(Σ_{K,g}[℘_{K}^{m}])^{red} ≅ (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})^{×}.

Proof: From the definitions we see that the scheme theoretic image of the morphism

Aut^{1}(Σ_{K,g}[℘_{K}^{m+1}]) ⟶ Aut^{1}(Σ_{K,g}[℘_{K}^{m}])

is just Aut^{1}(Σ_{K,g}[℘_{K}^{m}]). Suppose first that Aut^{1}(Σ_{K,g}[℘_{K}^{m}]) has an irreducible component V_{m} of dimension > 0. Then we can find irreducible components V_{m′} of Aut^{1}(Σ_{K,g}[℘_{K}^{m′}]) for m′ > m such that whenever m′′ ≥ m′ ≥ m then V_{m′′} maps to V_{m′} and is dominating. Let k(V_{m′}) denote the function field of V_{m′}^{red}, so that whenever m′′ ≥ m′ ≥ m we have k(V_{m′}) ↪ k(V_{m′′}). Let k be algebraically closed extension field of k(v_{K})^{ac} of uncountable transcendence degree. Then there are uncountably many maps k(V_{m}) ↪ k and each can be extended into a compatible series of injections k(V_{m′}) ↪ k for m′ > m. Thus Aut^{1}(Σ_{K,g}[℘_{K}^{m}])(k) has uncountably many points which can be lifted compatibly to each Aut^{1}(Σ_{K,g}[℘_{K}^{m′}])(k) with m′ > m. This implies that the image of

Aut (Σ_{K,g}/k) ⟶ Aut (Σ_{K,g}[℘_{K}^{m}]/k)

is uncountably infinite. On the other hand it follows from proposition 1.7 of [Dr] that this image is just (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})^{×}, which is finite. This contradiction shows that Aut^{1}(Σ_{K,g}[℘_{K}^{m}]) is zero dimensional.

As each Aut^{1}(Σ_{K,g}[℘_{K}^{m}]) is zero dimensional and as for m′ > m the morphism

Aut^{1}(Σ_{K,g}[℘_{K}^{m′}]) ⟶ Aut^{1}(Σ_{K,g}[℘_{K}^{m}])

is dominating we see that for m′ ≥ m

Aut^{1}(Σ_{K,g}[℘_{K}^{m′}])(k(v_{K})^{ac}) → → Aut^{1}(Σ_{K,g}[℘_{K}^{m}])(k(v_{K})^{ac}).

It follows that Aut^{1}(Σ_{K,g}[℘_{K}^{m}])(k(v_{K})^{ac}) equals the image of

Aut (Σ_{K,g}/k(v_{K})^{ac}) ⟶ Aut (Σ_{K,g}[℘_{K}^{m}]/k(v_{K})^{ac}).

Again by proposition 1.7 of [Dr] this is just (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})^{×} and so the lemma follows. □

We remark that for m > 1 the scheme Aut (Σ_{K,g}[℘_{K}^{m}]) has dimension > 0. By an explicit calculation with Dieudonne modules we checked in an earlier version of this paper that Aut^{1}(Σ_{K,g}[℘_{K}^{m}])^{red} coincides with the reduced subscheme of the image of Aut (Σ_{K,g}[℘_{K}^{m+1}]) → Aut (Σ_{K,g}[℘_{K}^{m}]). However we will not actually need that stronger result here, so we do not reproduce the argument.

Now suppose that S is a reduced k(v_{K})^{ac}-scheme and that H/S is a onedimensional formal Barsotti-Tate O_{K}-module of constant height g. We want to investigate how far H differs from Σ_{K,g} ×_{Spec k(v_{K})^{ac}} S. Consider the functor on S-schemes which sends T/S to the set of isomorphisms (over T)

j : Σ_{k,g}[℘_{K}^{m}] ×_{Spec k(v_{K})^{ac}} T ⟶ H[℘_{K}^{m}] ×_{S} T.

It is easy to see that this functor is represented by a scheme X_{m}(H/S) of finite type over S. (Think about j as a map of sheaves of Hopf algebras on T.) Then we define Y_{m}(H/S) to be the intersection of the scheme theoretic images of the

X_{m′}(H/S) ⟶ X_{m}(H/S)

for m′ ≥ m. Finally we set J^{(m)}(H/S) = Y_{m}(H/S)^{red}. We will also let j^{univ} denote the universal isomorphism

j^{univ} : Σ_{K,g}[℘_{K}^{m}] ⟶^{∼} H[℘_{K}^{m}]

over J^{(m)}(H/S).

For instance

J^{(m)}(Σ_{K,g}/k(v_{K})^{ac}) = Aut^{1}(Σ_{K,g}[℘_{K}^{m}])^{red} ≅ (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})^{×}.

In fact if S/k(v_{K})^{ac} is any reduced scheme then

J^{(m)}(Σ_{K,g}/S) = (J^{(m)}(Σ_{K,g}/k(v_{K})^{ac}) × S)^{red} = ((O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})_{S}^{×})^{red} = (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})_{S}^{×}.

Each of the schemes X_{m}(H/S), Y_{m}(H/S) and J^{(m)}(H/S) has a natural right action of (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})^{×}. (δ ∈ O_{D,g}^{×} takes j to j ∘ δ.) If S = T ×_{Spec k(v_{K})} Spec k(v_{K})^{ac} for a reduced scheme T/k(v_{K}) and if H = H_{0} ×_{T} S for a formal Barsotti-Tate O_{K}-module H_{0}/T, then this action extends to one of D_{K,g}^{×}/(1+℘_{K}^{m}O_{D_{K,g}}) on each of X_{m}(H/S), Y_{m}(H/S) and J^{(m)}(H/S) thought of as T -schemes. More precisely if δ ∈ D_{K,g}^{×} we get a commutative diagram

J^{(m)}(H/S)  ⟶^{δ}  J^{(m)}(H/S)
↓  ↓
T × Spec k(v_{K})^{ac} ⟶^{(1×Frob_{v_{K}}^{∗})^{−v_{K} (det δ)}} T × Spec k(v_{K})^{ac}.

commutes. (Let X/S denote the pull back of X_{m}(H/S) by

T × Spec k(v_{K})^{ac} ⟶^{(1×Frob_{v_{K}}^{∗})^{v_{K} (det δ)}} T × Spec k(v_{K})^{ac}.

Then over X we get an isomorphism j′ : Σ_{K,g}^{(p^{−f_{K} v_{K} (det δ)})}[℘_{K}^{m}] →^{∼} H[℘_{K}^{m}]. On the other hand δ gives an isomorphism

δ : Σ_{K,g} ⟶^{∼} Σ_{K,g}^{(p^{−f_{K} v_{K} (det δ)})}.

Thus over X we get

j′ ∘ δ : Σ_{K,g}[℘_{K}^{m}] →^{∼} H[℘_{K}^{m}].

This induces a map over S from X to X_{m}(H/S). Composing this with the inverse of the pull back of (1×Frob_{v_{K}}^{∗})^{v_{K}(det δ)} we get the desired automorphism of X_{m}(H/S). The following diagram

X_{m}(H/S)  ⟵  X  →  X_{m}(H/S)
↓  ↓  ↓
T × Spec k(v_{K})^{ac} ⟵ T × Spec k(v_{K})^{ac} = T × Spec k(v_{K})^{ac}

(where the leftwards arrow on the bottom row is (1×Frob_{v_{K}}^{∗})^{v_{K}(det δ)}) illustrates this construction.)

Before going on to the main result of this section let us take the opportunity to record a simple result in commutative algebra for which we do not know a reference.

Lemma 2.6 Let k be a perfect field, A/k a k-algebra of finite type, 𝔪 a maximal ideal of A and B/A a finite A-algebra.

1. Suppose also that B is reduced. Then the completion B_{𝔪}^{∧} of B with respect to the ideal 𝔪B is also reduced.

2. More generally if N is the nilradical of B then N_{𝔪}^{∧} is the nilradical of B_{𝔪}^{∧}.

Proof: Consider the first part of the lemma. B has finitely many maximal ideals 𝔫_{1}, ..., 𝔫_{r} above 𝔪 and we have that

B_{𝔪}^{∧} = B_{𝔫_{1}}^{∧} ⊕ ... ⊕ B_{𝔫_{r}}^{∧}

(see corollary 2 of section 7 of chapter VIII of [ZS]). Thus we can reduce to the case B = A.

Note that as A is reduced the same is true of the localisation A_{𝔪}. Let Q_{1}, ..., Q_{s} denote the minimal primes of A_{𝔪}. As A_{𝔪} is reduced we have a finite embedding

A_{𝔪} ↪ (A_{𝔪}/Q_{1}) ⊕ ... ⊕ (A_{𝔪}/Q_{s}).

Hence we also have an embedding

A_{𝔪}^{∧} ↪ (A_{𝔪}/Q_{1})^{∧} ⊕ ... ⊕ (A_{𝔪}/Q_{s})^{∧}

(use theorem 16 of section 6 of chapter VIII of [ZS] and the flatness of A_{𝔪}^{∧}/A_{𝔪}). Thus we may suppose that A_{𝔪} is an integral domain. In this case the result follows from lemmas 1 and 4 of section 13 of chapter VIII of [ZS].

For the second part consider the exact sequence

(0) ⟶ N ⟶ B ⟶ B′ ⟶ (0).

Again using theorem 16 of section 6 of chapter VIII of [ZS] and the flatness of A_{𝔪}^{∧}/A_{𝔪} we see that

(0) ⟶ N_{𝔪}^{∧} ⟶ B_{𝔪}^{∧} ⟶ (B′)_{𝔪}^{∧} ⟶ (0)

is also exact. By the first part of this lemma we know that (B′)_{𝔪}^{∧} is reduced and the second part follows. □

The following proposition is of key importance for us.

Proposition 2.7 Suppose that S/k(v_{K})^{ac} is a reduced scheme of finite type and let H/S be a one-dimensional formal Barsotti-Tate O_{K}-module of constant height g. Then for each m ≥ 1, J^{(m)}(H/S)/S is finite etale and Galois with Galois group (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})^{×}. (N.B. We are not asserting that J^{(m)}(H/S) is connected.)

Proof: It suffices to show that for any closed point s of S

• the group (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})^{×} has a faithful and transitive action on the points of J^{(m)}(H/S)_{s};

• and if t is any point of J^{(m)}(H/S)_{s} then J^{(m)}(H/S)_{t}^{∧} →^{∼} S_{s}^{∧}.

(See for instance theorem 3 of section 5 of chapter 3 of [Mu2].) Equivalently it suffices to check that for all closed points s of S we have

J^{(m)}(H/S) ×_{S} Spec O_{S,s}^{∧} ≅ (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})_{Spec O_{S,s}^{∧}}^{×}.

We note that by lemma 2.3

H ×_{S} Spec O_{S,s}^{∧} ≅ Σ_{K,g} ×_{Spec k(v_{K})^{ac}} Spec O_{S,s}^{∧}.

Because Spec O_{S,s}^{∧} is flat we see that for any closed point s of S we have

Y_{m}(H/S) ×_{S} Spec O_{S,s}^{∧} ≅ Y_{m}(H/Spec O_{S,s}^{∧}).

(The formation of scheme theoretic images commutes with flat base change.) Thus by lemma 2.3 we see that

Y_{m}(H/S) ×_{S} Spec O_{S,s}^{∧} ≅ Y_{m}(Σ_{K,g}/Spec O_{S,s}^{∧}) ≅ Aut^{1}(Σ_{K,g}[℘_{K}^{m}]) × Spec O_{S,s}^{∧}

is finite and flat over Spec O_{S,s}^{∧}. We conclude that Y_{m}(H/S) is finite and flat over S. (Use the fact that O_{S,s}^{∧} is faithfully flat over O_{S,s}. More precisely it suffices to show that for any closed point s of S the ring O_{Y_{m}(H/S)} ⊗_{O_{S}} O_{S,s} is finite and free over O_{S,s}. We have seen that this is true after tensoring with O_{S,s}^{∧}. Thus we can find a morphism of O_{S,s}-modules O_{S,s}^{a} → O_{Y_{m}(H/S)} ⊗_{O_{S}} O_{S,s} which becomes an isomorphism after tensoring with O_{S,s}^{∧}. Faithful flatness

implies that O_{S,s}^{a} →^{∼} O_{Y_{m}(H/S)} ⊗_{O_{S}} O_{S,s}.)

Thus J^{(m)}(H/S) is finite over S. The previous lemma then shows that for any closed point s of S

J^{(m)}(H/S) ×_{S} Spec O_{S,s}^{∧} ⟶^{∼} J^{(m)}(H/Spec O_{S,s}^{∧}).

Thus by lemma 2.3 we have that

J^{(m)}(H/S) ×_{S} Spec O_{S,s}^{∧} ≅ J^{(m)}(Σ_{K,g}/Spec O_{S,s}^{∧}) ≅ (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})_{Spec O_{S,s}^{∧}}^{×}.

This proves the proposition. □

Although we will not need it in this paper, it may be of interest to point out the following corollary of the proceeding proposition.

Corollary 2.8 Suppose that S/k(v_{K})^{ac} is a reduced, connected scheme of finite type, and suppose that s is a geometric point of S. If H/S is a onedimensional formal Barsotti-Tate O_{K}-module of height g then it gives rise to a continuous homomorphism

ρ_{H} : π_{1}^{alg}(S, s) ⟶ O_{D_{K,g}}^{×}.

This gives rise to a bijection between isomorphism classes of one-dimensional formal Barsotti-Tate O_{K}-module of height g on S and conjugacy classes of continuous homomorphisms

ρ : π_{1}^{alg}(S, s) ⟶ O_{D_{K,g}}^{×}.

Proof: As we will not use this result elsewhere in this paper we will simply sketch the proof.

First we explain the construction of ρ_{H} from H. Choose a compatible system of geometric points s_{m} of J^{(m)}(H/S) above s (i.e. if m′ > m then s_{m′} maps to s_{m} under J^{(m′)}(H/S) → J^{(m)}(H/S)). If σ ∈ π_{1}^{alg}(S, s) then σs_{m} = ρ_{H,m}(σ)s_{m} for some (unique) element ρ_{H,m}(σ) ∈ (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})^{×}. Moreover for m′ > m we have ρ_{H,m′}(σ) ≡ ρ_{H,m}(σ) mod ℘_{K}^{m}. We set

ρ_{H} = lim_{←} ρ_{H,m} : π_{1}^{alg}(S, s) → O_{D_{K,g}}^{×}.

It is a continuous homomorphism. The construction appears to depend on the choice of the system {s_{m}}, but a different choice simply changes ρ_{H} by conjugation in O_{D_{K,g}}^{×}.

Next we explain how one goes from a continuous homomorphism

ρ : π_{1}^{alg}(S, s) → O_{D_{K,g}}^{×}

to a one-dimensional formal Barsotti-Tate O_{K}-module H_{ρ}/S. The reduction ρ mod ℘_{K}^{m} gives rise to a Galois finite etale cover (not necessarily connected) S_{m} → S with Galois group (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})^{×}. Consider

Σ_{K,g}[℘_{K}^{m}] × S_{m} ⟶ S_{m}

with the diagonal action of (O_{D_{K,g}}/℘_{K}^{m}O_{D_{K,g}})^{×}. We may quotient out by the action of this finite group and we obtain a finite flat group scheme H_{m}/S. We set H_{ρ} = lim_{→} H_{m}.

We leave the reader both to check that H_{ρ}/S is a one-dimensional formal Barsotti-Tate O_{K}-module, and that these two constructions are inverse to each other. □

We end this section with some results about lifting extensions of Barsotti-Tate O_{K}-modules. We are very grateful to Johan de Jong for explaining to us how to prove corollary 2.10 below.

Lemma 2.9 Let A be a noetherian ring and J an ideal of A which contains some power of p and which satisfies J^{2} = (0). Suppose that over Spec A/J we have an exact sequence of Barsotti-Tate O_{K}-modules

(0) ⟶ H^{0} ⟶ H ⟶ H^{et} ⟶ (0)

with H^{0} formal and H^{et} ind-etale. Suppose moreover that H̃^{0} is a lift of H^{0} to a Barsotti-Tate O_{K}-module over Spec A. Then there is an exact sequence

(0) ⟶ H̃^{0} ⟶ H̃ ⟶ H̃^{et} ⟶ (0)

of Barsotti-Tate O_{K}-modules over Spec A, which reduces modulo J to the above exact sequence.

Proof: For the proof we use Grothendieck-Messing Dieudonne theory (see [Me] as completed by [I]). This associates to

(0) ⟶ H^{0} ⟶ H ⟶ H^{et} ⟶ (0)

an exact sequence of crystals in finite locally free modules

(0) ⟶ D(H^{0}) ⟶ D(H) ⟶ D(H^{et}) ⟶ (0)

on Spec A/J. (For exactness use [BBM] combined with the compatibility of the theories in [Me] and [BBM], which is proved in [BM].) Moreover we have locally free submodules V (H^{0}) ⊂ D(H^{0})_{A/J} (resp. V (H) ⊂ D(H)_{A/J}, resp. V (H^{et}) = D(H^{et})_{A/J}) with locally free quotients. To H̃^{0} we may associate a locally free submodule V (H̃^{0}) ⊂ D(H^{0})_{A} with a locally free quotient and with V (H̃^{0})/JV (H̃^{0}) = V (H^{0}). We will look for a locally free submodule V ⊂ D(H)_{A} with locally free quotient such that

• V (H̃^{0}) → V,

• V → → D(H^{et})_{A},

• and V/JV = V (H).

Assuming we can find such a V we would get a complex of Barsotti-Tate O_{K}-modules

(0) ⟶ H̃^{0} ⟶ H̃ ⟶ H̃^{et} ⟶ (0)

lifting

(0) ⟶ H^{0} ⟶ H ⟶ H^{et} ⟶ (0).

It follows from lemma 4.10 chapter II of [Me] that the lifted sequence is in fact exact.

It remains to construct such a V. This is equivalent to constructing a splitting for the exact sequence

(0) ⟶ D(H^{0})_{A}/V (H̃^{0}) ⟶ D(H)_{A}/V (H̃^{0}) ⟶ D(H^{et})_{A} ⟶ (0)

above the splitting of

(0) ⟶ D(H^{0})_{A/J}/V (H^{0}) ⟶ D(H)_{A/J}/V (H^{0}) ⟶ D(H^{et})_{A/J} ⟶ (0)

provided by V (H)/V (H^{0}). As D(H^{et})_{A} is locally free we can find such a splitting Zariski locally on Spec A. It is not unique but determined up to an element of

Hom (D(H^{et})_{A/J}, J(D(H^{0})/V (H̃^{0}))).

Thus the obstruction to the existence of a global splitting lies in

H^{2}(Spec A/J, Hom (D(H^{et})_{A/J}, J(D(H^{0})/V (H̃^{0})))).

Because Spec A/J is affine this group vanishes and so we can find such a splitting globally. □

As an immediate consequence we obtain the following corollary.

Corollary 2.10 Let A be a noetherian ring complete with respect to the topology defined by an ideal I. Suppose that I contains a power of p. Suppose also that over Spec A/I we have an exact sequence of Barsotti-Tate O_{K}-modules

(0) ⟶ H^{0} ⟶ H ⟶ H^{et} ⟶ (0)

with H^{0} formal and H^{et} ind-etale. Suppose moreover that H̃^{0} is a lift of H^{0} to a Barsotti-Tate O_{K}-module over Spf A. Then there is an exact sequence

(0) ⟶ H̃^{0} ⟶ H̃ ⟶ H̃^{et} ⟶ (0)

of Barsotti-Tate O_{K}-modules over Spf A, which reduces modulo I to the previous exact sequence.

3 Drinfeld level structures

Suppose that H/S is a Barsotti-Tate O_{K}-module of constant height h over a scheme S. By a Drinfeld ℘_{K}^{m}-structure on H/S we shall mean a morphism of abelian groups

α : (℘_{K}^{−m}/O_{K})^{h} ⟶ H[℘_{K}^{m}](S)

such that the set of α(x) for x ∈ (℘_{K}^{−m}/O_{K})^{h} forms a full set of sections of H[℘_{K}^{m}] in the sense of [KM] section 1.8. We will collect together here some of the basic properties of Drinfeld level structures.

Lemma 3.1 In this lemma S will denote an O_{K}-scheme and H/S will be a Barsotti-Tate O_{K}-module of constant height h.

1. Suppose that α : (℘_{K}^{−m}/O_{K})^{h} ⟶ H[℘_{K}^{m}](S) is a Drinfeld ℘_{K}^{m}-structure and that T/S is any scheme. Then the composite

α_{T} : (℘_{K}^{−m}/O_{K})^{h} ⟶^{α} H[℘_{K}^{m}](S) ⟶ H[℘_{K}^{m}](T)

is a Drinfeld ℘_{K}^{m}-structure for H ×_{S} T.

2. Suppose that S/𝔽_{p} is reduced. If H/S is one-dimensional and formal then H contains a unique finite flat subgroup scheme of any order p^{s}, namely the kernel of F^{s}.

3. Suppose that S/𝔽_{p} is reduced. If H/S is one-dimensional and formal then there is a unique Drinfeld ℘_{K}^{m}-structure on H/S, namely the trivial homomorphism

\begin{matrix}α^{triv} : (℘_{K}^{−m}/O_{K})^{h} & ⟶ & H(S) \\ x & ⟼ & 0 \end{matrix}

for all x ∈ (℘_{K}^{−m}/O_{K})^{h}. (We will refer to this as the trivial Drinfeld ℘_{K}^{m}-structure.)

4. Suppose that there is an exact sequence of Barsotti-Tate O_{K}-modules

(0) ⟶ H^{0} ⟶ H ⟶ H^{et} ⟶ (0),

over S with H^{0} formal, H^{et} ind-etale and both of constant height. Then

α : (℘_{K}^{−m}/O_{K})^{h} → H[℘_{K}^{m}](S)

is a Drinfeld ℘_{K}^{m}-structure if and only if there is a direct summand O_{K}- submodule M ⊂ (℘_{K}^{−m}/O_{K})^{h} such that • α|_{M} : M → H^{0}[℘_{K}^{m}](S) is a Drinfeld ℘_{K}^{m}-structure,

• α induces an isomorphism

α : ((℘_{K}^{−m}/O_{K})^{h}/M)_{S} →^{∼} H^{et}[℘_{K}^{m}].

5. Suppose that S is reduced and that p = 0 on S. Suppose also that there is an exact sequence of Barsotti-Tate O_{K}-modules

(0) ⟶ H^{0} ⟶ H ⟶ H^{et} ⟶ (0),

over S with H^{0} formal, H^{et} ind-etale and both of constant height. If H/S admits a Drinfeld ℘_{K}^{m}-level structure (with m ≥ 1) then there is a unique splitting

H[℘_{K}^{m}] ≅ H^{0}[℘_{K}^{m}] × H^{et}[℘_{K}^{m}]

over S. On the other hand if there is a splitting H[℘_{K}^{m}] ≅ H^{0}[℘_{K}^{m}] × H^{et}[℘_{K}^{m}]/S then to give a Drinfeld ℘_{K}^{m}-structure α : (℘_{K}^{−m}/O_{K})^{h} → H[℘_{K}^{m}](S) is the same as giving a direct summand M ⊂ (℘_{K}^{−m}/O_{K})^{h} and an isomorphism

((℘_{K}^{−m}/O_{K})^{h}/M)_{S} →^{∼} H^{et}[℘_{K}^{m}].

6. For any m ≥ 0 there is a scheme S(m) which is finite over S and a Drinfeld ℘_{K}^{m}-structure

α^{univ} : (℘_{K}^{−m}/O_{K})^{h} ⟶ H[℘_{K}^{m}](S(m))

on H ×S(m), which is universal in the sense that if T/S is any S-scheme and if

α : (℘_{K}^{−m}/O_{K})^{h} ⟶ H[℘_{K}^{m}](T)

is any Drinfeld ℘_{K}^{m}-structure on H × T then T → S factors uniquely through S(m) in such a way that α^{univ} pulls back to α. Moreover S(m)/S has a right action of GL_{h}(O_{K}/℘_{K}^{m}), which can be characterised as follows. If g ∈ GL_{h}(O_{K}/℘_{K}^{m}) then under the morphism g : S(m) → S(m), α^{univ} pulls back to α^{univ} ∘ g.

7. Suppose that S = Spec R with R a noetherian local ring and that H/S is one-dimensional and formal. Then H ≅ Spf R[[T]]. Choose a uniformiser ϖ_{K} ∈ O_{K} and let f_{ϖ_{K}^{m}}(T) ∈ R[[T]] be the power series representing multiplication by ϖ_{K}^{m} (i.e. f_{ϖ_{K}^{m}}(T) = (ϖ_{K}^{m})^{∗}(T)). Suppose that α : (℘_{K}^{−m}/O_{K})^{h} → H[℘_{K}^{m}](R). Then the following are equivalent • α is a Drinfeld ℘_{K}^{m}-level structure,

• ∏_{x}(T − T (α(x)))|f_{ϖ_{K}^{m}}(T),

• f_{ϖ_{K}^{m}}(T) = g(T) ∏_{x}(T − T (α(x))) for some unit g(T) ∈ R[[T]]^{×}.

Proof: 1. This follows from proposition 1.9.1 of [KM].

2. We may suppose that S is connected. From the discussion on page 26 of [Me] we see that ker F^{s} locally isomorphic to Spec O_{S}[T]/(T^{p^{s}}). Thus ker F^{s} is indeed a finite flat group scheme of rank p^{s}. If A ⊂ H is any finite flat subgroup scheme then we will show that A = ker F^{s} for some s. Choose s maximal such that ker F^{s} ⊂ A. We must show we actually have equality. Modding out by ker F^{s} we may suppose that s = 0. We must show that A = (0). If for any point s of S, ker F |_{A_{s}} = (0) then as A_{s} is connected we must have A_{s} = (0). As S is connected the rank of A is constant and we would then have that A = (0) as desired. Thus suppose that for all points s of S we have that (ker F_{A})_{s} ≠ (0). As ker F is finite flat of rank p we see that we must have (ker F_{A})_{s} = (ker F)_{s} for all points s of S. Over S we have ker F |_{A} is a closed subscheme of ker F which becomes equal to ker F when pulled back to any point of S. As S is reduced we must have ker F = ker F |_{A} ⊂ A, a contradiction.

3. To see that there is no more than one Drinfeld ℘_{K}^{m}-structure it suffices to check that H[℘_{K}^{m}](S) = {0}. This follows because S is reduced and H[℘_{K}^{m}]/S is radicial (see proposition 4.4 of chapter II of [Me]). (If f : T → S is finite and radicial and if S is reduced then there is at most one section to f. To see this one reduces to the case that S = Spec A and T = Spec B. We are looking for sections to f^{∗} : A → B. Suppose g_{1}^{∗} and g_{2}^{∗} are two such sections. As S is reduced we can embed A into a product of fields. Thus if g_{1}^{∗} ≠ g_{2}^{∗} we can find a field k and a homomorphism ϕ^{∗} : A → k such that ϕ^{∗} ∘ g_{1}^{∗} ≠ ϕ^{∗} ∘ g_{2}^{∗}. On the other hand we must have g_{1}^{∗} ∘f^{∗} = g_{2}^{∗} ∘f^{∗} and so ϕ^{∗} ∘g_{1}^{∗} ∘f^{∗} = ϕ^{∗} ∘g_{2}^{∗} ∘f^{∗}. This contradicts the fact that f is radicial. (The finiteness hypothesis is presumably unnecessary, but this additional hypothesis does us no harm.))

It remains to show that α^{triv} is indeed a Drinfeld ℘_{K}^{m}-structure. We must have that H[℘_{K}^{m}] = ker F^{f_{K}hm} and hence H[℘_{K}^{m}] is locally isomorphic to O_{S}[T]/(T^{p^{f_{K} hm}}). If f ∈ O_{S}[T]/(T^{p^{f_{K} hm}}) and we write f = f_{0} + f_{1}T + ... + f_{p^{f_{K} hm}−1}T^{p^{f_{K} hm}−1}, then the norm down to O_{S} of f is f_{0}^{p^{f_{K} hm}} = f(0)^{p^{f_{K} hm}}. This verifies condition (2) on page 33 of [KM].

4. This follows from proposition 1.11.2 and lemma 1.8.3 of [KM].

5. Suppose that α : (℘_{K}^{−m}/O_{K})^{h} → H[℘_{K}^{m}](S) is a Drinfeld ℘_{K}^{m}-structure. Let M = ker α. By parts 2 and 3 we see that the composite

α : ((℘_{K}^{−m}/O_{K})/M)_{S} ⟶ H[℘_{K}^{m}] ⟶ H^{et}[℘_{K}^{m}]

is an isomorphism. A splitting of H^{et}[℘_{K}^{m}] into H[℘_{K}^{m}] is provided by the image of ((℘_{K}^{−m}/O_{K})/M)_{S} in H[℘_{K}^{m}]. To see the splitting is unique we argue as follows. To give a splitting is the same as giving a morphism γ : ((℘_{K}^{−m}/O_{K})/M)_{S} → H[℘_{K}^{m}] such that the composite

γ : ((℘_{K}^{−m}/O_{K})/M)_{S} ⟶ H[℘_{K}^{m}] ⟶ H^{et}[℘_{K}^{m}]

coincides with the map induced by α. To give γ is the same as giving

γ : ((℘_{K}^{−m}/O_{K})/M) ⟶ H[℘_{K}^{m}](S) ⟶^{∼} H^{et}[℘_{K}^{m}](S).

Thus there is only one possible choice for γ. The second assertion of this part now follows from parts 3 and 4.

6. Let S(m) be the closed subscheme of T = H[℘_{K}^{m}]^{(℘_{K}^{−m}/O_{K})^{h}} where the tautological map

(℘_{K}^{−m}/O_{K})^{h} ⟶ H[℘_{K}^{m}](T)

is a Drinfeld ℘_{K}^{m}-structure (use lemma 1.9.1 of [KM]).

7. That H ≅ Spf R[[T]] follows from page 26 of [Me]. Let T^{s} be the first power of T whose coefficient in f_{ϖ_{K}^{m}}(T) is not in the maximal ideal of R. Consider the map

⨁_{i=0}^{s−1} RT^{i} ⟶ R[[T]]/(f_{ϖ_{K}^{m}}(T)) ≅ R^{f_{K}mh}.

After tensoring with the residue field of R we get an isomorphism. Thus this map is already an isomorphism and s = f_{K}mh. We conclude that T^{p^{f_{K} mh}} is a linear combination of 1, T, ..., T^{p^{f_{K} mh}−1} in R[[T]]/f_{ϖ_{K}^{m}}(T), and again by reducing modulo the maximal ideal of R we see that T^{p^{f_{K} mh}} is a linear combination of 1, T, ..., T^{p^{f_{K} mh}−1} with coefficients in the maximal ideal of R. Put another way we can find a monic polynomial h(T) of degree p^{f_{K}mh} over R all whose nonleading coefficients are in the maximal ideal of R, and a power series g(T) ∈ R[[T]] such that h(T) = g(T)f_{ϖ_{K}^{m}}(T). We see at once that the constant term of g(T) is a unit in R and hence that g(T) ∈ R[[T]]^{×}. We see at once that the second and third conditions are equivalent. The first and third conditions are equivalent by lemma 1.10.2 of [KM]. □

Suppose that X is a locally noetherian formal scheme with ideal of definition I. We will let X_{n} denote the scheme with underlying topological space X and structure sheaf O_{X} /I^{n}. By a Drinfeld ℘_{K}^{m}-structure on a Barsotti-Tate O_{K}-module H/X we shall mean a compatible system of Drinfeld ℘_{K}^{m}- structures

α_{n} : (℘_{K}^{−m}/O_{K})^{h} ⟶ H[℘_{K}^{m}](X_{n})

for H×_{X} X_{n}. This is easily checked to be canonically independent of the choice of ideal of definition I.

If A is a noetherian ring complete with respect to the I-adic topology for some ideal I and if H/Spec A is a Barsotti-Tate O_{K}-module with Drinfeld ℘_{K}^{m}-structure α, then we obtain a a natural Drinfeld ℘_{K}^{m}-structure α on the corresponding Barsotti-Tate O_{K}-module H/Spf A (take the push forward of α on X_{n} for each n). This establishes a bijection from Drinfeld ℘_{K}^{m}-structures on H/Spec A to Drinfeld ℘_{K}^{m}-structures on H/Spf A. (Given a compatible system

α_{n} : (℘_{K}^{−m}/O_{K})^{h} ⟶ H[℘_{K}^{m}](A/I^{n})

using the completeness of A we get in the limit a homomorphism

α : (℘_{K}^{−m}/O_{K})^{h} ⟶ H[℘_{K}^{m}](A).

Using lemma 1.9.1 of [KM] we see that α is in fact a Drinfeld ℘_{K}^{m}-structure.)

We will next recall some results of Drinfeld about formal deformations of one-dimensional Barsotti-Tate O_{K}-modules with Drinfeld level structures. Consider first the case of a formal Barsotti-Tate O_{K}-module. More precisely consider the functor which associates to any local Artinian O_{K}-algebra A with residue field k(v)^{ac} the set of isomorphism classes of triples (H, j, α) where H/A is a Barsotti-Tate O_{K}-module, where j : Σ_{K,g} →^{∼} H ×_{Spec A} Spec k(v_{K})^{ac} and where α : (℘_{K}^{−m}/O_{K})^{g} ⟶ H[℘_{K}^{m}](Spec A) is a Drinfeld ℘_{K}^{m}-structure. Proposition 4.3 of [Dr] tells us that this functor is pro-represented by a regular finite flat local R_{K,g}-algebra, which we will denote R_{K,g,m}. We will denote by (Σ̃_{K,g}, j̃, α̃) the universal triple. Again by lemma 4.16 of chapter II of [Me] we see that (Σ̃_{K,g}, j̃) is actually defined over Spec R_{K,g,m}. In fact α̃ : (℘_{K}^{−m}/O_{K})^{g} → Σ̃_{K,g}[℘_{K}^{m}](Spec R_{K,g,m}) (as Σ_{K,g}[℘_{K}^{m}]/Spec R_{K,g,m} is finite) and by proposition 1.9.1 of [KM] α̃ is a Drinfeld ℘_{K}^{m}-structure over Spec R_{K,g,m}. Thus we get a map Spec R_{K,g,m} → (Spec R_{K,g})(m), which is an isomorphism after tensoring with any Artinian quotient of R_{K,g} and hence is an isomorphism. We also have that R_{K,g,m} is degree #GL_{g}(O_{K}/℘_{K}^{m}) over R_{K,g}. (To see this it suffices to look at the generic fibre R_{K,g} ⊗_{ℤ_{p}} ℚ_{p}, where the degree is easy to calculate because Σ̃_{K,g} becomes ind-etale and a Drinfeld ℘_{K}^{m}-structure is nothing but an isomorphism

α : (℘_{K}^{−m}/O_{K})_{S}^{h} →^{∼} H[℘_{K}^{m}].)

We next turn to deformations of H_{0} = Σ_{K,g} × (K/O_{K})^{h}/k(v_{K})^{ac}. Fix a surjection

δ : (℘_{K}^{−m}/O_{K})^{g+h} → → ℘_{K}^{−m}T H_{0}/T H_{0};

and let GL_{g+h}(O_{K}/℘_{K}^{m})_{δ} denote the group of x ∈ GL_{g+h}(O_{K}/℘_{K}^{m}) such that δ ∘ x = δ. Then there is a short exact sequence

(0) → Hom (℘_{K}^{−m}T H_{0}/T H_{0}, ker δ) → GL_{g+h}(O_{K}/℘_{K}^{m})_{δ} → Aut (ker δ) → (0).

Consider the functor which associates to any local Artinian O_{K}-algebra A with residue field k(v)^{ac} the set of isomorphism classes of triples (H, j, α) where H/A is a Barsotti-Tate O_{K}-module, where j : Σ_{K,g} →^{∼} H ×_{Spec A} Spec k(v_{K})^{ac} and where α : (℘_{K}^{−m}/O_{K})^{g+h} ⟶ H[℘_{K}^{m}](Spec A) is a Drinfeld ℘_{K}^{m}-structure such that the composite

(℘_{K}^{−m}/O_{K})^{g+h} ⟶^{α} H[℘_{K}^{m}](Spec A) ⟶ H[℘_{K}^{m}](Spec k(v_{K})^{ac}) ⟶^{j^{−1}}

⟶ H_{0}[℘_{K}^{m}](Spec k(v_{K})^{ac}) = ℘_{K}^{−m}T H_{0}/T H_{0}

equals δ. This functor is pro-representable by (H̃, j̃, α̃_{H})/Spf R_{δ}^{univ} (by proposition 4.5 of [Dr]).

To describe how it is pro-represented it is convenient to also fix a homomorphism γ : (℘_{K}^{−m}/O_{K})^{g+h} ⟶ (℘_{K}^{−m}/O_{K})^{g} such that

δ ⊕ γ : (℘_{K}^{−m}/O_{K})^{g+h} ⟶^{∼} (℘_{K}^{−m}T H_{0}/T H_{0}) ⊕ (℘_{K}^{−m}/O_{K})^{g}.

This first of all gives rise to a splitting Aut (ker δ) ↪ GL_{g+h}(O_{K}/℘_{K}^{m})_{δ}. Secondly it gives rise to an isomorphism of Spf R_{δ}^{univ} with

Hom (℘_{K}^{−m}T H_{0}, Σ̃_{K,g}) ×_{Spec R_{K,g}} Spec R_{K,g,m}.

Over this ring we have the pull back from Hom (T H_{0}, Σ̃_{K,g}) of the tautological extension

(0) ⟶ Σ̃_{K,g} ⟶ H̃ ⟶ T H_{0} ⊗ (K/O_{K}) ⟶ (0),

as well as a second extension

(0) ⟶ Σ̃_{K,g} ⟶ H̃′ ⟶ T H_{0} ⊗ (K/℘_{K}^{−m}) ⟶ (0).

There is a natural isogeny H̃ ⟶ H̃′ whose kernel projects isomorphically to ℘_{K}^{−m}T H_{0}/T H_{0}, and so we get a splitting

i : ℘_{K}^{−m}T H_{0}/T H_{0} ↪ H̃.

Then

α̃_{H} = α̃ ∘ γ + i ∘ δ.

(See the (rather sketchy) proof of proposition 4.5 in [Dr].)

The deformation space Spf R_{δ}^{univ} has a right action of GL_{g+h}(O_{K}/℘_{K}^{m})_{δ} which may be described as follows. The splitting γ allows one to write the group GL_{g+h}(O_{K}/℘_{K}^{m})_{δ} as a semidirect product of Hom (℘_{K}^{−m}T H_{0}/T H_{0}, ker δ) and Aut (ker δ). It thus suffices to describe the action of both these groups on Spf R_{δ}^{univ}. Now Aut (ker δ) is isomorphic via γ to GL_{h}(O_{K}/℘_{K}^{m}) and simply acts on the factor Spf R_{K,g,m}. On the other hand composition with α̃ gives a map

Hom (℘_{K}^{−m}T H_{0}/T H_{0}, ker δ) ⟶ Hom (℘_{K}^{−m}T H_{0}, Σ̃_{K,g}(R_{K,g,m}))
= Hom (℘_{K}^{−m}T H_{0}, Σ̃_{K,g})(R_{K,g.m}).

The action of an element ϕ ∈ Hom (℘_{K}^{−m}T H_{0}/T H_{0}, ker δ) on

Hom (℘_{K}^{−m}T H_{0}, Σ̃_{K,g}) ×_{Spec R_{K,g}} Spec R_{K,g,m}

is simply by translation by the image of ϕ in Hom (℘_{K}^{−m}T H_{0}, Σ̃_{K,g})(R_{K,g.m}).

If we write Hom (T H_{0}, Σ̃_{K,g}) = Spf R then H̃ is defined over Spec R. Moreover as before we can identify (Spec R)(m) with

∐_{δ} Spec R_{δ}^{univ},

where the disjoint union is over surjections

δ : (℘_{K}^{−m}/O_{K})^{g+h} → → ℘_{K}^{−m}T H_{0}/T H_{0}.

If x ∈ GL_{g+h}(O_{K}/℘_{K}^{m}) then x takes R_{δ}^{univ} isomorphically to R_{δ∘x}^{univ}. If x ∈ GL_{g+h}(O_{K}/℘_{K}^{m})_{δ} then the two actions of x on Spec R_{δ}^{univ} coincide. In particular we see that (Spec R)(m) is regular and is finite and flat over Spec R of degree #GL_{g+h}(O_{K}/℘_{K}^{m}).

We now record a few more basic facts about Drinfeld level structures which can be proved by reduction to the universal case.

Lemma 3.2 In this lemma S will denote an O_{K}-scheme which we will assume is locally noetherian with a dense set of points with residue field algebraic over k(v). Also H/S will be a one-dimensional Barsotti-Tate O_{K}-module of constant height h.

1. S(m)/S is finite, flat of degree #GL_{h}(O_{K}/℘_{K}^{m}).

2. Suppose that

α : (℘_{K}^{−m}/O_{K})^{h} ⟶ H[℘_{K}^{m}](S)

is a Drinfeld ℘_{K}^{m}-structure. Suppose also that M ⊂ (℘_{K}^{−m}/O_{K})^{h} is a O_{K}-submodule. Then there is a unique O_{K}-invariant finite flat subgroup scheme N ⊂ H[℘_{K}^{m}] such that the set of α(x) for x ∈ M form a full set of sections for N/S. If moreover

δ : (℘_{K}^{−m′}/O_{K}^{h}) ↪ (℘_{K}^{−m}/O_{K})^{h}/M

is a map of O_{K}-modules, then α ∘ δ is a Drinfeld ℘_{K}^{m′} structure for H/N.

The construction of N is compatible with base change in the following sense. If T/S is a locally noetherian S-scheme with a dense set of points with residue field algebraic over k(v) and if

α_{T} : (℘_{K}^{−m}/O_{K})^{h} ⟶^{α} H[℘_{K}^{m}](S) ⟶ (H ×_{S} T)[℘_{K}^{m}](T),

then the set of α_{T} (x) for x ∈ M is a full set of sections for N ×_{S} T.

Proof: The first part is proved by a straightforward reduction to the universal formal case. We will prove the second part only, as the argument in this case is slightly more difficult. The last paragraph of the second part follows from lemma 1.9.1 of [KM]. Thus we concentrate on the proof of the first paragraph of part 2.

By corollary 1.30.3 of [KM] there is a unique closed subscheme N ⊂ H[℘_{K}^{m}] which is locally free over S and for which the set of x ∈ M form a full set of sections. From the uniqueness it follows that N is invariant by the action of O_{K}^{×}. Thus it suffices to check that

1. N is a subgroup scheme;

2. if

δ : (℘_{K}^{−m′}/O_{K}^{h}) ↪ (℘_{K}^{−m}/O_{K})^{h}/M

is a map of O_{K}-modules, then α ∘ δ is a Drinfeld ℘_{K}^{m′} structure for H/N.

There is a closed subscheme S′ ⊂ S such that for any scheme T/S, N ×_{S} T ⊂ H ×_{S} T has the two properties above if and only if T → S factors through S′. (H[℘_{K}^{m}] is the spectrum of a sheaf of locally free Hopf algebras H/S. Let I denote the subsheaf of ideals defining N. The first property above is equivalent to the composite map

I ↪ H ⟶ H ⊗ H ⟶ H/I ⊗ H/I,

where the middle map is the comultiplication, being zero. It follows that there exists a closed S′′ ⊂ S universal for the truth of the first property. The existence of S′ ⊂ S′′ follows from proposition 1.9.1 of [KM].) What we must show is that S′ = S.

We may at once reduce to the case that S = Spec A for an Artinian local ring A. (Look at O_{S,s}/𝔪_{s}^{a} as s runs over a dense set of points and a runs over positive integers.) Then by tensoring with W (k(v_{K})^{ac}) we may assume that the residue field of A is k(v_{K})^{ac}. Next we may replace H/A by the universal deformation of H × Spec k(v_{K})^{ac} together with its Drinfeld ℘_{K}^{m}-structure, and so we may suppose that S = Spec R for a complete noetherian local ring which is flat over OK̂^{nr}. Then we may replace S by S × Spec ℚ_{p}, which is dense in S. But H/S × Spec ℚ_{p} is ind-etale and the result is easy. □

The following corollary follows readily.

Corollary 3.3 In this lemma X will denote a locally noetherian O_{K}-formal scheme. We will assume that X^{red} has a dense set of points with residue field algebraic over k(v). Also H/X will denote a one-dimensional Barsotti-Tate O_{K}-module of constant height h.

1. There is a formal scheme X (m)/X and a Drinfeld ℘_{K}^{m}-level structure α^{univ} on H ×_{X} X (m) which is universal in the following sense. If Y → X is any morphism of formal schemes and if δ is a Drinfeld ℘_{K}^{m}-level structure on H ×_{X} Y then there is a unique morphism Y → X (m) over X under which α^{univ} pulls back to δ. Moreover X (m)/X finite, flat of degree #GL_{h}(O_{K}/℘_{K}^{m}).

2. Suppose that α is a Drinfeld ℘_{K}^{m}-structure on H/X. Suppose also that M ⊂ (℘_{K}^{−m}/O_{K})^{h} is a O_{K}-submodule. Then we can find a Barsotti-Tate O_{K}-module H/α(M) over X and a morphism H → H/α(M) over X such that when restricted to any closed subscheme X ⊂ X the morphism

H|_{X} ⟶ (H/α(M))|_{X}

is surjective and the set of α|_{X}(x) for x ∈ M form a complete set of sections for the kernel. The construction of H/α(M) is compatible with base change in the following sense. If Y/X is a locally noetherian X -formal scheme such that Y^{red} has a dense set of points with residue field algebraic over k(v) and if H_{Y} (resp. α_{Y}) denotes the pull back of H (resp. α) to Y, then H_{Y}/α_{Y}(M) is canonically the pull back of H/α(M) to Y.

We have seen that R_{K,g} has a natural continuous left action of O_{D_{K,g}}^{×}. The same is true of R_{K,g,m} and so in fact R_{K,g,m} has a continuous left action of

GL_{g}(O_{K}) × O_{D_{K,g}}^{×} → → GL_{g}(O_{K}/℘_{K}^{m}) × O_{D_{K,g}}^{×}.

In fact we can extend this action to a continuous left action of GL_{g}(K) × D_{K,g}^{×} on the direct system of the R_{K,g,m} such that

R_{K,g,m_{1}}  ⟶^{(γ,δ)}  R_{K,g,m_{2}}
↑  ↑

W (k(v_{K})^{ac}) Frob_{vK}^{vK (det γ)−vK (det δ))} ⟶ W (k(v_{K})^{ac})

commutes if m_{2} >> m_{1}.

To describe this first suppose that (γ, δ) ∈ GL_{g}(K) × D_{K,g}^{×}, that γ^{−1} ∈ M_{g}(O_{K}), that v_{K}(det(δ)) ≤ 0 and that γO_{K}^{g} ⊂ ℘_{K}^{m_{1}−m_{2}}O_{K}^{g}. It suffices to define a W (k(v_{K})^{ac})-linear map

(γ, δ) : R_{K,g,m_{1}} ⟶ R_{K,g,m_{2}} ˆ⊗_{W (k(v)^{ac}),Frob_{v_{K}}^{v_{K} (det δ)−v_{K} (det γ)}}W (k(v)^{ac}).

Next, by the universal property of R_{K,g,m_{1}} it suffices to give a deformation of (Σ_{K,g}, j, α^{triv}) to R_{K,g,m_{2}} ˆ⊗_{W (k(v)^{ac}),Frob_{v_{K}}^{v_{K} (det δ)−v_{K} (det γ)}}W (k(v)^{ac}). By lemma 3.2

there is a unique finite flat subgroup scheme A ⊂ Σ̃_{K,g} over R_{K,g,m_{1}} such that the set of α^{univ}(x) for x ∈ (γO_{K}^{g})/O_{K}^{g} are a complete set of sections for A. For our deformation of Σ_{K,g} we will take

(Σ̃_{K,g}/A) ×_{Spf W (k(v_{K})^{ac}),(Frob_{v_{K}}^{∗})^{v_{K} (det δ)−v_{K} (det γ)}} Spf W (k(v_{K})^{ac}).

It has a Drinfeld ℘_{K}^{m_{1}}-structure coming from

(℘_{K}^{−m_{1}}/O_{K})^{g} ↪^{γ} ℘_{K}^{−m_{2}}O_{K}^{g}/(γO_{K}^{g}) ⟶^{α^{univ}} (Σ̃_{K,g}/A)(R_{K,g,m_{2}}).

(Use lemma 3.2.) Reducing modulo the maximal ideal of

R_{K,g,m_{2}} ˆ⊗_{W (k(v)^{ac}),Frob_{v_{K}}^{v_{K} (det δ)−v_{K} (det γ)}}W (k(v)^{ac})

and using j̃ we obtain

Σ_{K,g}^{(p^{−f_{K} v_{K} (det γ)})} ×_{Spec k(v_{K})^{ac},(Fr^{∗})^{f_{K} (v_{K} (det γ)−v_{K} (det δ))}} Spec k(v_{K})^{ac} ≅ Σ_{K,g}^{(p^{−f_{K} v_{K} (det δ)})}.

Finally we identify this with Σ_{K,g} via

δ^{−1} : Σ_{K,g}^{(p^{−v_{K} (det δ)})} →^{∼} Σ_{K,g}.

We note that if x ∈ O_{K} and x ≠ 0 then the element (x^{−1}, x^{−1}) acts trivially. Thus we obtain an action of

GL_{g}(K) × D_{K,g}^{×} → → (GL_{g}(K) × D_{K,g}^{×})/K^{×},

where K^{×} is embedded diagonally.

Lemma 3.4 Let us fix K, g and m. For each positive integer s we can find an integer N(s) such that

• N(s) increases monotonically to infinity as s → ∞;

• any element of ϖ_{K}^{s−m}O_{D_{K,g}} lifts to an endomorphism of

Σ̃_{K,g} × (R_{K,g,m}/𝔪_{R_{K,g,m}}^{N(s)})

• and (1 + ϖ_{K}^{s}O_{D_{K,g}}) acts trivially on R_{K,g,m}/𝔪_{R_{K,g,m}}^{N(s)}.

Proof: Note that if we can choose N(s) satisfying the first two conditions, then the third will also be satisfied. (If δ ∈ O_{D_{g}}^{×} with δ ≡ 1 mod p^{s} then (δ−1)/ϖ_{K}^{m} and (δ^{−1}−1)/ϖ_{K}^{m} lift to endomorphisms of Σ̃_{K,g}×(R_{K,g,m}/𝔪_{R_{K,g,m}}^{N(s)}). By the uniqueness of such lifts (see part 2 of lemma 1.1.3 of [Kat]) we see that δ lifts to an automorphism of Σ̃_{K,g} × (R_{K,g,m}/𝔪_{R_{K,g,m}}^{N(s)}) which is the identity on ϖ_{K}^{m}-torsion.)

Now take N(s) to be the integer part of \sqrt{(s/e_{K/ℚ_{p}} − m)} (or 0 if this is not defined). As p^{N(s)} is zero on (R_{K,g,m}/𝔪_{R_{K,g,m}}^{N(s)}) the second condition on N(s) follows from part 3 of lemma 1.1.3 of [Kat]. □

The following lemma will be proved in section 5.

Lemma 3.5 We can find an inverse system of proper schemes of finite type X_{m}/OK̂^{nr} with compatible actions of GL_{g}(O_{K}) and a closed point x ∈ X_{m} such that

1. ker(GL_{g}(O_{K}) → GL_{g}(O_{K}/℘^{m})) acts trivially on X_{m},

2. the generic fibre of X_{m}/X_{0} is finite, etale and Galois with Galois group ker(GL_{g}(O_{K}) → GL_{g}(O_{K}/℘^{m})),

3. x is totally ramified in each X_{m},

4. and the inverse system of formal schemes (X_{m})_{x}^{∧} with action of GL_{g}(O_{K}) is isomorphic to the inverse system of the Spf R_{K,g,n}.

Suppose that l is a prime not equal to p. We will let Ψ_{K,l,g,m}^{i} denote the i^{th} vanishing cycle sheaf with coefficients in ℚ_{l}^{ac} for Spf R_{K,g,m} in the sense of Berkovich (see appendix II). Thus Ψ_{K,l,g,m}^{i} is a finite dimensional ℚ_{l}^{ac}-vector space equal to ℚ_{l}^{ac} ⊗_{ℤ_{l}} lim_{→s} R^{i}Ψ_{η}(ℤ/l^{s}ℤ). We set Ψ_{K,l,g}^{i} = lim_{→m} Ψ_{K,l,g,m}^{i} and we may drop the subscripts K and/or l when no confusion can arise. Let A_{K,g} denote the set of (γ, δ, σ) ∈ GL_{g}(K) × D_{K,g}^{×} × W_{K} such that

v_{K}(γ) = v_{K}(δ) + v_{K}(σ).

The action of GL_{g}(K) × D_{K,g}^{×} on the tower of the R_{K,g,m} gives rise to an action of A_{K,g} on the tower R_{K,g,m} ⊗_{OK̂^{nr}} L for any finite extension L/K̂^{nr}. More precisely (γ, δ, σ) ∈ A_{K,g} acts as (γ, δ) ⊗ σ. In this way we get a left action of A_{K,g} on Ψ_{K,l,g}^{i}.

Lemma 3.6 The action of A_{K,g} on Ψ_{K,l,g}^{i} is admissible/continuous.

Proof: The action of GL_{g}(O_{K}) is smooth from the definitions. It follows from lemma 3.4, corollary 4.5 of [Berk3] and lemma 3.5 that the action of O_{D_{K,g}}^{×} is smooth. Let X_{m} denote the kernel of GL_{g}(O_{K}) → GL_{g}(O_{K}/℘_{K}^{m}). It follows from lemmas 3.5 and II.3 that Ψ_{K,l,g,m}^{i} →^{∼} (Ψ_{K,l,g}^{i})^{X_{m}}. Finally it follows from lemma 3.5, the comparison theorem of [Berk3] and lemma II.1 that Ψ_{K,l,g,m}^{i} is finite dimensional and has a continuous action of I_{K}. □

If ρ is a irreducible admissible representation of D_{K,g}^{×} over ℚ_{l}^{ac} (and hence necessarily finite dimensional) then we set

Ψ_{K,l,g}^{i}(ρ) = Hom_{O_{D_{K,g}}^{×}} (ρ, Ψ_{K,l,g}^{i}).

This is naturally an admissible GL_{g}(K) × W_{K}-module. More precisely

((γ, σ)ϕ)(x) = (γ, δ, σ)ϕ(ρ(δ)^{−1}x),

for any δ ∈ D_{K,g}^{×} with v_{K}(det δ) = v_{K}(γ) − v_{K}(σ). Define a homomorphism d : GL_{g}(K) × W_{K} → ℤ by d_{g}(γ, σ) = v_{K}(det γ) − v_{K}(σ). The following lemma is immediate.

Lemma 3.7 If ψ : ℤ → (ℚ_{l}^{ac})^{×} then

Ψ_{K,l,g}^{i}(ρ ⊗ (ψ ∘ v_{K} ∘ det)) ≅ Ψ_{K,l,g}^{i}(ρ) ⊗ (ψ^{−1} ∘ d_{g}).

There is a natural map of A_{K,g}-modules

Ψ_{K,l,g}^{i}(ρ) ⊗ ρ ⟶ Ψ_{K,l,g}^{i},

which sends f ⊗ v to f(v). We will denote the image of this map Ψ_{K,l,g}^{i}[ρ] and we will let Ψ_{K,l,g,t}^{i}[ρ] denote the preimage of Ψ_{K,l,g}^{i}[ρ] in Ψ_{K,l,g,t}^{i}. We will call irreducible admissible representations of D_{g}^{×} inertially equivalent if they differ by twisting by a character of the form ψ ∘ v_{K} ∘ det for some character ψ : ℤ → (ℚ_{l}^{ac})^{×}. The submodule Ψ_{K,l,g}^{i}[ρ] only depends on the inertial equivalence class of ρ. Because O_{D_{K,g}}^{×} is compact it is easy to check that

Ψ_{K,l,g}^{i} ≅ ⨁ ρ ⊗ Ψ_{K,l,g}^{i}[ρ]

where the sum is over one representative of each inertial equivalence class of irreducible admissible representation of D_{K,g}^{×}.

We will let e[ρ] denote the number of irreducible components of ρ|_{O_{D_{K,g}}^{×}}.

It also equals the number of characters ψ : ℤ ⟶ (ℚ_{l}^{ac})^{×} such that ρ ≅ ρ ⊗ (ψ ∘ v_{K} ∘ det). Let Δ[ρ] be a set of e[ρ] elements δ ∈ D_{K,g}^{×} such that the set of v_{K}(det δ) for δ ∈ Δ[ρ] run over a set of representatives of the congruence classes mode[ρ]. If δ ∈ D_{K,g}^{×} we will we will let

Ψ_{K,l,g}^{i}[ρ]^{δ}

denote Ψ_{K,l,g}^{i}[ρ] but with its A_{K,g} action twisted so that (γ, ϵ, σ) acts via (γ, δ^{−1}ϵδ, σ). Then there is an isomorphism of A_{K,g}-modules

Ψ_{K,l,g}^{i}(ρ) ⟶^{∼} ⨁_{δ∈Δ[ρ]} Ψ_{K,l,g}^{i}[ρ]^{δ},

which sends f ⊗ v to (f(δ^{−1}v))_{δ}.

We also introduce the virtual GL_{g}(K) × W_{K}-module

[Ψ_{K,l,g}(ρ)] = ∑_{i=0}^{g−1}(−1)^{g−1−i}[Ψ_{K,l,g}^{i}(ρ)].

The following lemma is proved in [Car3].

Lemma 3.8 If ρ is a character of K^{×} then

Ψ_{K,l,1}(ρ) ≅ ρ^{−1} ⊗ ρ ∘ Art _{K}^{−1}.

4 Some simple Shimura varieties

In this section we shall introduce some Shimura varieties which will be the main object of study in this paper. This class (or one close to it) of Shimura varieties, which are particularly simple in a number of respects, were first singled by Kottwitz (see [Ko4]).

We will use without comment notation established in section 1. Let U be a sufficiently small open compact subgroup of G(𝔸^{∞}). By sufficiently small we shall mean that the projection of U to G(ℚ_{x}) for some prime x contains no element of finite order other than 1. Let S be a connected, locally noetherian F -scheme and s a geometric point of S. Consider the functor that to (S, s) associated the set of equivalence classes of quadruples (A, λ, i, \overline{η}) where

• A is an abelian scheme of dimension dn^{2};

• λ : A → A^{∨} is a polarisation;

• i : B ↪ End^{0}(A) such that λ ∘ i(b) = i(b^{∗})^{∨} ∘ λ for all b ∈ B and

tr (b|_{Lie(A)}) = (c ∘ tr_{F/E} ∘ tr_{B/F})(nb) + tr_{B/F} (b) − (c ∘ tr_{B/F})(b)

for all b ∈ B;

• \overline{η} is a π_{1}(S, s)-invariant U-orbit of isomorphisms of B ⊗_{ℚ} 𝔸^{∞}-modules η : V ⊗_{ℚ} 𝔸^{∞} → V A_{s} which take the standard pairing ( , ) on V to a (𝔸^{∞})^{×}-multiple of the λ-Weil pairing on V A_{s} (see [Ko3] pages 390-391).

We consider two quadruples (A, λ, i, \overline{η}) and (A′, λ′, i′, \overline{η}′) equivalent if there is an isogeny α : A → A′ which takes λ to a ℚ^{×} multiple of λ′, takes i to i′ and takes \overline{η} to \overline{η}′ (see [Ko3] page 390). If s′ is a second geometric point of S then there is a canonical bijection between the image of (S, s) and of (S, s′). Thus we obtain a functor from connected, locally noetherian schemes S/F to sets (see [Ko3] page 391).

Because U is sufficiently small this functor is represented by a smooth projective scheme X_{U}/F (see [Ko3] page 391). If V ⊂ U there is a natural finite etale map X_{V} → → X_{U}. There is also a natural right action of G(𝔸^{∞}) on the inverse system of the X_{U}: if g^{−1}V g ⊂ U then g : X_{V} → X_{U} by (A, λ, i, \overline{η}) ↦ (A, λ, i, \overline{η ∘ g}). If U and V are sufficiently small open compact subgroups of G(𝔸^{∞}) and if V is a normal subgroup of U then the finite etale cover X_{V} → X_{U} is Galois with group U/V. Thus if x is a geometric point of X_{U} we obtain a continuous homomorphism

π_{1}^{alg}(X_{U}, x) ⟶ U.

(This map is only determined up to conjugation unless one chooses a compatible choice of liftings of x to all the covers X_{V}.)

If ξ is a finite dimensional irreducible representation of G on a ℚ_{l}^{ac}-vector space W_{ξ} then we obtain a representation

ξ : π_{1}(X_{U}, x) ⟶ U ⟶ G(ℚ_{l}) ⟶^{ξ} Aut (W_{ξ}).

Thus we also obtain a lisse etale ℚ_{l}^{ac}-sheaf L_{ξ}/X_{U}. Although the construction of L_{ξ} appears to depend on a choice of base points this is illusory (as we will explain below). If g ∈ G(𝔸^{∞}) and U, V are sufficiently small open compact subgroups of G(𝔸^{∞}) with g^{−1}V g ⊂ U then we have the morphism

g : X_{V} ⟶ X_{U}

and ξ(g_{l}) also gives rise to a morphism of sheaves

g : g^{∗}L_{ξ} ⟶ L_{ξ}.

(We will explain the construction of this map g and why the choice of base point does not effect the construction of L_{ξ} by looking at the case of a locally constant etale sheaf of (ℤ/l^{r}ℤ)-modules. We will leave to the reader the standard and rather tiresome extension to ℚ_{l}^{ac}-sheaves. Thus suppose that a finite group G acts freely on a scheme Y (on the right). Suppose also that H_{1} and H_{2} are subgroups of G and that g ∈ G with gH_{1}g^{−1} ⊂ H_{2}. Let X_{i} = Y/H_{i} so that we get a morphism g : X_{1} → X_{2}. Suppose that V is a finite free ℤ/l^{r}ℤ-module and that ρ : G → Aut (V) is a homomorphism. As above we obtain a sheaf L_{ρ} on each X_{i}. Let us describe this sheaf without reference to base points. If U → X_{i} is a finite etale cover of a Zariski open subset of X_{i} then L_{ρ}(U) is the set of functions

f : π_{0}(Y ×_{X_{i}} U) ⟶ V

such that for all σ ∈ H_{i} and C ∈ π_{0}(Y ×_{X_{i}} U) we have

f(Cσ) = ρ(σ)^{−1}f(C).

To define a map g^{∗} : L_{ρ} → L_{ρ} over X_{1} it suffices to give compatible maps

g : L_{ρ}(U_{2}) ⟶ L_{ρ}(U_{1}),

whenever we have a commutative diagram

\begin{matrix}U_{1} & ⟶ & U_{2} \\ ↓ & & ↓ \\ X_{1} & ⟶^{g} & X_{2} \end{matrix}

with the vertical maps being finite etale covers of Zariski opens. If f : π_{0}(Y ×_{X_{2}} U_{2}) → V then we let g(f) : π_{0}(Y ×_{X_{1}} U_{1}) → V be the function defined by

g(f)(C) = ρ(g)f(Cg).)

We will set

H^{i}(X, L_{ξ}) = lim_{→U} H_{et}^{i}(X_{U} ×_{F} F^{ac}, L_{ξ}).

If g ∈ G(𝔸^{∞}) and U, V are sufficiently small open compact subgroups of G(𝔸^{∞}) with g^{−1}V g ⊂ U then we get a morphism

g : H_{et}^{i}(X_{U} ×_{F} F^{ac}, L_{ξ}) ⟶ H_{et}^{i}(X_{V} ×_{F} F^{ac}, L_{ξ}).

If V ⊂ U then we see that

H_{et}^{i}(X_{U} ×_{F} F^{ac}, L_{ξ}) ≅ H_{et}^{i}(X_{V} ×_{F} F^{ac}, L_{ξ})^{U/V} .

Thus H^{i}(X, L_{ξ}) becomes an admissible G(𝔸^{∞})-module, in fact an admissible/continuous G(𝔸^{∞}) × Gal (F^{ac}/F)-module. We will let [H(X, L_{ξ})] denote the virtual G(𝔸^{∞}) × Gal (F^{ac}/F)-module

∑_{i}(−1)^{n−1−i}[H^{i}(X, L_{ξ})]

(see appendix I).

Our distinguished embedding τ_{0} : F ↪ ℂ allows us to speak of the complex points of X_{U}, which we will denote X_{U}(ℂ), a smooth manifold. We get an isomorphism

H_{et}^{i}(X_{U} ×_{F} F^{ac}, L_{ξ}) ⟶^{∼} H^{i}(X_{U}(ℂ), L_{ξ}).

There is an element I ∈ B⊗_{ℚ}ℝ (unique if n > 2) with the following properties.

• I^{2} = −1.

• βI^{∗} = −Iβ.

• The pairing b_{1} × b_{2} ↦ (b_{1}, b_{2}I) is a positive definite symmetric form on V ⊗_{ℚ} ℝ.

• For any embedding σ : F^{+} ↪ ℝ the space V ⊗_{F^{+},σ} ℝ is an (E ⊗_{ℚ} ℝ)[I]/(I^{2})-module. Our choice of τ_{0} gives rise to an isomorphism (E ⊗_{ℚ} ℝ)[I]/(I^{2}) ≅ ℂ[I]/(I^{2}). If σ ≠ τ_{0}|_{F^{+}} we require that I = −i on V ⊗_{F^{+},σ}ℝ. On the other hand we require that the i eigenspace of I on V ⊗_{F^{+},τ_{0}} ℝ has ℂ-dimension n.

We will let U_{∞} denote the centraliser of I in G(ℝ). It is a maximal connected compact mod centre subgroup of G(ℝ). We will let Ũ_{∞} denote a maximal compact mod centre subgroup of G(ℝ) containing U_{∞}. As on page 400 of [Ko3] we can give a more explicit description of the smooth manifold X_{U}(ℂ). It is the disjoint union of

# ker^{1}(ℚ, G) = # ker((F^{+})^{×}/ℚ^{×}N_{F/F^{+}}(F^{×}) ⟶ 𝔸_{F^{+}}^{×}/𝔸^{×}N_{F^{+}/F} (𝔸_{F}^{×}))

copies of the manifold

G(ℚ)\(G(𝔸^{∞})/U × G(ℝ)/U_{∞}).

Under this identification the right action of G(𝔸^{∞}) on inverse system of the X_{U}’s corresponds to the action by right translation on the

G(ℚ)\(G(𝔸^{∞})/U × G(ℝ)/U_{∞}).

Using in addition our identification ı : ℚ_{l}^{ac} →^{∼} ℂ we see from Matsushima’s formula that we get an isomorphism

H^{i}(X, L_{ξ}) ≅ ⨁_{π} π^{∞} ⊗ H^{i}(Lie G(ℝ), U_{∞}, π_{∞} ⊗ ξ),

where π runs over irreducible constituents of the space of automorphic forms for G(𝔸), each taken with its multiplicity in the space of automorphic forms. (See [Ko4] page 655.) Thus we obtain the following lemma.

Lemma 4.1 As virtual G(𝔸^{∞})-modules we have an equality

[H(X, L_{ξ})] =
= # ker^{1}(ℚ, G) ∑_{π}[π^{∞}] ∑_{i}(−1)^{n−1−i} dim H^{i}(Lie G(ℝ), U_{∞}, π_{∞} ⊗ ξ),

where π runs over irreducible constituents of the space of automorphic forms for G(𝔸), each taken with its multiplicity in the space of automorphic forms.

Let us now give a slightly modified definition of X_{U}. For this purpose suppose that U = U^{p} × U_{p,0} × ∏_{i=1}^{r} U_{w_{i}}, where U^{p} ⊂ G(𝔸^{∞,p}), U_{p,0} ⊂ ℚ_{p}^{×} and for i ≥ 1, U_{w_{i}} ⊂ (B_{w_{i}}^{op})^{×}. We give an equivalent moduli problem represented by X_{U}/F_{w}. We consider the functor which takes a connected locally noetherian F_{w}-scheme and a geometric point s to equivalence classes of (r + 5)-tuples (A, λ, i, \overline{η}^{p}, \overline{η}_{p,0}, \overline{η}_{w_{i}})/S where

• A/S is an abelian scheme of dimension dn^{2};

• λ : A ⟶ A^{∨} is a polarisation; • i : B ↪ End (A) ⊗_{ℤ} ℚ such that λ ∘ i(b) = i(b^{∗})^{∨} ∘ λ for all b ∈ B, for

i > 1 we have Lie (A) ⊗_{F_{p}} F_{w_{i}} = (0) and Lie (A) ⊗_{F_{p}} F_{w_{i}^{c}} is a projective sheaf of rank [F_{v_{i}}^{+} : ℚ_{p}]n^{2}, Lie (A) ⊗_{F_{p}} F_{w} is a projective sheaf of rank n, and Lie (A) ⊗_{F_{p}} F_{w^{c}} is a projective sheaf of rank ([F_{w} : ℚ_{p}]n − 1)n;

• \overline{η}^{p} is a π_{1}(S, s)-invariant U^{p}-orbit of isomorphisms of B ⊗_{ℚ}𝔸^{∞,p}-modules η^{p} : V ⊗_{ℚ} 𝔸^{∞,p} → V^{p}A_{s} which take the standard pairing ( , ) on V ⊗_{ℚ} 𝔸^{∞,p} to a (𝔸^{∞,p})^{×} multiple of the λ-Weil pairing on V^{p}A_{s};

• \overline{η}_{p,0} denotes a π_{1}(S, s)-invariant U_{p}^{0}-orbit of isomorphisms ℚ_{p} →^{∼} ℚ_{p}(1);

• \overline{η}_{w_{1}} is a π_{1}(S, s)-invariant U_{w_{1}}-orbit of isomorphisms of F_{w}-modules \overline{η}_{w_{1}} :

Λ_{11} ⊗_{ℤ_{p}} ℚ_{p} →^{∼} εV_{w_{1}}A_{s};

• for i > 1, \overline{η}_{w_{i}} is a π_{1}(S, s)-invariant U_{w_{1}}-orbit of isomorphisms of B_{w_{i}}-

modules \overline{η}_{w_{i}} : Λ_{i} ⊗_{ℤ_{p}} ℚ_{p} →^{∼} V_{w_{i}}A_{s}.

We call two (r + 5)-tuples (A, λ, i, \overline{η}^{p}, \overline{η}_{p,0}, \overline{η}_{w_{i}}) and (A′, λ′, i′, (\overline{η}^{p})′, \overline{η}′_{p,0}, \overline{η}′_{w_{i}}) equivalent if there exists an isogeny α : A → A′ and γ ∈ ℚ^{×} such that α carries λ to γλ′, i to i′, \overline{η}^{p} to (\overline{η}^{p})′, and \overline{η}_{w_{i}} to \overline{η}′_{w_{i}}; and such that \overline{η}_{p,0} = γ\overline{η}′_{p,0}. The image of this functor is canonically independent of the base point s so we can think of it as a functor on connected locally noetherian F_{w}-schemes. Then this functor is also represented by X_{U} ×_{F} F_{w}. The isomorphism between these two moduli problems is given by mapping (A, λ, i, \overline{η}^{p}, \overline{η}_{p,0}, \overline{η}_{w_{i}}) to (A, λ, i, \overline{η}′) where

η′ = η^{p} × (((Id_{O_{F,w}^{n}} ⊗ η_{w_{1}}) ⊕ ⨁_{i>1} η_{w_{i}}) ⊕ ((Id_{O_{F,w}^{n}} ⊗ η_{w_{1}}) ⊕ ⨁_{i>1} η_{w_{i}})^{∨_{η_{p,0}}}).

The dual ∨_{η_{p,0}} is taken with respect to the canonical pairing on Λ ⊗_{ℤ_{p}} ℚ_{p} and η_{p,0}^{−1} composed with the λ-Weil pairing on V_{p}A. The action of G(ℚ_{p}) in this picture may be described as follows. An element (g_{0}, g_{1}, ..., g_{r}) ∈ ℚ_{p}^{×} × ∏_{i=1}^{r}(B_{w_{i}}^{op})^{×} maps (A, λ, i, \overline{η}^{p}, \overline{η}_{p,0}, \overline{η}_{w_{i}}) to (A, λ, i, \overline{η}^{p}, g_{0}\overline{η}_{p,0}, \overline{η}_{w_{i}} ∘ g_{i}).

5 Integral models

Next we wish to describe an integral model for X_{U} over O_{F,w}. More precisely suppose that U^{p} ⊂ G(𝔸^{∞,p}) is a sufficiently small open compact subgroup and suppose that m = (m_{1}, ..., m_{r}) ∈ ℤ_{≥0}^{r}. Then we will let U^{p}(m) ⊂ G(𝔸^{∞}) denote the product

U^{p} × ℤ_{p}^{×} × ∏_{i=1}^{r} ker((O_{B_{w_{i}}}^{op})^{×} ⟶ (O_{B_{w_{i}}}^{op}/w_{i}^{m_{i}})^{×}).

For simplicity we will restrict attention to open compact subgroups of this form. Given U^{p} and m as above we shall consider the functor from locally noetherian connected O_{F,w}-schemes S with a geometric point s to sets which takes (S, s) to the set of equivalence classes of (r + 4)-tuples (A, λ, i, \overline{η}^{p}, α_{i}) where

• A/S is an abelian scheme of dimension dn^{2};

• λ : A ⟶ A^{∨} is a prime to p polarisation;

• i : O_{B} ↪ End (A) ⊗_{ℤ} ℤ_{(p)} such that λ ∘ i(b) = i(b^{∗})^{∨} ∘ λ for all b ∈ O_{B}, for i > 1 we have Lie (A) ⊗_{O_{F,p}} O_{F,w_{i}} = (0) and Lie (A) ⊗_{O_{F,p}} O_{F,w_{i}^{c}} is a projective sheaf of rank [F_{v_{i}}^{+} : ℚ_{p}]n^{2}, Lie (A) ⊗_{O_{F,p}} O_{F,w} is a projective sheaf of rank n, and Lie (A) ⊗_{O_{F,p}} O_{F,w^{c}} is a projective sheaf of rank ([F_{w} : ℚ_{p}]n − 1)n;

• \overline{η}^{p} is a π_{1}(S, s)-invariant U^{p}-orbit of isomorphisms of B ⊗_{ℚ}𝔸^{∞,p}-modules η^{p} : V ⊗_{ℚ} 𝔸^{∞,p} → V^{p}A_{s} which take the standard pairing ( , ) on V ⊗_{ℚ} 𝔸^{∞,p} to a (𝔸^{∞,p})^{×} multiple of the λ-Weil pairing on V^{p}A_{s};

• α_{1} : w_{1}^{−m_{1}}Λ_{11}/Λ_{11} → εA[w_{1}^{m_{1}}](S) is a Drinfeld w_{1}^{m_{1}}-structure;

• for i > 1, α_{i} : (w_{i}^{−m_{i}}Λ_{i}/Λ_{i})_{S} →^{∼} A[w_{i}^{m_{i}}] is an isomorphism of S-schemes with O_{B}-actions.

Two (r+4)-tuples (A, λ, i, \overline{η}^{p}, α_{i}) and (A′, λ′, i′, (\overline{η}^{p})′, α′_{i}) are equivalent if there exists a prime to p isogeny δ : A → A′ and γ ∈ ℤ_{(p)}^{×} such that δ carries λ to γλ′, i to i′, \overline{η}^{p} to (\overline{η}^{p})′, and α_{i} to α′_{i}. Again this functor is canonically independent of the base point s so we can think of it as a functor from connected locally noetherian O_{F,w}-schemes to sets. On connected locally noetherian F_{w}-schemes it is naturally isomorphic to the functor defined in the last section. (Note that because we are now assuming that ℤ_{p}^{×} ⊂ U, we no longer require an analogue of \overline{η}_{p,0}.)

If m_{1} = 0 then it is known that this functor is represented by a projective scheme X_{U^{p},m}/O_{F,w}. (Representability and quasi-projectivity follow as on page 391 of [Ko3] or as in section 5.3 of [Car1]. Properness follows from the valuative criterion as in section 5.5 of [Car1], the point being that if A is an abelian variety of dimension dn^{2} with an action of an order in B over the field of fractions of a DVR and if A has semistable reduction then A has good reduction (otherwise the toric part of the reduction has too small a dimension to have an action of an order in B). The level structure then extends uniquely to the Neron model Ã of A, because Ã[𝔫] is etale over the DVR for 𝔫 supported on w_{2}, ..., w_{r} and the primes not dividing p (use the fact that Lie A[w_{i}]^{∞} = (0) for i > 1).) Hence by 3.1, this functor is represented for all m by a projective scheme X_{U^{p},m}/O_{F,w}.

The inverse system of the X_{U^{p},m}/O_{F,w} again has an action of G(𝔸^{∞}). The action of g ∈ G(𝔸^{∞,p}) just sends (A, λ, i, \overline{η}^{p}, α_{i}) to (A, λ, i, \overline{η}^{p} ∘ g, α_{i}). The action of (g_{0}, g_{1}, ..., g_{r}) ∈ G(ℚ_{p}) is slightly trickier to describe. To do so let us suppose that for each i ≥ 1 we have the following integrality conditions

• g_{i}^{−1} ∈ O_{B,w_{i}}^{op},

• g_{0}^{−1}g_{i} ∈ O_{B,w_{i}}^{op},

• w_{i}^{m_{i}−m′_{i}}g_{i} ∈ O_{B,w_{i}}^{op}.

Under these assumptions we will define a morphism

(g_{i}) : X_{U^{p},m} ⟶ X_{U^{p},m′}.

It will send (A, λ, i, \overline{η}^{p}, α_{i}) to (A/(C ⊕ C^{⊥}), p^{val_{p}(g_{0})}λ, i, \overline{η}^{p}, α_{i} ∘ g_{i}), where

• C_{1} ⊂ εA[w_{1}^{m_{1}}] is the unique closed subscheme for which the set of α_{1}(x) with x ∈ g_{1}Λ_{11}/Λ_{11} is a complete set of sections;

• for i > 1, C_{i} = α_{i}(g_{i}Λ_{i}/Λ_{i});

• C = (O_{F,w}^{n} ⊗_{O_{F,w}} C_{1}) ⊕ ⨁_{i=2}^{r} C_{i} ⊂ A[u^{−val_{p}(g_{0})}];

• C^{⊥} is the annihilator of C ⊂ A[u^{−val_{p}(g_{0})}] inside A[(u^{c})^{−val_{p}(g_{0})}] under the λ-Weil pairing;

• p^{val_{p}(g_{0})}λ is the polarisation A/(C ⊕C^{⊥}) → (A/(C ⊕C^{⊥}))^{∨} which makes the following diagram commute

A  ⟶^{p^{−val p(g_{0})}λ}  A^{∨}
↓  ↑
A/(C ⊕ C^{⊥}) ⟶ (A/(C ⊕ C^{⊥}))^{∨};

• α_{1} ∘ g_{1} : w_{1}^{−m′_{1}}Λ_{11}/Λ_{11} → (εA[w_{1}^{∞}]/C_{1})(S) is the homomorphism making the following diagram commute

w_{1}^{−m′_{1}}Λ_{11}/Λ_{11} ⟶  εA[w_{1}^{∞}]/C_{1}(S)
↓  ↓
w_{1}^{−m′_{1}}g_{1}Λ_{11}/g_{1}Λ_{11} ⟶ (εA[w_{1}^{∞}]/C_{1})[w_{1}^{m′_{1}}](S)
↓  ↓
w_{1}^{−m_{1}}Λ_{11}/g_{1}Λ_{11} ⟶ (εA[w_{1}^{m_{1}}]/C_{1})(S)
↑  ↑
w_{1}^{−m_{1}}Λ_{11}/Λ_{11}  ⟶^{α_{1}}  εA[w_{1}^{m_{1}}](S);

• for i > 1, α_{i} ∘ g_{i} : w_{i}^{−m′_{i}}Λ_{i}/Λ_{i} → A[w_{i}^{∞}]/C_{i} is the homomorphism making the following diagram commute

w_{i}^{−m′_{i}}Λ_{i}/Λ_{i} ⟶  A[w_{i}^{∞}]/C_{i}
↓  ↓
w_{i}^{−m′_{i}}g_{i}Λ_{i}/g_{i}Λ_{i} ⟶^{∼} (A[w_{i}^{∞}]/C_{i})[w_{i}^{m′_{i}}]
↓  ↓
w_{i}^{−m_{i}}Λ_{i}/g_{i}Λ_{i} ⟶^{∼}  A[w_{i}^{m_{i}}]/C_{i}
↑  ↑
w_{i}^{−m_{i}}Λ_{i}/Λ_{i}  ⟶^{α_{i}}  A[w_{i}^{m_{i}}].

It is tedious but straightforward to check that this does define an action. We see that (p^{−2}, p^{−1}, ..., p^{−1}) acts in the same way as p ∈ G(𝔸^{∞,p}) and so acts invertibly on the inverse system. Thus this definition can be extended to the whole of G(ℚ_{p}). We also see that on the generic fibre (i.e. over F_{w}) this definition (when it makes sense) agrees with the action previously defined. (A less tedious argument is to first note that this definition coincides with the previously defined action on the generic fibre and then use the fact that the generic fibre is Zariski dense in X_{U^{p},m} to check that first two assertions. That the generic fibre is indeed dense follows at once from lemma 5.1 below.)

We remark that the lisse ℚ_{l}^{ac} sheaf L_{ξ} can be defined over the whole of X_{U^{p},m} in exactly the same manner it was defined over the generic fibre X_{U^{p}(m)}. If g ∈ G(𝔸^{∞}) maps X_{U^{p},m} to X_{(U^{p})′,m′} then again ξ(g_{l}) induces a morphism of sheaves

g : g^{∗}L_{ξ} ⟶ L_{ξ}

over X_{U^{p},m}.

We next establish some important pieces of notation. We will let A/X_{U^{p},m} denote the universal abelian variety. If A/S is an abelian scheme and if i :

O_{B} ↪ End (A/S) ⊗_{ℤ} ℤ_{(p)} then we will let G_{A} denote the Barsotti-Tate O_{F,w}- module εA[w^{∞}]. We will simply use G/X_{U^{p},m} to denote G_{A}. If s is a closed geometric point of X_{U^{p},m} we will let h(s) denote the height of G_{s}^{et}. We will let \overline{X}_{U^{p},m} denote the reduction X_{U^{p},m} ×_{Spec O_{F,w}} Spec k(w). We will let \overline{X}_{U^{p},m}^{[h]} denote the reduced closed subscheme of \overline{X}_{U^{p},m} which is the closure of the set of closed geometric points s with h(s) ≤ h. We will also let

\overline{X}_{U^{p},m}^{(h)} = \overline{X}_{U^{p},m}^{[h]} − \overline{X}_{U^{p},m}^{[h−1]}.

The action of G(𝔸^{∞}) on the inverse system of the X_{U^{p},m} takes the inverse system of locally closed subschemes \overline{X}_{U^{p},m}^{(h)} to itself (because they are defined in an invariant manner).

Lemma 5.1 Throughout this lemma we suppose that U^{p} is sufficiently small. Suppose also that m_{1} = 0 and that m′_{i} = m_{i} for i > 1. Let s be a closed point

of \overline{X}_{U^{p},m} ×_{Spec k(w)} k(w)^{ac} and fix an isomorphism G_{s}^{0} →^{∼} Σ_{F_{w},n−h(s)}.

1. The formal completion of X_{U^{p},m} ×_{Spec O_{F,w}} Spec OF̂_{w}^{nr} at s is isomorphic to the universal formal deformation space for the Barsotti-Tate O_{F,w}- module G_{s}. Thus we get an identification

(X_{U^{p},m} ×_{Spec O_{F,w}} Spec OF̂_{w}^{nr})_{s}^{∧} ≅ Hom (T G_{s}, Σ̃_{F_{w},n−h(s)});

while (\overline{X}_{U^{p},m}^{(h(s))} ×_{Spec k(w)} Spec k(w)^{ac})_{s}^{∧} is identified to the closed formal subscheme

Hom (T G_{s}, Σ_{F_{w},n−h(s)}) ⊂ Hom (T G_{s}, Σ̃_{F_{w},n−h(s)}).

2. X_{U^{p},m}/Spec O_{F,w} is smooth. Moreover each \overline{X}_{U^{p},m}^{(h)}/Spec k(w) is either empty or smooth of dimension h.

3. The closed points of \overline{X}_{U^{p},m′} ×_{Spec k(w)} k(w)^{ac} above s are in natural bijection with the surjective homomorphisms

δ : w^{−m′_{1}}Λ_{11}/Λ_{11} → → G_{s}^{et}[w^{m′_{1}}](k(s)).

We will write s_{δ} for the point corresponding to δ. Then we can identify the formal completion of X_{U^{p},m′} ×_{Spec O_{F,w}} Spec OF̂_{w}^{nr} at s_{δ} with

Hom (w^{−m′_{1}}T G_{s}, Σ̃_{F_{w},n−h(s)}) ×_{Spf R_{Fw,n−h(s)}} Spf R_{F_{w},n−h(s),m′_{1}},

such that the morphism

(X_{U^{p},m′} ×_{Spec O_{F,w}} Spec OF̂_{w}^{nr})_{s_{δ}}^{∧} ⟶ (X_{U^{p},m} ×_{Spec O_{F,w}} Spec OF̂_{w}^{nr})_{s}^{∧}

corresponds to the natural morphism

Hom (w^{−m′_{1}}T G_{s}, Σ̃_{F_{w},n−h(s)}) ×_{Spf R_{Fw,n−h(s)}} Spf R_{F_{w},n−h(s),m′_{1}}
↓
Hom (T G_{s}, Σ̃_{F_{w},n−h(s)}).

Moreover the formal completion of \overline{X}_{U^{p},m′}^{(h(s))} at s_{δ} corresponds to the closed formal subscheme Hom (w^{−m′_{1}}T G_{s}, Σ_{F_{w},n−h(s)}) inside

Hom (w^{−m′_{1}}T G_{s}, Σ̃_{F_{w},n−h(s)}) ×_{Spf R_{Fw,n−h(s)}} Spf R_{F_{w},n−h(s),m′_{1}}.

4. X_{U^{p},m′}/O_{F,w} is regular and flat.

5. \overline{X}_{U^{p},m′}^{(h)}/k(w) is smooth and the morphism \overline{X}_{U^{p},m′}^{(h)} → \overline{X}_{U^{p},m}^{(h)} is finite and flat of degree #GL_{n}(O_{F,w}/w^{m′_{1}})/#GL_{n−h}(O_{F,w}/w^{m′_{1}}).

6. Suppose that (U^{p})′′ ⊂ U^{p} and that for all i we have m′′_{i} ≥ m′_{i}. Then the

natural morphism

X_{(U^{p})′′,m′′} ⟶ X_{U^{p},m′}

is finite and flat of degree

[U^{p} : (U^{p})′′] ∏_{i=1}^{r} #GL_{n}(O_{F,w_{i}}/w_{i}^{m′′_{i}}O_{F,w_{i}})/(∏_{i=1}^{r} #GL_{n}(O_{F,w_{i}}/w_{i}^{m′_{i}}O_{F,w_{i}})).

If m′′_{1} = m′_{1} then this morphism is in fact etale.

Proof: First of all it is standard that (X_{U^{p},m} ×_{Spec O_{F,w}} Spec OF̂_{w}^{nr})_{s}^{∧} is the formal deformation space for (r + 2)-tuples deforming (A_{s}, λ_{s}, i_{s}, α_{i,s}) (where i > 1). By the Serre-Tate theorem this is the same as deformations of the (r + 2)-tuple (A_{s}[p^{∞}], λ_{s}, i_{s}, α_{i,s}). As λ : A_{s}[u^{∞}] →^{∼} A_{s}[(u^{c})^{∞}] we see that this is the same as deformations of the (r + 1)-tuple (A_{s}[u^{∞}], i_{s}, α_{i,s}). As A_{s}[w_{i}^{∞}] is ind-etale for i > 1 it has a unique deformation over any Artinian local ring with residue field k(s) as does α_{i,s}. Thus we need only consider deformations of the pair (A_{s}[w^{∞}], i_{s}). As O_{B,w} ≅ M_{n}(O_{F,w}) this is the same as deformations of G_{s} = εA_{s}[w^{∞}] with its O_{F,w}-action. This proves the first assertion of the lemma.

The rest of the first part of the lemma follows from the discussion before lemma 2.3 and from corollary 2.4. The second part of the lemma follows from the first.

The first assertion of the third part of the lemma follows from lemma 3.1. The second assertion follows from the discussion proceeding lemma 3.2. \overline{X}_{U^{p},m′}^{(h(s))}

can be constructed as the reduced subscheme of the fibre product of \overline{X}_{U^{p},m}^{(h(s))}

and X_{U^{p},m′} over X_{U^{p},m}. Thus (\overline{X}_{U^{p},m′}^{(h(s))})_{s_{δ}}^{∧} is the reduced formal subscheme of the fibre product over Hom (T G_{s}, Σ̃_{F_{w},n−h(s)}) of Hom (T G_{s}, Σ_{F_{w},n−h(s)}) and Hom (w^{−m′_{1}}T G_{s}, Σ̃_{F_{w},n−h(s)}) ×_{Spf R_{Fw,n−h(s)}} Spf R_{F_{w},n−h(s),m′_{1}}. (Here we make use

of lemma 2.6.) Hence (\overline{X}_{U^{p},m′}^{(h(s))})_{s_{δ}}^{∧} is the reduced formal subscheme of

Hom (w^{−m′_{1}}T G_{s}, Σ_{F_{w},n−h(s)}) ×_{Spf R_{Fw,n−h(s)}} Spf R_{F_{w},n−h(s),m′_{1}},

i.e. Hom (w^{−m′_{1}}T G_{s}, Σ_{F_{w},n−h(s)}).

The fourth part now follows on applying proposition 4.3 of [Dr] because both these properties can be detected on formal completions at closed points. (If A is a noetherian local ring with maximal ideal 𝔪 then dim A_{𝔪}^{∧} = dim A, 𝔪/𝔪^{2} →^{∼} 𝔪^{∧}/(𝔪^{∧})^{2} and A_{𝔪}^{∧}/A is faithfully flat.) As for the fifth part, finiteness follows from lemma 3.1. Smoothness and flatness follow from the computation of the formal completions. The degree can also be computed on formal completions: suppose that s is a closed point of \overline{X}_{U^{p},m}^{(h)} × Spec k(w)^{ac}. The number

of closed points of \overline{X}_{U^{p},m′}^{(h)} × Spec k(w)^{ac} above s is the number of surjective homomorphisms from (O_{F,w}/w^{m′_{1}})^{n} to (O_{F,w}/w^{m′_{1}})^{h}. If s_{δ} is one of these points the degree of (\overline{X}_{U^{p},m′}^{(h)} × Spec k(w)^{ac})_{s_{δ}}^{∧} over (\overline{X}_{U^{p},m}^{(h)} × Spec k(w)^{ac})_{s}^{∧} is the rank

of Σ_{F_{w},n−h}^{h}[w^{m′_{1}}]. Thus the degree of \overline{X}_{U^{p},m′}^{(h)} over \overline{X}_{U^{p},m}^{(h)} is

\begin{matrix}(#O_{F,w}/w^{m′_{1}})^{h(n−h)}(#GL_{n}(O_{F,w}/w^{m′_{1}})/#GL_{n}(O_{F,w}/w^{m′_{1}})_{δ}) = \\ = #GL_{n}(O_{F,w}/w^{m′_{1}})/#GL_{n−h}(O_{F,w}/w^{m′_{1}}). \end{matrix}

We can divide the proof of the sixth part into two cases: the case where m′′_{1} = m′_{1} and the case where U^{p} = (U^{p})′′ and m′′_{i} = m′_{i} for i > 1. In the second of these two cases it is standard the morphism is etale of the stated degree. In the first case it follows from lemma 3.2. □

(We remark that one can use the results of Drinfeld’s paper [Dr] to show that in fact if m_{1} = 0 then \overline{X}_{U^{p},m}^{[h]} is smooth. We will not give details here as we will not need this result. It seems to us an interesting question whether this remains true for m_{1} > 0.)

The next lemma will be proved in section 8 below.

Lemma 5.2 The scheme \overline{X}_{U^{p},m}^{(0)} is non-empty.

As a first application of this lemma we have the following corollary.

Corollary 5.3 The scheme \overline{X}_{U^{p},m}^{(h)} for h = 0, ..., n − 1 is smooth of pure dimension h.

As a second application we now provide the postponed proof of lemma 3.5.

Proof of lemma 3.5: Choose a totally real field F^{+} with a place w above p such that F_{w}^{+} ≅ K and choose an imaginary quadratic field E in which p splits. We may then choose u, B, ∗, τ_{0}, β and Λ_{i} as in section 1 and such that dim_{F} B = g^{2}. Also choose a sufficiently small open compact subgroup U^{p} ⊂ G(𝔸^{∞,p}). Let

X_{m} = X_{U^{p},(m,0,...,0)} ×_{Spec O_{F,w}} Spec OF̂_{w}^{nr}

and let x be any closed point of

\overline{X}_{U^{p},(0,0,...,0)}^{(g)} ×_{Spec k(w)} Spec k(w)^{ac} ⊂ X_{0}.

(The existence of x follows from the last lemma.) That the collection of the X_{m} and the x follows from lemma 5.1. □

Now let Φ^{i} denote the vanishing cycles for \overline{X}_{U^{p},m} ⊂ X_{U^{p},m}. Then we have a spectral sequence

H^{i}(\overline{X}_{U^{p},m} × k(w)^{ac}, Φ^{j} ⊗ L_{ξ}) ⇒ H^{i+j}(X_{U^{p},m} × F_{w}^{ac}, L_{ξ}).

(See lemma II.2.) If (g, σ) ∈ G(𝔸^{∞}) × W_{F_{w}} then we have a natural map

(g, σ) : (g × Frob_{w}^{w(σ)})^{∗}Φ^{j} = (g ∘ (F r^{∗})^{f_{1}w(σ)} × 1)^{∗}Φ^{j} ⊗ L_{ξ} ⟶ Φ^{j} ⊗ L_{ξ}.

Thus we get a smooth/continuous action of G(𝔸^{∞}) × W_{F_{w}} on

lim_{→U^{p},m} H^{i}(\overline{X}_{U^{p},m} × k(w)^{ac}, Φ^{j} ⊗ L_{ξ})

which is compatible with the action on H^{i+j}(X, L_{ξ}) and the above spectral sequence.

For any 0 < h ≤ n − 1 we get a long exact sequence (see for example [FK] I.8.7 (3))

\begin{matrix}... ⟶ H_{c}^{i}(\overline{X}_{U^{p},m}^{(h)} × k(w)^{ac}, Φ^{j} ⊗ L_{ξ}) ⟶ H^{i}(\overline{X}_{U^{p},m}^{[h]} × k(w)^{ac}, Φ^{j} ⊗ L_{ξ}) ⟶ \\ ⟶ H^{i}(\overline{X}_{U^{p},m}^{[h−1]} × k(w)^{ac}, Φ^{j} ⊗ L_{ξ}) ⟶ ... \end{matrix}

Combining these two observations we obtain the following lemma.

Lemma 5.4 Suppose that for each 0 ≤ h ≤ n − 1, 0 ≤ i ≤ 2h and 0 ≤ j ≤ n − 1 the G(𝔸^{∞}) × W_{F_{w}}-module lim_{→U^{p},m} H_{c}^{i}(\overline{X}_{U^{p},m}^{(h)} × k(w)^{ac}, Φ^{j} ⊗ L_{ξ}) is admissible/continuous. Then the same is true for each lim_{→U^{p},m} H^{i}(\overline{X}_{U^{p},m} × k(w)^{ac}, Φ^{j} ⊗ L_{ξ}) and we have an equality of virtual G(𝔸^{∞}) × W_{F_{w}}-modules

∑_{i}(−1)^{i}[H^{i}(X, L_{ξ})^{ℤ_{p}^{×}}] = ∑_{i,j,h}(−1)^{i+j}[ lim_{→U^{p},m} H_{c}^{i}(\overline{X}_{U^{p},m}^{(h)} × k(w)^{ac}, Φ^{j} ⊗ L_{ξ})].

We wish to further analyse the structure of \overline{X}_{U^{p},m}. For s a closed point of \overline{X}_{U^{p},m}^{(h)} let M_{s} denote the kernel of the composite

α_{1} : w^{−m_{1}}Λ_{11}/Λ_{11} ⟶ G_{s}[w_{1}^{m_{1}}](k(s)) ⟶ G_{s}^{et}[w_{1}^{m_{1}}](k(s)).

Then M_{s} is a direct summand of w^{−m_{1}}Λ_{11}/Λ_{11} which is free over O_{F,w}/w^{m_{1}} of rank (n − h). The function s ↦ M_{s} is locally constant on \overline{X}_{U^{p},m}^{(h)}. (Suppose that H is a finite abelian group, that S is a connected scheme and that F is a lisse etale sheaf on S. If α : H_{S} → F is a morphism of etale sheaves then there is a subgroup H′ ⊂ H such that ker α = H_{S} (cf page 49 of [KM]).) Thus we have a decomposition

\overline{X}_{U^{p},m}^{(h)} = ∐_{M} \overline{X}_{U^{p},m,M},

where M runs over free O_{F,w}/w^{m_{1}}-submodules of w^{−m_{1}}Λ_{11}/Λ_{11} of rank n − h and where M_{s} = M for s a closed point of \overline{X}_{U^{p},m,M}. If g ∈ (O_{B,w}^{op})^{×} then g gives an isomorphism

g : \overline{X}_{U^{p},m,M} ⟶^{∼} \overline{X}_{U^{p},m,g^{−1}M}.

The following lemma follows at once (using lemma 5.1).

Lemma 5.5 Suppose that m_{1} = 0 and m′_{i} = m_{i} for i > 1. Then \overline{X}_{U^{p},m′,M}^{(h)} is

smooth of dimension h and \overline{X}_{U^{p},m′,M}^{(h)}/\overline{X}_{U^{p},m}^{(h)} is finite flat of degree

\begin{matrix}#P_{M}(O_{F,w}/w^{m′_{1}})/#GL_{n−h}(O_{F,w}/w^{m′_{1}}) = \\ = #(O_{F,w}/w^{m′_{1}})^{h(n−h)}#GL_{h}(O_{F,w}/w^{m′_{1}}). \end{matrix}

Now suppose that M ⊂ Λ_{11} is a O_{F,w}-submodule which is both a direct summand and free of rank n − h. We will let P_{M} ⊂ Aut (Λ_{11}) denote the maximal parabolic subgroup which stabilises M. Then we will set

G_{M}(𝔸^{∞}) = G(𝔸^{∞,p}) × ℚ_{p}^{×} × P_{M}(F_{w}) × ∏_{r=2}^{r}(B_{w_{i}}^{op})^{×}.

We will also set

\overline{X}_{U^{p},m,M} = \overline{X}_{U^{p},m,w^{−m_{1}}M/M}.

For fixed M the inverse system of the \overline{X}_{U^{p},m,M} inherits an action of G_{M}(𝔸^{∞}). Then

H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ}) = lim_{→U^{p},m} H_{c}^{i}(\overline{X}_{U^{p},m,M} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ})

is a smooth/continuous G_{M}(𝔸^{∞}) × W_{F_{w}}-module.

We will next describe a natural map

Ind _{P_{M} (O_{F,w}/w^{m_{1}})}^{(O_{B,w}^{op}/w^{m_{1}})^{×}}H_{c}^{i}(\overline{X}_{U^{p},m,M} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ})
↓
H_{c}^{i}(\overline{X}_{U^{p},m}^{(h)} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ}).

A typical element of Ind _{P_{M} (O_{F,w}/w^{m_{1}})}^{(O_{B,w}^{op}/w^{m_{1}})^{×}}H_{c}^{i}(\overline{X}_{U^{p},m,M} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ}) is represented by a function

f : (O_{B,w}^{op}/w^{m_{1}})^{×} ⟶ H_{c}^{i}(\overline{X}_{U^{p},m,M} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ})

such that

f(γg) = γf(g)

for all γ ∈ P_{M}(O_{F,w}/w^{m_{1}}) and all g ∈ (O_{B,w}^{op}/w^{m_{1}})^{×}. We map f to

(#P_{M}(O_{F,w}/w^{m_{1}})/#(O_{B,w}^{op}/w^{m_{1}})^{×}) ∑_{g∈P_{M} (O_{F,w}/w^{m_{1}})\(O_{B,w}^{op}/w^{m_{1}})^{×}} g^{−1}f(g).

This map is easily checked to be a well defined isomorphism of (O_{B,w}^{op})^{×}- modules.

Similarly we can define a natural map

Ind _{P_{M} (F_{w})}^{(B_{w}^{op})^{×}}H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ}) ⟶ lim_{→U^{p},m} H_{c}^{i}(\overline{X}_{U^{p},m}^{(h)} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ})

as follows. A typical element of Ind _{P_{M} (F_{w})}^{(B_{w}^{op})^{×}}H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ}) is represented by a locally constant function

f : (B_{w}^{op})^{×} ⟶ H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ})

such that

f(γg) = γf(g)

for all γ ∈ P_{M}(F_{w}) and all g ∈ (B_{w}^{op})^{×}. We map f to

∫_{P_{M} (F_{W})/(B_{w}^{op})^{×}} g^{−1}f(g)dg,

where we choose Haar measures to give P_{M}(F_{w})/(B_{w}^{op})^{×} volume 1, and where the integral makes sense as it is in fact a finite sum (as f is locally constant and P_{M}(F_{W})/(B_{w}^{op})^{×} is compact). This is a morphism of G(𝔸^{∞}) × W_{F_{w}}-modules.

Lemma 5.6 Suppose that H_{c}^{i}(\overline{X}_{M}, Φ^{j}⊗L_{ξ}) is an admissible G_{M}(𝔸^{∞})-module. Then the above map gives an isomorphism of G(𝔸^{∞}) × W_{F_{w}}-modules

Ind _{P_{M} (F_{w})}^{(B_{w}^{op})^{×}}H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ}) ⟶ lim_{→U^{p},m} H_{c}^{i}(\overline{X}_{U^{p},m}^{(h)} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ}).

In particular H_{c}^{i}(\overline{X}_{U^{p},m}^{(h)} ×Spec k(w)^{ac}, Φ^{j} ⊗L_{ξ}) is an admissible G(𝔸^{∞})×W_{F_{w}}- module.

Proof: Recall the Iwasawa decomposition (B_{w}^{op})^{×} = P_{M}(F_{w})(O_{B,w}^{op})^{×}. It follows that

P_{M}(F_{w})\(B_{w}^{op})^{×} ≅ P_{M}(O_{F,w})\(O_{B,w}^{op})^{×} → → P_{M}(O_{F,w}/w^{m_{1}})\(O_{B,w}^{op}/w^{m_{1}})^{×}.

This gives rise to maps

Ind _{P_{M} (O_{F,w}/w^{m_{1}})}^{(O_{B,w}^{op}/w^{m_{1}})^{×}}H_{c}^{i}(\overline{X}_{U^{p},m,M} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ})
↓
Ind _{P_{M} (F_{w})}^{(B_{w}^{op})^{×}}H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ})

which are compatible with the maps

H_{c}^{i}(\overline{X}_{U^{p},m}^{(h)} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ})
↓
lim_{→U^{p},m} H_{c}^{i}(\overline{X}_{U^{p},m}^{(h)} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ}).

As each of the maps

Ind _{P_{M} (O_{F,w}/w^{m_{1}})}^{(O_{B,w}^{op}/w^{m_{1}})^{×}}H_{c}^{i}(\overline{X}_{U^{p},m,M} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ})

↓

H_{c}^{i}(\overline{X}_{U^{p},m}^{(h)} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ}).

is an isomorphism, the lemma will follow on passing to the limit as long as we can check that the map

lim_{→U^{p},m} Ind _{P_{M} (O_{F,w}/w^{m_{1}})}^{(O_{B,w}^{op}/w^{m_{1}})^{×}}H_{c}^{i}(\overline{X}_{U^{p},m,M} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ})
↓
Ind _{P_{M} (F_{w})}^{(B_{w}^{op})^{×}}H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ})

is an isomorphism. Injectivity is straightforward. As for surjectivity any f ∈ Ind _{P_{M} (F_{w})}^{(B_{w}^{op})^{×}}H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ}) being locally constant factors through one of the finite quotients P_{M}(O_{F,w}/w^{m_{1}})\(O_{B,w}^{op}/w^{m_{1}})^{×}. Then f will be in the image of

Ind _{P_{M} (O_{F,w}/w^{m′_{1}})}^{(O_{B,w}^{op}/w^{m′_{1}})^{×}}H_{c}^{i}(\overline{X}_{U^{p},m′,M} × Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ}) for some U^{p} and m′. □

Putting together the analysis of this section we obtain the following proposition.

Proposition 5.7 For h = 0, ..., n − 1 choose a direct summand M_{h} ⊂ Λ_{11} of rank n − h. Suppose that for each 0 ≤ h ≤ n − 1, 0 ≤ j ≤ n − 1 and 0 ≤ i ≤ 2h the G_{M_{h}}(𝔸^{∞})-module H_{c}^{i}(\overline{X}_{M_{h}}, Φ^{j} ⊗ L_{ξ}) is admissible. Then we have an equality of virtual G(𝔸^{∞}) × W_{F_{w}}-modules

[H(X, L_{ξ})^{ℤ_{p}^{×}}] = ∑_{h,j,i}(−1)^{n−1+i+j}Ind _{P_{M_{h}}(F_{w})}^{(B_{w}^{op})^{×}}[H_{c}^{i}(\overline{X}_{M_{h}}, Φ^{j} ⊗ L_{ξ})].

We will let (X_{U^{p},m}^{(h)})^{∧} (resp. X_{U^{p},m,M}^{∧}) denote the formal completion of

X_{U^{p},m} along the locally closed subscheme \overline{X}_{U^{p},m}^{(h)} (resp. \overline{X}_{U^{p},m,M}). The comparison theorem of [Berk3] implies that Φ^{i}|\overline{X}{(h) \atop U^{p},m} (resp. Φ^{i}|_{\overline{X}_{U^{p},m,M}}) coincides with

the formal vanishing cycles for \overline{X}_{U^{p},m}^{(h)} ⊂ (X_{U^{p},m}^{(h)})^{∧} (resp. \overline{X}_{U^{p},m,M} ⊂ X_{U^{p},m,M}^{∧})

(see appendix II). In terms of G/\overline{X}_{U^{p},m}^{(h)} (resp. G/\overline{X}_{U^{p},m,M}) the formal completion is completely characterised by the following useful universal property.

Lemma 5.8 Suppose that X is a locally noetherian O_{F,w}-formal scheme and assume p = 0 on X^{red}. Suppose also that H/X is a Barsotti-Tate O_{F,w}-module and that γ is a Drinfeld w^{m_{1}}-structure on H/X. Moreover suppose that we are given a morphism f : X^{red} → \overline{X}_{U^{p},m}^{(h)} (resp. \overline{X}_{U^{p},m,M}) under which G with its canonical Drinfeld w^{m_{1}}-structure pulls back to H|_{X^{red}} with the Drinfeld w^{m_{1}}-structure γ|_{X^{red}}. Then there is a unique extension of f to a morphism f̃ : X → X_{U^{p},m,M}^{∧} under which G with its canonical Drinfeld w^{m_{1}}-structure pulls back to H and γ respectively.

Proof: Let (A, λ, i, \overline{η}^{p}, α_{i}) be the pull back to X^{red} of the universal object over X_{U^{p},m}. Exactly as in the first paragraph of the proof of lemma 5.1 we see that deformations of (A, λ, i, \overline{η}^{p}, α_{i}) to X are in natural bijection with the deformations of (f^{∗}G, f^{∗}α_{1}). Thus we have a unique deformation (A′, λ′, i′, (\overline{η}^{p})′, α′_{i}) over X of (A, λ, i, \overline{η}^{p}, α_{i}) which gives rise to (H, γ). Thus we have a unique morphism f̃: X → X_{U^{p},m} such that the universal (r + 4)-tuple pulls back to (A′, λ′, i′, (\overline{η}^{p})′, α′_{i}). This morphism restricts on X^{red} to f and so must factor through X_{U^{p},m,M}^{∧}. We see that f̃ is also the unique such morphism extending f under which (G, α_{1}) pulls back to (H, γ). □

6 Igusa varieties of the first kind

In our setting there seem to be two natural analogues of the familiar Igusa curves in the theory of elliptic modular curves. We will call these Igusa varieties of the first and second kind. When we speak of these Igusa varieties we will refer only to the analogue of the ordinary locus on the usual Igusa curves. We have not looked at the question of whether our Igusa varieties admit natural smooth compactifications, although we feel this is a natural and interesting question. In the case of elliptic modular curves, the Weil pairing on the pdivisible group of an elliptic curve allows one to identify these two kinds of Igusa variety.

In this section we introduce the more naive notion of Igusa variety of the first kind in the context of the Shimura varieties we are studying. To this end fix an integer h in the range 0 ≤ h ≤ n − 1. Also if m = (m_{1}, ..., m_{r}) ∈ ℤ_{≥0}^{r} then let \overline{m} denote (0, m_{2}, ..., m_{r}).

By an Igusa variety of the first kind I_{U^{p},m}^{(h)}/\overline{X}_{U^{p},\overline{m}}^{(h)} we shall mean the moduli space for isomorphisms α_{1}^{et} : (w^{−m_{1}}O_{F,w}/O_{F,w})_{\overline{X}_{U^{p},\overline{m}}^{(h)}}^{h} →^{∼} G^{et}[w^{m_{1}}]. Thus

I_{U^{p},m}^{(h)}/\overline{X}_{U^{p},\overline{m}}^{(h)} is Galois (and in particular finite etale, but not necessarily con-

nected) with Galois group GL_{h}(O_{F,w}/w^{m_{1}}). The morphism I_{U^{p},m}^{(h)} → \overline{X}_{U^{p},\overline{m}}^{(h)} factors naturally through I_{U^{p},m′}^{(h)} if m′_{1} < m_{1} and m′_{i} = m_{i} for i > 1. The inverse system of the I_{U^{p},m}^{(h)} has a natural action of G(𝔸^{∞,p}) × GL_{h}(O_{F,w}) × ∏_{i=2}^{r}(O_{B,w_{i}}^{op})^{×}.

Let (ℤ × GL_{h}(F_{w}))^{+} denote the sub-semigroup of elements (c, g) ∈ ℤ × GL_{h}(F_{w}) such that w to the integral part of −c/(n − h) times g is integral. Then the inverse system of the I_{U^{p},m}^{(h)} has an action of G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} extending that of G(𝔸^{∞,p}) × GL_{h}(O_{F,w}) × ∏_{i=2}^{r}(O_{B,w_{i}}^{op})^{×}. We leave the action of G(𝔸^{∞,p}) to the reader. First suppose that (g_{0}, c, g_{1}^{et}, g_{i}) ∈ ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} also satisfies

• for i > 1 we have g_{i}^{−1} ∈ O_{B,w_{i}}^{op} and g_{0}^{−1}g_{i} ∈ O_{B,w_{i}}^{op},

• (g_{1}^{et})^{−1} ∈ M_{h}(O_{F,w}) and g_{0}^{−1}g_{1}^{et} ∈ M_{h}(O_{F,w}),

• (n − h)w(g_{0}) ≤ c ≤ 0,

• for i > 1 we have w_{i}^{m_{i}−m′_{i}}g_{i} ∈ O_{B,w_{i}}^{op},

• w_{1}^{m_{1}−m′_{1}}g_{1}^{et} ∈ M_{h}(O_{F,w}).

Under these assumptions we will define a morphism

(g_{0}, c, g_{1}^{et}, g_{i}) : I_{U^{p},m}^{(h)} ⟶ I_{U^{p},m′}^{(h)}.

It will send (A, λ, i, \overline{η}^{p}, α_{1}^{et}, α_{i}) to (A/(C ⊕ C^{⊥}), p^{val_{p}(g_{0})}λ, i, \overline{η}^{p}, α_{1}^{et} ∘ g_{1}^{et}, α_{i} ∘ g_{i}), where we have set

• C_{1} ⊂ εA[w_{1}^{m_{1}}] is the unique closed subscheme for which there is an exact sequence

(0) ⟶ ker F^{−f_{1}c} ⟶ C_{1} ⟶ α_{1}^{et}(F_{w}^{h}/O_{F,w}^{h}[(g_{1}^{et})^{−1}]) ⟶ (0),

(this makes sense as if d denotes the integral part of −c/(n − h) then ker F^{−f_{1}c} ⊃ G_{A}^{0}[w^{d}] and G_{A}^{et}[w^{d}] ⊃ α(F_{w}^{h}/O_{F,w}^{h}[(g_{1}^{et})^{−1}]) (we are using the fact that (c, g_{1}) ∈ (ℤ × GL_{h}(F_{w}))^{+}));

• for i > 1, C_{i} = α_{i}(g_{i}Λ_{i}/Λ_{i});

• C = (O_{F,w}^{n} ⊗_{O_{F,w}} C_{1}) ⊕ ⨁_{i=2}^{r} C_{i} ⊂ A[u^{−val_{p}(g_{0})}];

• C^{⊥} is the annihilator of C ⊂ A[u^{−val_{p}(g_{0})}] inside A[(u^{c})^{−val_{p}(g_{0})}] under the λ-Weil pairing;

• p^{val_{p}(g_{0})}λ is the polarisation A/(C ⊕C^{⊥}) → (A/(C ⊕C^{⊥}))^{∨} which makes the following diagram commute

A  ⟶^{p^{−val p(g_{0})}λ}  A^{∨}
↓  ↑
A/(C ⊕ C^{⊥}) ⟶ (A/(C ⊕ C^{⊥}))^{∨};

• α_{1}^{et} ∘g_{1}^{et} : (w_{1}^{−m′_{1}}O_{F,w}/O_{F,w})_{I_{U^{p},m}^{(h)}}^{h} → (εA[w_{1}^{∞}]/C_{1})^{et} is the homomorphism

making the following diagram commute

(w_{1}^{−m′_{1}}O_{F,w}/O_{F,w})^{h} ⟶  (εA[w_{1}^{∞}]/C_{1})^{et}
↓  ↑
w_{1}^{−m′_{1}}g_{1}^{et}O_{F,w}^{h}/g_{1}^{et}O_{F,w}^{h} ⟶  (εA[w_{1}^{∞}]/C_{1})^{et}[w_{1}^{m′_{1}}]
↓  ↓
w_{1}^{−m_{1}}O_{F,w}^{h}/g_{1}^{et}O_{F,w}^{h} ⟶  (εA[w_{1}^{m_{1}}]/C_{1})^{et}
↓  ↓
w_{1}^{−m_{1}}O_{F,w}^{h}/g_{1}^{et}O_{F,w}^{h} ⟶ εA[w_{1}^{m_{1}}]^{et}/α_{1}^{et}((F_{w}/O_{F,w})^{h}[(g_{1}^{et})^{−1}])
↑  ↑
(w_{1}^{−m_{1}}O_{F,w}/O_{F,w})^{h}  ⟶^{α_{1}^{et}}  εA[w_{1}^{m_{1}}]^{et};

• for i > 1, α_{i} ∘ g_{i} : w_{i}^{−m′_{i}}Λ_{i}/Λ_{i} → A[w_{i}^{∞}]/C_{i} is the homomorphism making the following diagram commute

w_{i}^{−m′_{i}}Λ_{i}/Λ_{i} ⟶  A[w_{i}^{∞}]/C_{i}
↓  ↑
w_{i}^{−m′_{i}}g_{i}Λ_{i}/g_{i}Λ_{i} ⟶^{∼} (A[w_{i}^{∞}]/C_{i})[w_{i}^{m′_{i}}]
↓  ↓
w_{i}^{−m_{i}}Λ_{i}/g_{i}Λ_{i} ⟶^{∼}  A[w_{i}^{m_{i}}]/C_{i}
↑  ↑
w_{i}^{−m_{i}}Λ_{i}/Λ_{i}  ⟶^{α_{i}}  A[w_{i}^{m_{i}}].

It is tedious but straightforward to check that this does define an action. We see that (p^{−2}, p^{−1}, ..., p^{−1}) acts in the same way as p ∈ G(𝔸^{∞,p}) and so acts invertibly on the inverse system. Thus this definition can be extended to the whole of ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}. (A less tedious argument is to use the compatibility described below with the action of G(𝔸^{∞,p}) × ℚ_{p}^{×} × P_{M}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} on the inverse system of the \overline{X}_{U^{p},m,M}.)

We will denote by F̃r^{∗}^{f_{1}} the element (p^{−f_{1}}, −1, 1) ∈ ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+}. We see that

1. ℚ_{p}^{×} × ℤ × GL_{h}(F_{w}) is generated by ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} and F̃r^{∗}^{−f_{1}};

2. F̃r^{∗}^{f_{1}} : I_{U^{p},m}^{(h)} → I_{(U^{p}),m}^{(h)} is just (Fr^{∗})^{f_{1}}. (Note that according to the

definitions above F̃r^{∗}^{f_{1}} does take I_{(U^{p}),m}^{(h)} to itself.)

Now fix j : Λ_{11} → → O_{F,w}^{h} with kernel M. This induces a homomorphism

\begin{matrix}j_{∗} : P_{M}(F_{w}) & → → & ℤ × GL_{h}(F_{w}) \\ g & ⟼ & w ∘ det(g|_{M}) × j ∘ g ∘ j^{−1}). \end{matrix}

We will define a morphism j^{∗} : I_{U^{p},m}^{(h)} → \overline{X}_{U^{p},m,M} such that

I_{U^{p},m}^{(h)}  ⟶^{j^{∗}}  \overline{X}_{U^{p},m,M}
↓  ↓
\overline{X}_{U^{p},\overline{m}}^{(h)} ⟶^{(Fr^{∗})^{f_{1}(n−h)m_{1}}}  \overline{X}_{U^{p},\overline{m}}^{(h)}

commutes. More precisely j^{∗} is the map which takes (A, λ, i, \overline{η}^{p}, α_{1}^{et}, α_{i}) to (A^{(p^{m_{1}f_{1}(n−h)})}, λ^{(p^{m_{1}f_{1}(n−h)})}, i, \overline{η}^{p}, α_{1}, F^{m_{1}f_{1}(n−h)} ∘ α_{i}) where

α_{1}(x) = F^{m_{1}f_{1}(n−h)} ∘ α_{1}^{et} ∘ j(x).

(To see that α_{1} is well defined and that it is a Drinfeld level structure use lemma 3.1.) Because \overline{X}_{U^{p},m,M}/\overline{X}_{U^{p},\overline{m}}^{(h)} is finite flat of degree

#(O_{F,w}/w^{m′_{1}})^{h(n−h)}#GL_{h}(O_{F,w}/w^{m′_{1}})

(see lemma 5.5), because \overline{X}_{U^{p},m,M} is smooth and hence normal (see lemma 5.5), and because the composite

I_{U^{p},m}^{(h)} ⟶ \overline{X}_{U^{p},\overline{m}}^{(h)} ⟶^{(Fr^{∗})^{f_{1}(n−h)m_{1}}} \overline{X}_{U^{p},\overline{m}}^{(h)}

is also finite flat of the same degree we see that j^{∗} is an isomorphism.

Suppose that g ∈ G(𝔸^{∞,p}) × ℚ_{p}^{×} × P_{M}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} and that j_{∗}(g) ∈ G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}. Suppose also that (U^{p})′ ⊃ g^{−1}U^{p}g. If for each i we have m_{i} >> m′_{i} then

I_{U^{p},m}^{(h)}  ⟶^{j_{∗}(g)F̃r^{∗}^{f_{1}(n−h)(m_{1}−m′_{1})}} I_{(U^{p})′,m′}^{(h)}
↓  ↓
\overline{X}_{U^{p},m,M}  ⟶^{g}  \overline{X}_{(U^{p})′,m′,M}

commutes.

We now look at natural formal extensions of these Igusa varieties. In particular we will let (I_{U^{p},m}^{(h)})^{∧}/(X_{U^{p},\overline{m}}^{(h)})^{∧} denote the unique etale covering with

reduced subschemes I_{U^{p},m}^{(h)}/\overline{X}_{U^{p},\overline{m}}^{(h)} (see [Berk2]). If t ∈ ℤ_{≥0} then we will let (I_{U^{p},m}^{(h)})^{∧}(t)/(I_{U^{p},m}^{(h)})^{∧} denote the moduli space for Drinfeld w^{t}-structures on G/(I_{U^{p},m}^{(h)})^{∧}.

Lemma 6.1 1. The natural morphism (I_{U^{p},m}^{(h)})^{∧}(t) ⟶ (I_{U^{p},m}^{(h)})^{∧} is finite and flat of degree #GL_{n−h}(O_{F,w}/w^{t}).

2. (I_{U^{p},m}^{(h)})^{∧}(t)/(I_{U^{p},\overline{m}}^{(h)})^{∧}(t) is the unique etale cover with reduced subschemes

I_{U^{p},m}^{(h)}/\overline{X}_{U^{p},\overline{m}}^{(h)}.

3. (I_{U^{p},m}^{(h)})^{∧}(t) has the following universal property. Suppose that X is a locally noetherian O_{F,w}-formal scheme and assume p = 0 on X^{red}. Suppose also that H/X is a Barsotti-Tate O_{F,w}-module and that we are given a morphism f : X^{red} → I_{U^{p},m}^{(h)} under which G pulls back to H|_{X^{red}}. Then we have an exact sequence

(0) → H^{0} → H → H^{et} → (0)

over X, with H^{0} formal and H^{et} ind-etale. Suppose finally that γ is a Drinfeld w^{t}-structure on H^{0}/X. Then there is a unique extension of f to a morphism f̃: X → (I_{U^{p},m}^{(h)})^{∧} under which G pulls back to H and the canonical Drinfeld w^{t}-structure on G/(I_{U^{p},m}^{(h)})^{∧}(t) pulls back to γ.

Proof: The first part follows from corollary 3.3. The second part follows because

(I_{U^{p},m}^{(h)})^{∧}(t) ⟶^{∼} (I_{U^{p},m}^{(h)})^{∧} ×_{(I_{U^{p},\overline{m}}^{(h)})^{∧}} (I_{U^{p},\overline{m}}^{(h)})^{∧}(t).

From the definition of (I_{U^{p},m}^{(h)})^{∧}(t) the third part reduces to the special case t = 0. In this case, by lemma 5.8, we obtain a unique morphism X → (I_{U^{p},\overline{m}}^{(h)})^{∧} under which G pulls back to H. The third part of the lemma now follows from the following standard result (which in turn follows easily from, for instance, proposition 1.1 of section I of [Arti]).

Lemma 6.2 Suppose that Z → Y is a finite etale morphism of locally noetherian formal schemes. Suppose that X is also a locally noetherian formal scheme and that we have a commutative diagram

\begin{matrix}X^{red} & ↪ & X \\ ↓ & & ↓ \\ Z & ⟶ & Y. \end{matrix}

Then there is a unique diagonal morphism X → Z making the diagram still commute.

□

We will let (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} denote the set of pairs (g^{0}, g^{et}) in GL_{n−h}(F_{w}) × GL_{h}(F_{w}) for which there exists a scalar a ∈ F_{w}^{×} such that both ag^{et} ∈ M_{h}(O_{F,w}) and (ag^{0})^{−1} ∈ M_{n−h}(O_{F,w}). This is a subsemigroup of GL_{n−h}(F_{w}) × GL_{h}(F_{w}). There is a natural homomorphism from

G(𝔸^{∞,p}) × ℚ_{p}^{×} × GL_{n−h}(F_{w}) × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}

to

G(𝔸^{∞,p}) × ℚ_{p}^{×} × ℤ × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}

under which

(g^{p}, g_{p,0}, g_{w}^{0}, g_{w}^{et}, g_{w_{i}}) ⟼ (g^{p}, g_{p,0}, w(det g_{w}^{0}), g_{w}^{et}, g_{w_{i}}).

We will denote this map g ↦ [g]. Under this homomorphism

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}

is taken to

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}.

If ϖ is a uniformiser in O_{F,w} then we will let

F̃r_{ϖ}^{∗}^{f_{1}(n−h)} = (1, p^{−f_{1}(n−h)}, ϖ^{−1}, 1, 1)

in

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}.

Then

[F̃r_{ϖ}^{∗}^{f_{1}(n−h)}] = F̃r^{∗}^{f_{1}(n−h)},

and G(𝔸^{∞,p}) × ℚ_{p}^{×} × GL_{n−h}(F_{w}) × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} is generated as a semi-group by G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} and

F̃r_{ϖ}^{∗}^{−f_{1}(n−h)}.

The inverse system of the (I_{U^{p},m}^{(h)})^{∧}(t) has a natural action of G(𝔸^{∞,p}) × ℚ_{p}^{×}×(GL_{n−h}(F_{w})×GL_{h}(F_{w}))^{+}×∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}, which is compatible via [ ] with the action of G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} on the inverse system of the I_{U^{p},m}^{(h)}. We will leave the action of G(𝔸^{∞,p}) to the reader and describe the action of ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}. To this end suppose that

(g_{0}, g_{1}^{0}, g_{1}^{et}, g_{i}) ∈ ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×},

and that

• for i > 1 we have g_{i}^{−1} ∈ O_{B,w_{i}}^{op} and g_{0}^{−1}g_{i} ∈ O_{B,w_{i}}^{op},

• (g_{1}^{et})^{−1} ∈ M_{h}(O_{F,w}) and g_{0}^{−1}g_{1}^{et} ∈ M_{h}(O_{F,w}),

• g_{0}^{−1}g_{1}^{0} ∈ M_{n−h}(O_{F,w}),

• for i > 1 we have w_{i}^{m_{i}−m′_{i}}g_{i} ∈ O_{B,w_{i}}^{op},

• w_{1}^{m_{1}−m′_{1}}g_{1}^{et} ∈ M_{h}(O_{F,w}),

Also choose a ∈ ℤ_{≥0} such that w^{a}g_{1}^{et} ∈ M_{h}(O_{F,w}) and (w^{a}g_{1}^{0})^{−1} ∈ M_{n−h}(O_{F,w}) (if there is a choice, choose the maximal such a). Finally also suppose that

• w^{t−t′+a}g_{0} ∈ M_{n−h}(O_{F,w}).

We will define a morphism

(g_{0}, g_{1}^{0}, g_{1}^{et}, g_{i}) : (I_{U^{p},m}^{(h)})^{∧}(t) ⟶ (I_{U^{p},m′}^{(h)})^{∧}(t′),

which extends

[(g_{0}, g_{1}^{0}, g_{1}^{et}, g_{i})] : I_{U^{p},m}^{(h)} ⟶ I_{U^{p},m′}^{(h)}.

Let \overline{C}_{1} be the unique closed subscheme of G[w^{m_{1}}]/I_{U^{p},m}^{(h)} for which there is a short exact sequence

(0) ⟶ ker F^{−f_{1}w(det g_{1}^{0})} ⟶ \overline{C}_{1} ⟶ α_{1}^{et}(F_{w}^{h}/O_{F,w}^{h}[(g_{1}^{et})^{−1}]) ⟶ (0).

To define the desired extension (g_{0}, g_{1}^{0}, g_{1}^{et}, g_{i}) of [(g_{0}, g_{1}^{0}, g_{1}^{et}, g_{i})] it suffices (by the universal property of (I_{U^{p},m′}^{(h)})^{∧}(t′)) to specify a lifting G′ of G/\overline{C_{1}} from I_{U^{p},m}^{(h)} to (I_{U^{p},m}^{(h)})^{∧}(t) together with a Drinfeld w^{t′}-level structure on (G′)^{0}.

We now explain the construction of G′ and the Drinfeld w^{t′}-structure on (G′)^{0}. To do so fix a uniformiser ϖ of O_{F,w}. Note that we have an embedding

(F_{w}/O_{F,w})^{h}[(g_{1}^{et})^{−1}] ⟶^{α_{1}^{et}} G^{et}[w^{a}] ⟶ G/G^{0}[w^{a}].

(By an embedding we mean a compatible system of embeddings over each closed subscheme of (I_{U^{p},m}^{(h)})^{∧}(t).) We also have a Drinfeld w^{t}-structure ϖ^{−a}α_{1}^{0} on (G/G^{0}[w^{a}])^{0}. We set

G′ = (G/G^{0}[w^{a}])/((ϖ^{−a}α_{1}^{0})(F_{w}^{n−h}/O_{F,w}^{n−h}[(w^{a}g_{1}^{0})^{−1}])+α_{1}^{et}((F_{w}/O_{F,w})^{h}[(g_{1}^{et})^{−1}])).

This does not depend on the choice of ϖ. By corollary 3.3 we see that the composite of ϖ^{−a}α_{1}^{0} with

(w^{−t′}O_{F,w}/O_{F,w})^{n−h} ↪^{ϖ^{a}g_{1}^{0}} (w^{−t}O_{F,w}/O_{F,w})^{n−h}/((F_{w}/O_{F,w})^{n−h}[(w^{a}g_{1}^{0})^{−1}])

gives a Drinfeld w^{t′}-structure on (G′)^{0}. This Drinfeld w^{t′}-structure is also independent of the choice of ϖ.

It is tedious but straightforward to check that this does define an action. We see that (p^{−2}, p^{−1}, ..., p^{−1}) acts in the same way as p ∈ G(𝔸^{∞,p}) and so acts invertibly on the inverse system. Thus this definition can be extended to the whole of ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}. (A less tedious argument is to use the compatibility described below with the action of G(𝔸^{∞,p}) × ℚ_{p}^{×} × P_{M}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} on the inverse system of the

X_{U^{p},m,M}^{∧}.) Note that F̃r_{ϖ}^{∗}^{f_{1}(n−h)} maps (I_{U^{p},m}^{(h)})^{∧}(t) to itself and defines a lifting of (Fr^{∗})^{f_{1}(n−h)}, which is analogous to the canonical lifting of Frobenius in the theory of elliptic modular curves.

Now fix homomorphisms j^{et} : Λ_{11} → → O_{F,w}^{h} and j^{0} : Λ_{11} → → O_{F,w}^{n−h} such that j^{0} ⊕ j^{et} is an isomorphism. Let M = ker j^{et}. These choices induce homomorphisms define a Levi component L_{(j^{0},j^{et})} ⊂ P_{M}, i.e. the elements of P_{M} which also preserve ker j^{0}. They also induce an isomorphism

(j^{0}, j^{et})_{∗} : L_{(j^{0},j^{et})} →^{∼}  GL_{n−h}(F_{w}) × GL_{h}(F_{w})
g  ⟼ (j^{0} ∘ g ∘ (j^{0})^{−1}, j^{et} ∘ g ∘ (j^{et})^{−1}).

If ϖ is a uniformiser in O_{F,w}, we will define a morphism

(j^{0}, j^{et}, ϖ)^{∗} : (I_{U^{p},m}^{(h)})^{∧}(m_{1}) ⟶ X_{U^{p},m,M}^{∧}

which extends the morphism

(j^{et})^{∗} : I_{U^{p},m}^{(h)} ⟶ \overline{X}_{U^{p},m,M}.

To define such a morphism it suffices (by lemma 5.8) to specify a deformation of the pair

(G^{(p^{m_{1}f_{1}(n−h)})}, F^{m_{1}f_{1}(n−h)} ∘ α_{1}^{et} ∘ j^{et})/I_{U^{p},m}^{(h)}

to (I_{U^{p},m}^{(h)})^{∧}(m_{1}). As a deformation of G^{(p^{m_{1}f_{1}(n−h)})} we take G/G^{0}[w^{m_{1}}]. Then we have the identification

\begin{matrix}G^{0}[w^{m_{1}}] × G^{et}[w^{m_{1}}] & →^{∼} & G/G^{0}[w^{m_{1}}] \\ (x, y) & ⟼ & ϖ^{−m_{1}}x + y. \end{matrix}

As a deformation of F^{m_{1}f_{1}(n−h)}∘α_{1}^{et}∘j^{et} we take (α_{1}^{0}∘j^{0})⊕(α_{1}^{et}∘j^{et}). (Note that over I_{U^{p},m}^{(h)} we are identifying G^{(p^{m_{1}f_{1}(n−h)})} and G/G^{0}[w^{m_{1}}] so that F^{m_{1}f_{1}(n−h)} : G → G^{(p^{m_{1}f_{1}(n−h)})} corresponds to the natural projection G → G/G^{0}[w^{m_{1}}].)

Lemma 6.3 1. The morphism (j^{0}, j^{et}, ϖ)^{∗} is an isomorphism.

2. Suppose that g ∈ G(𝔸^{∞,p})×ℚ_{p}^{×} ×L_{(j^{0},j^{et})} ×∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} and suppose that (j^{0}, j^{et})_{∗}(g) ∈ G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}. Suppose also that (U^{p})′ ⊃ g^{−1}U^{p}g and that for each i we have m_{i} >> m′_{i}. Then

(I_{U^{p},m}^{(h)})^{∧}(m_{1}) ⟶^{(j^{0},j^{et})_{∗}(g)F̃r_{ϖ}^{∗}^{f_{1}(n−h)(m_{1}−m′_{1})}} (I_{(U^{p})′,m′}^{(h)})^{∧}(m′_{1})
↓  ↓
X_{U^{p},m,M}^{∧}  ⟶^{g}  X_{(U^{p})′,m′,M}^{∧}

commutes, where the vertical maps are (j^{0}, j^{et}, ϖ)^{∗}.

Proof: The second part is formal. To prove the first part we will verify that X_{U^{p},m,M}^{∧} has the same universal property as (I_{U^{p},m,M}^{(h)})^{∧}(m_{1}). That is we will show that if X is a locally noetherian O_{F,w}-formal scheme, if p = 0 on X^{red}, if H/X is a Barsotti-Tate O_{F,w}-module, if f : X^{red} → \overline{X}_{U^{p},m,M} is a morphism under which G pulls back to (H/H^{0}[w^{m_{1}}])|_{X^{red}} and if γ is a Drinfeld w^{m_{1}}-structure on H^{0}/X, then there is a unique extension of f to a morphism f̃: X → (I_{U^{p},m}^{(h)})^{∧} under which G pulls back to H/H^{0}[w^{m_{1}}] and the canonical Drinfeld w^{t}-structure α_{1} ∘ j^{0} on G^{0}/X_{U^{p},m,M}^{∧} pulls back to ϖ^{−m_{1}}γ. To see that X_{U^{p},m,M}^{∧} has this universal property we use lemma 5.8 and note that there is a natural bijection between

• Drinfeld w^{m_{1}}-structures γ on H^{0}/X

• and Drinfeld w^{m_{1}}-structures δ : w^{−m_{1}}Λ_{11}/Λ_{11} → (H/H^{0}[w^{m_{1}}])[w^{m_{1}}] over X which restrict to α_{1} ∘ j^{et} on X^{red}.

This bijection sends δ to ϖ^{m_{1}}δ ∘(j^{0}|_{w^{−m_{1}}M/M}^{−1}) and γ to ϖ^{−m_{1}}γ ∘j^{0} +α̃_{1}^{et}, where α̃_{1}^{et} : (w^{−m_{1}}Λ_{11}/Λ_{11}) → → H^{et}[w^{m_{1}}] is the unique lifting over X of the pull back from X_{U^{p},m,M}^{∧} to X^{red} of α_{1} : (w^{−m_{1}}Λ_{11}/Λ_{11}) → → G[w^{m_{1}}]^{et}. □

We will let Φ^{i}(t)/I_{U^{p},m}^{(h)} ×_{Spec k(w)} Spec k(w)^{ac} denote the formal vanishing cycles for I_{U^{p},m}^{(h)} ⊂ (I_{U^{p},m}^{(h)})^{∧}(t). (If t = m_{1} then it follows from the last lemma that (I_{U^{p},m}^{(h)})^{∧}(t) is isomorphic to the completion of a proper scheme of finite type over O_{F,w} along a locally closed subscheme of the special fibre. In general (I_{U^{p},m}^{(h)})^{∧}(t) is etale locally isomorphic to (I_{U^{p},m′}^{(h)})^{∧}(t) with m′_{1} = t. Thus Φ^{i}(t)/I_{U^{p},m}^{(h)} ×_{Spec k(w)} Spec k(w)^{ac} is well defined.)

Note that, if U^{p} ⊃ (U^{p})′ and for each i we have m_{i} ≤ m′_{i}, then the restriction of Φ^{i}(t)/I_{U^{p},m}^{(h)} ×_{Spec k(w)} Spec k(w)^{ac} to I_{(U^{p})′,m′}^{(h)} ×_{Spec k(w)} Spec k(w)^{ac}

is canonically isomorphic to Φ^{i}(t)/I_{(U^{p})′,m′}^{(h)} ×_{Spec k(w)} Spec k(w)^{ac} (see [Berk3]).

Suppose that x is a closed point of I_{U^{p},m}^{(h)} × Spec k(w)^{ac} and suppose that j_{x} : Σ_{n−h} →^{∼} G_{x}^{0}. Then we obtain a natural map

j_{x}^{∗} : ((I_{U^{p},m}^{(h)})^{∧}(t) ×_{Spf O_{F,w}} Spf OF̂_{w}^{nr})_{x}^{∧} ⟶ Spf R_{F_{w},n−h,t},

and hence a homomorphism

(j_{x}^{∗})^{∗} : Ψ_{F_{w},l,n−h,t}^{j} ⟶ Φ^{j}(t)_{x}.

Lemma 6.4

(j_{x}^{∗})^{∗} : Ψ_{F_{w},l,n−h,t}^{j} →^{∼} Φ^{j}(t)_{x}.

Proof: We will let Spf R(G_{x}) (resp. Spf R_{t}(G_{x}^{0})) denote the universal deformation space for G_{x} (resp. G_{x}^{0} with its (unique) Drinfeld w^{t} level structure). Then we have

((I_{U^{p},m}^{(h)})^{∧}(t) ×_{Spf O_{F,w}} Spf OF̂_{w}^{nr})_{x}^{∧} ≅ Spf R(G_{x}) ×_{Spf R_{0}(G_{x}^{0})} Spf R_{t}(G_{x}^{0})

→^{∼} Hom (T G_{x}, Σ̃_{F_{w},n−h}) ×_{Spf R_{Fw,n−h}} Spf R_{F_{w},n−h,t}.

As Hom (T G_{x}, Σ̃_{F_{w},n−h}) and Spf R_{F_{w},n−h} are formally smooth we see that

Hom (T G_{x}, Σ̃_{F_{w},n−h}) ×_{Spf R_{Fw,n−h}} Spf R_{F_{w},n−h,t} ⟶ Spf R_{F_{w},n−h,t}

induces an isomorphism on vanishing cycles (see lemma II.4). The lemma follows. □

The inverse system of sheaves

Φ^{i}(t)/I_{U^{p},m}^{(h)} ×_{Spec k(w)} Spec k(w)^{ac}

has an action of G(𝔸^{∞,p})×ℚ_{p}^{×}×(GL_{n−h}(F_{w})×GL_{h}(F_{w}))^{+}×∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}×W_{F_{w}} in the following sense. If

(g, σ) ∈ G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × W_{F_{w}}

and if [g] : I_{U^{p},m}^{(h)} → I_{(U^{p})′,m′}^{(h)} then for t >> t′ we get a natural map

(g, σ) : ([g] × (Frob_{w}^{w(σ)})^{∗})^{∗}Φ(t′) ⟶ Φ(t)

on I_{U^{p},m}^{(h)} × Spec k(w)^{ac}.

We now wish to describe the action of (g, σ) on stalks. Thus let x be a closed point of I_{U^{p},m}^{(h)} × Spec k(w)^{ac} and let y = ([g] × (Frob_{w}^{w(σ)})^{∗})x, a closed point of

I_{(U^{p})′,m′}^{(h)} × Spec k(w)^{ac}. Suppose also that j_{x} : Σ_{n−h} →^{∼} G_{x}^{0}. If δ ∈ D_{F_{w},n−h}^{×} and

if w(det δ) = w(det g_{w}^{0}) − w(σ) then we will define

([g] × (Frob_{w}^{w(σ)})^{∗} × δ)(j_{x}) : Σ_{F_{w},n−h} →^{∼} G_{y}^{0}.

To do so it suffices to give an isomorphism

Σ_{F_{w},n−h}^{(p^{f_{1}(w(det g^{0}_{w})−w(σ))})} →^{∼} G_{x}^{0}.

We simply take j_{x} ∘ δ. We will write simply

j_{y} = ([g] × (Frob_{w}^{w(σ)})^{∗} × δ)(j_{x}),

but recall that it depends on a choice of δ. We see that, for t >> t′,

((I_{U^{p},m}^{(h)})^{∧}(t) ×_{Spf O_{F,w}} Spf OF̂_{w}^{nr})_{x}^{∧} ⟶^{j_{x}^{∗}} Spf R_{F_{w},n−h,t}
↓  ↓
((I_{U^{p},m}^{(h)})^{∧}(t′) ×_{Spf O_{F,w}} Spf OF̂_{w}^{nr})_{y}^{∧} ⟶^{j_{y}^{∗}} Spf R_{F_{w},n−h,t′}

commutes (the left vertical arrow being g × (Frob_{w}^{w(σ)})^{∗} and the right vertical arrow (g_{w}^{0}, δ)). Hence

Φ^{j}(t)_{x}  ⟵^{g×σ}  Φ^{j}(t′)_{y}
↑  ↑
Ψ_{F_{w},l,n−h,t}^{j} ⟵^{(g_{w}^{0},δ,σ)} Ψ_{F_{w},l,n−h,t′}^{j}

also commutes (the left vertical arrow being (j_{x}^{∗})^{∗} and the right vertical arrow being (j_{y}^{∗})^{∗} = ([g] × (Frob_{w}^{w(σ)})^{∗} × δ)(j_{x})^{∗∗}).

If we set

H_{c}^{i}(I^{(h)}, Φ^{j} ⊗ L_{ξ}) = lim_{→U^{p},m,t} H_{c}^{i}(I_{U^{p},m}^{(h)} ×_{Spec k(w)} Spec k(w)^{ac}, Φ^{j}(t) ⊗ L_{ξ}),

then H_{c}^{i}(I^{(h)}, Φ^{j} ⊗ L_{ξ}) becomes a smooth

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × W_{F_{w}}

module. Then we have the following lemma.

Lemma 6.5 1. If Γ_{t} denotes the kernel of the homomorphism

GL_{n−h}(O_{F,w}) → → GL_{n−h}(O_{F,w}/w^{t})

then for t ≤ t′ we have a canonical isomorphism

Φ^{i}(t) ⟶^{∼} Φ^{i}(t′)^{Γ_{t}}.

2. The action of the semigroup

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × W_{F_{w}}

on H_{c}^{i}(I^{(h)}, Φ^{j} ⊗ L_{ξ}) is admissible/continuous.

Proof: The first part follows by calculating on stalks using isomorphisms (j_{x}^{∗})^{∗} and the above compatibility. The second part follows from the first. □

We also have the following lemma which we will prove in the next section.

Lemma 6.6 F̃r_{ϖ}^{∗}^{f_{1}(n−h)} acts invertibly on each

H_{c}^{i}(I_{U^{p},m}^{(h)} ×_{Spec k(w)} Spec k(w)^{ac}, Φ^{j}(t) ⊗ L_{ξ}).

We are now in a position to prove the following result.

Proposition 6.7 1. The action of the semi-group

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × W_{F_{w}}

on H_{c}^{i}(I^{(h)}, Φ^{j} ⊗L_{ξ}) extends uniquely to an admissible/continuous action of

G(𝔸^{∞,p}) × ℚ_{p}^{×} × GL_{n−h}(F_{w}) × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × W_{F_{w}}.

2. If we fix a triple (j^{0}, j^{et}, ϖ) as above then we get an isomorphism of smooth

\begin{matrix}G(𝔸^{∞,p}) × ℚ_{p}^{×} × GL_{n−h}(F_{w}) × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × W_{F_{w}} ≅ \\ ≅ G(𝔸^{∞,p}) × ℚ_{p}^{×} × L_{(j^{0},j^{et})} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × W_{F_{w}} \end{matrix}

modules

H_{c}^{i}(I^{(h)}, Φ^{j} ⊗ L_{ξ}) ⟶^{∼} H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ}).

3. The unipotent radical of P_{M}(F_{w}) acts trivially on

H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ}).

Proof: The first part follows from the lemma 6.6. For the second part we consider the maps

F̃r_{ϖ}^{∗}^{f_{1}(n−h)m_{1}} ∘ ((j^{0}, j^{et}, ϖ)^{∗})^{−1} : X_{U^{p},m,M}^{∧} ⟶ (I_{U^{p},m}^{(h)})^{∧}(m_{1})

and the induced maps

(((j^{0}, j^{et}, ϖ)^{∗})^{−1})^{∗} ∘ (F̃r_{ϖ}^{∗}^{f_{1}(n−h)m_{1}})^{∗}

from

H_{c}^{i}(I_{U^{p},m}^{(h)} ×_{Spec k(w)} Spec k(w)^{ac}, Φ^{j}(m_{1}) ⊗ L_{ξ})

to

H_{c}^{i}(\overline{X}_{U^{p},m,M} ×_{Spec k(w)} Spec k(w)^{ac}, Φ^{j} ⊗ L_{ξ}).

Combining lemmas 6.3 and 6.5 we see that these latter maps are isomorphisms. Again by lemma 6.3 they are compatible as U^{p} and m vary and give in the limit an isomorphism

H_{c}^{i}(I^{(h)}, Φ^{j} ⊗ L_{ξ}) ⟶^{∼} H_{c}^{i}(\overline{X}_{M}, Φ^{j} ⊗ L_{ξ}).

That this isomorphism is compatible with

\begin{matrix}G(𝔸^{∞,p}) × ℚ_{p}^{×} × GL_{n−h}(F_{w}) × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × W_{F_{w}} ≅ \\ ≅ G(𝔸^{∞,p}) × ℚ_{p}^{×} × L_{(j^{0},j^{et})} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × W_{F_{w}} \end{matrix}

actions again follows from lemma 6.3.

The final part follows from the second and from lemma I.1. □

Corollary 6.8 For h = 0, ..., n − 1 choose homomorphisms j_{h}^{0} : Λ_{11} → → O_{F,w}^{n−h} and j_{h}^{et} : Λ_{11} → → O_{F,w}^{h} such that j_{h}^{0} ⊕ j_{h}^{et} is an isomorphism. Let M_{h} = ker j_{h}^{et}. Then we have an equality of virtual G(𝔸^{∞}) × W_{F_{w}}-modules

[H(X, L_{ξ})^{ℤ_{p}^{×}}] = ∑_{h,j,i}(−1)^{n−1+i+j}[Ind _{P_{M_{h}}(F_{w})}^{(B_{w}^{op})^{×}}H_{c}^{i}(I^{(h)}, Φ^{j} ⊗ L_{ξ})].

7 Igusa varieties of the second kind

We now come to the slightly less obvious generalisation of Igusa curves. More precisely we define the Igusa variety of the second kind,

J_{U^{p},m,s}^{(h)} = J^{(s)}(G^{0}/I_{U^{p},m}^{(h)}).

Then J_{U^{p},m,s}^{(h)}/I_{U^{p},m}^{(h)} × Spec k(w)^{ac} is Galois with group (O_{D_{Fw,n−h}}/w^{s})^{×} (acting on the right).

We will describe an action of the semigroup

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × D_{F_{w},n−h}^{×}

on the inverse system of the J_{U^{p},m,s}^{(h)}. Consider an element

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ) ∈ G(𝔸^{∞,p})×ℚ_{p}^{×}×(ℤ×GL_{h}(F_{w}))^{+}×∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}×D_{F_{w},n−h}^{×}.

Choose a ∈ F_{w}^{×} with c + (n − h) > (n − h)w(a) ≥ c. If (U^{p})′ ⊃ (g^{p})^{−1}U^{p}g^{p} and if for all i we have m_{i} >> m′_{i}, then we let

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ) : I_{U^{p},m}^{(h)} × Spec k(w)^{ac} ⟶ I_{(U^{p})′,m′}^{(h)} × Spec k(w)^{ac}

be the map

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}) × (Frob_{w}^{∗})^{c−w(det δ)}.

We will extend this to a compatible series of morphisms

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ) : J_{U^{p},m,s}^{(h)} ⟶ J_{(U^{p})′,m′,s′}^{(h)}

for s > s′. For this it suffices to give compatible isomorphisms

((Frob_{w}^{∗})^{c−w(det δ)}Σ_{F_{w},n−h})[w^{s′}] →^{∼} ((g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}})^{∗}G^{0})[w^{s′}]

over J_{U^{p},m,s}^{(h)}. First note that a^{−1}δ gives an isomorphism

a^{−1}δ : (Frob_{w}^{∗})^{c−w(det δ)}Σ_{F_{w},n−h} →^{∼} Σ_{F_{w},n−h}/Σ_{F_{w},n−h}[aF^{−f_{1}c}].

Also note that

a : G^{0}/G^{0}[aF^{−cf_{1}}] →^{∼} (g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}})^{∗}G^{0}.

Thus for our isomorphism

((Frob_{w}^{∗})^{c−w(det δ)}Σ_{F_{w},n−h})[w^{s′}] →^{∼} ((g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}})^{∗}G^{0})[w^{s′}]

we may simply take a^{−1}δ, followed by the map induced by the universal isomorphism

Σ_{F_{w},n−h}[w^{s}] →^{∼} G^{0}[w^{s}]

over J_{U^{p},m,s}^{(h)}, in turn followed by a. It is straight forward but tedious to check this is independent of the choice of a and does define an action.

Note that the element (F̃r^{∗}^{f_{1}}, 1) simply acts as (Fr^{∗})^{f_{1}}. (So for instance on I_{U^{p},m}^{(h)} × Spec k(w)^{ac} it acts as (Fr^{∗})^{f_{1}} = (Fr^{∗})^{f_{1}} × (Frob_{w}^{∗})^{−1}.)

We will be most interested in the part of this action which is an action of k(w)^{ac}-schemes. To this end define

(D_{F_{w},n−h}^{×} × GL_{h}(F_{w}))^{+}

to be the set of elements

(δ, γ) ∈ D_{F_{w},n−h}^{×} × GL_{h}(F_{w})

such that (w(det δ), γ) ∈ (ℤ × GL_{h}(F_{w}))^{+}. Also set

G^{(h)}(𝔸^{∞}) = G(𝔸^{∞,p}) × ℚ_{p}^{×} × D_{F_{w},n−h}^{×} × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}

and

G^{(h)}(𝔸^{∞})^{+} = G(𝔸^{∞,p}) × ℚ_{p}^{×} × (D_{F_{w},n−h}^{×} × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}.

If we embed G^{(h)}(𝔸^{∞})^{+} into

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × D_{F_{w},n−h}^{×}

by sending (g^{p}, g_{p,0}, δ, g_{w}^{et}, g_{w_{i}}) to (g^{p}, g_{p,0}, w(det δ), g_{w}^{et}, g_{w_{i}}, δ), then G^{(h)}(𝔸^{∞}) acts on the inverse system of the J_{U^{p},m,s}^{(h)} over k(w)^{ac}. We note that this action is compatible with w ∘ det : D_{F_{w},n−h}^{×} ⟶ ℤ and the action of

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}

on the inverse system of the I_{U^{p},m}^{(h)}.

If ρ is an irreducible admissible representation of D_{F_{w},n−h}^{×} over ℚ_{l}^{ac} we get a lisse etale sheaf F_{ρ}/J_{U^{p},m,s}^{(h)} coming from the restriction of ρ to

ker(O_{D_{Fw,n−h}}^{×} ⟶ (O_{D_{Fw,n−h}}/w^{s})^{×}).

If

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ) ∈ G(𝔸^{∞,p})×ℚ_{p}^{×} ×(ℤ×GL_{h}(F_{w}))^{+} ×∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} ×D_{F_{w},n−h}^{×}

defines a map

J_{U^{p},m,s}^{(h)} ⟶ J_{(U^{p})′,m′,s′}^{(h)}

then we obtain a morphism

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ) : (g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ)^{∗}(F_{ρ} ⊗ L_{ξ}) ⟶ F_{ρ} ⊗ L_{ξ}.

Note moreover that

ker(O_{D_{Fw,n−h}}^{×} ⟶ (O_{D_{Fw,n−h}}/w^{s})^{×}

acts trivially on F_{ρ} ⊗ L_{ξ} over J_{U^{p},m,s}^{(h)}.

We remark that as a sheaf (without the action of any groups) F_{ρ} only depends on ρ up to twists by unramified characters. Thus F_{ρ} ≅ F_{ρ′} for some ρ′ with finite image. The representation ρ′ can be thought of a representation of the fundamental group of I_{U^{p},m}^{(h)} and hence defines a lisse etale sheaf over I_{U^{p},m}^{(h)} which has base change to I_{U^{p},m}^{(h)} × Spec k(w)^{ac} isomorphic to F_{ρ}.

We will set

H_{c}^{i}(I^{(h)}, F_{ρ} ⊗ L_{ξ}) = lim_{→U^{p},m} H_{c}^{i}(I_{U^{p},m}^{(h)} × Spec k(w)^{ac}, F_{ρ} ⊗ L_{ξ}).

Then H_{c}^{i}(I^{(h)}, F_{ρ} ⊗ L_{ξ}) has an admissible action of

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × (D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×})

and the element (F̃r^{∗}^{f_{1}}, 1) acts trivially. Thus this action factors through the surjection from

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × (D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×})

to G^{(h)}(𝔸^{∞})/CO_{D_{Fw,n−h}}^{×} which sends

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ)

to

(g^{p}, g_{p,0}p^{f_{1}(w(det δ)−c)}, δ, g_{w}^{et}, g_{w_{i}}).

Thus we may and will consider H_{c}^{i}(I^{(h)}, F_{ρ} ⊗ L_{ξ}) as an admissible G^{(h)}(𝔸^{∞})- module. This is compatible with the action of G^{(h)}(𝔸^{∞})^{+} on the inverse system of the J_{U^{p},m,s}^{(h)} over k(w)^{ac}.

Similarly the action of O_{D_{Fw,n−h}}^{×} on Ψ_{F_{w},l,n−h,t}^{j} (resp. Ψ_{F_{w},l,n−h,t}^{j}[ρ], where ρ is an irreducible admissible representation of D_{F_{w},n−h}^{×}) and on the inverse system of the J_{U^{p},m,s}^{(h)} defines a sheaf F(Ψ_{F_{w},l,n−h,t}^{j}) (resp. F(Ψ_{F_{w},l,n−h,t}^{j}[ρ])) on each J_{U^{p},m,s}^{(h)}. If

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ) ∈ G(𝔸^{∞,p})×ℚ_{p}^{×} ×(ℤ×GL_{h}(F_{w}))^{+} ×∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} ×D_{F_{w},n−h}^{×}

defines a map

J_{U^{p},m,s}^{(h)} ⟶ J_{(U^{p})′,m′,s′}^{(h)},

and if

(δ, γ, σ) ∈ A_{F_{w},n−h}

defines a map

Ψ_{F_{w},l,n−h,t′}^{j} ⟶ Ψ_{F_{w},l,n−h,t}^{j}

then we obtain a morphism (g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, (δ, γ, σ)) :

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ)^{∗}(F(Ψ_{F_{w},l,n−h,t′}^{j}) ⊗ L_{ξ}) ⟶ F(Ψ_{F_{w},l,n−h,t}^{j}) ⊗ L_{ξ} (resp.

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ)^{∗}(F(Ψ_{F_{w},l,n−h,t′}^{j}[ρ]) ⊗ L_{ξ}) ⟶ F(Ψ_{F_{w},l,n−h,t}^{j}[ρ]) ⊗ L_{ξ}.) Note moreover that

• ker(O_{D_{Fw,n−h}}^{×} ⟶ (O_{D_{Fw,n−h}}/w^{s})^{×}) acts trivially on F(Ψ_{F_{w},l,n−h,t}^{j}) ⊗ L_{ξ} and F(Ψ_{F_{w},l,n−h,t}^{j}[ρ]) ⊗ L_{ξ} over J_{U^{p},m.s}^{(h)},

• that if Γ_{t} = ker(GL_{n−h}(O_{F,w}) → GL_{n−h}(O_{F,w}/w^{t})) then

F(Ψ_{F_{w},l,n−h,t}^{j}) →^{∼} F(Ψ_{F_{w},l,n−h,t′}^{j})^{Γ_{t}}

and

F(Ψ_{F_{w},l,n−h,t}^{j}[ρ]) →^{∼} F(Ψ_{F_{w},l,n−h,t′}^{j}[ρ])^{Γ_{t}},

• and that

F(Ψ_{F_{w},l,n−h,t}^{j}) = ⨁_{ρ} F(Ψ_{F_{w},l,n−h,t}^{j}[ρ]),

where ρ runs over a set representatives of the inertial equivalence classes of representations of D_{F_{w},n−h}^{×}.

We will set

H_{c}^{i}(I^{(h)}, F(Ψ^{j}) ⊗ L_{ξ}) = lim_{→U^{p},m,t} H_{c}^{i}(I_{U^{p},m}^{(h)} × Spec k(w)^{ac}, F(Ψ_{F_{w},l,n−h,t}^{j}) ⊗ L_{ξ})

and

H_{c}^{i}(I^{(h)}, F(Ψ^{j}[ρ])⊗L_{ξ}) = lim_{→U^{p},m,t} H_{c}^{i}(I_{U^{p},m}^{(h)}×Spec k(w)^{ac}, F(Ψ_{F_{w},l,n−h,t}^{j}[ρ])⊗L_{ξ}).

Then H_{c}^{i}(I^{(h)}, F(Ψ^{j})⊗L_{ξ}) and H_{c}^{i}(I^{(h)}, F(Ψ^{j}[ρ])⊗L_{ξ}) have admissible actions of

\begin{matrix}G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × (A_{F_{w},n−h}/O_{D_{Fw,n−h}}^{×}) ≅ \\ ≅ G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × GL_{n−h}(F_{w}) × W_{F_{w}}. \end{matrix}

Again the element (F̃r^{∗}^{f_{1}}, 1) acts trivially. We see that

H_{c}^{i}(I^{(h)}, F(Ψ^{j}) ⊗ L_{ξ}) = ⨁_{ρ} H_{c}^{i}(I^{(h)}, F(Ψ^{j}[ρ]) ⊗ L_{ξ})

where again ρ runs over a set of representatives of the inertial equivalence classes of representations of D_{F_{w},n−h}^{×}.

One also has that

F_{ρ} ⊗ Ψ_{F_{w},l,n−h,t}^{j}(ρ) ⟶^{∼} ⨁_{δ∈Δ[ρ]} δ^{∗}F(Ψ^{j}[ρ]).

Hence

H_{c}^{i}(I^{(h)}, F_{ρ} ⊗ L_{ξ}) ⊗ Ψ_{F_{w},l,n−h}^{j}(ρ) ≅ H_{c}^{i}(I^{(h)}, F(Ψ^{j}[ρ]) ⊗ L_{ξ})^{e[ρ]},

where the action of

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, γ, σ)

on H_{c}^{i}(I^{(h)}, F(Ψ^{j}[ρ]) ⊗ L_{ξ})^{e[ρ]} is compatible with the action of (g^{p}, g_{p,0}p^{f_{1}(w(det γ)−w(σ)−c)}, δ, g_{w}^{et}, g_{w_{i}}) ⊗ (γ, σ) ∈ G^{(h)}(𝔸^{∞,p}) × GL_{n−h}(F_{w}) × W_{F_{w}} on H_{c}^{i}(I^{(h)}, F_{ρ} ⊗ L_{ξ}) ⊗ Ψ_{F_{w},l,n−h}^{j}(ρ), where δ ∈ D_{F_{w},n−h}^{×} satisfies

w(det δ) = w(det γ) − w(σ).

We record this as the following lemma. Lemma 7.1 We have a natural isomorphism

H_{c}^{i}(I^{(h)}, F(Ψ^{j}) ⊗ L_{ξ})^{n−h} ≅ ⨁_{ρ}(H_{c}^{i}(I^{(h)}, F_{ρ} ⊗ L_{ξ}) ⊗ Ψ_{F_{w},l,n−h}^{j}(ρ))^{(n−h)/e[ρ]}

under which the action of

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, γ, σ)

on H_{c}^{i}(I^{(h)}, F(Ψ^{j}) ⊗ L_{ξ}) is compatible with the action of (g^{p}, g_{p,0}p^{f_{1}(w(det γ)−w(σ)−c)}, δ, g_{w}^{et}, g_{w_{i}}) ⊗ (γ, σ) ∈ G^{(h)}(𝔸^{∞,p}) × GL_{n−h}(F_{w}) × W_{F_{w}} on H_{c}^{i}(I^{(h)}, F_{ρ} ⊗ L_{ξ}) ⊗ Ψ_{F_{w},l,n−h}^{j}(ρ), where δ ∈ D_{F_{w},n−h}^{×} satisfies

w(det δ) = w(det γ) − w(σ).

By a closed point x_{∞} of J_{U^{p},m,∞}^{(h)} we shall mean a compatible system of closed points x_{s} of J_{U^{p},m,s}^{(h)} as s varies. If x is a closed point of I_{U^{p},m}^{(h)}×Spec k(w)^{ac} then the following are equivalent

• the choice of an isomorphism j : Σ_{F_{w},n−h} →^{∼} G_{x}^{0}

• and the choice a closed point x_{∞} of J_{U^{p},m,∞}^{(h)} above x.

We will write j_{x_{∞}} for the isomorphism corresponding to x_{∞}. Note that if

(g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}) ∈ G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×},

if δ ∈ D_{F_{w},n−h}^{×} and if e = c − w(det δ) then

((g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}) × (Frob_{w}^{e})^{∗} × δ)(j_{x_{∞}}) = j_{(g^{p},g_{p,0},c,g_{w}^{et},g_{w_{i}},δ)x_{∞}}.

If x_{∞} is a closed point of J_{U^{p},m,∞}^{(h)} above a closed point x of I_{U^{p},m}^{(h)} × Spec k(w)^{ac} then we set

F(Ψ_{F_{w},l,n−h,t}^{j})_{x_{∞}} = lim_{→s} F(Ψ_{F_{w},l,n−h,t}^{j})_{x_{s}}.

We then have canonical isomorphisms (the composite depending on the choice of x_{∞} above x)

F(Ψ_{F_{w},l,n−h,t}^{j})_{x} →^{∼} F(Ψ_{F_{w},l,n−h,t}^{j})_{x_{∞}} →^{∼} Ψ_{F_{w},l,n−h,t}^{j}.

Moreover for (g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, (δ, γ, σ)) in

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (ℤ × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × A_{F_{w},n−h},

for

y = ((g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}) × (Frob_{w}^{∗})^{c−w(det δ)})x

and for

y_{∞} = (g^{p}, g_{p,0}, c, g_{w}^{et}, g_{w_{i}}, δ)x_{∞}

we have a commutative diagram

F(Ψ_{F_{w},l,n−h,t′}^{j})_{y}  ⟶^{(g^{p},g_{p,0},c,g_{w}^{et},g_{w_{i}},γ,σ)} F(Ψ_{F_{w},l,n−h,t}^{j})_{x}
↓  ↓
F(Ψ_{F_{w},l,n−h,t′}^{j})_{y_{∞}} ⟶^{(g^{p},g_{p,0},c,g_{w}^{et},g_{w_{i}},(δ,γ,σ))} F(Ψ_{F_{w},l,n−h,t}^{j})_{x_{∞}}
↓  ↓
Ψ_{F_{w},l,n−h,t′}^{j}  ⟶^{(δ,γ,σ)}  Ψ_{F_{w},l,n−h,t}^{j}.

The following proposition is of key importance for us. In the original version of this paper we reduced its proof to an abstract result on formal vanishing cycles which Vladimir Berkovich kindly proved for us. This result with Berkovich’s proof is reproduced in his appendix to this paper. In this version of the paper we have found it simpler to incorporate Berkovich’s argument directly into this paper. Thus the latter half of the proof given below is entirely due to to Berkovich.

Proposition 7.2 There is a unique homomorphism

κ : Φ^{j}(t) ⟶ F(Ψ_{F_{w},l,n−h,t}^{j})

over I_{U^{p},m}^{(h)} × Spec k(w)^{ac} such that if x_{∞} is a closed point of J_{U^{p},m,∞}^{(h)} above a closed point x of I_{U^{p},m}^{(h)} × Spec k(w)^{ac}, then

κ : Φ^{j}(t)_{x} ⟶ F(Ψ_{F_{w},l,n−h,t}^{j})_{x}

coincides with the inverse of the composite

F(Ψ_{F_{w},l,n−h,t}^{j})_{x} ⟶ F(Ψ_{F_{w},l,n−h,t}^{j})_{x_{∞}} ⟶ Ψ_{F_{w},l,n−h,t}^{j} ⟶^{(j_{x∞}^{∗})^{∗}} Φ^{j}(t)_{x}.

Proof: Because of the uniqueness it suffices to work locally on I_{U^{p},m}^{(h)} × Spec k(w)^{ac}. Thus let W ⊂ I_{U^{p},m}^{(h)} × Spec k(w)^{ac} be an open subset such that W^{∧}, the restriction of (I_{U^{p},m}^{(h)})^{∧}(t) ×_{Spf O_{F,w}} Spf OF̂_{w}^{nr} to W, is affine. For any positive integer s let W_{s} denote the pull back of W to J_{U^{p},m,s}^{(h)} and let W_{s}^{∧}/W^{∧} denote the unique etale cover with reduced subschemes W_{s}/W.

Choose a lifting of W/k(w)^{ac} to a smooth scheme of finite type Y/OF̂_{w}^{nr}. Set

Y^{∧} = Y_{W}^{∧} ×_{Spf OF̂{nr \atop w}} Spf R_{F_{w},n−h,t},

and more generally

Y_{s}^{∧} = (Y_{W}^{∧})_{s} ×_{Spf OF̂{nr \atop w}} Spf R_{F_{w},n−h,t},

where (Y_{W}^{∧})_{s}/Y_{W}^{∧} is the unique etale cover with reduced subschemes W_{s}/W. Also set

Y^{∧}\{N\} = Y_{W}^{∧} ×_{Spf OF̂{nr \atop w}} Spf R_{F_{w},n−h,t}/𝔪_{F_{w},n−h,t}^{N},

and more generally

Y_{s}^{∧}\{N\} = (Y_{W}^{∧})_{s} ×_{Spf OF̂{nr \atop w}} Spf R_{F_{w},n−h,t}/𝔪_{F_{w},n−h,t}^{N}.

Note also that the formal vanishing cycles for Y_{s}^{∧} are just the constant sheaves Ψ_{F_{w},l,n−h,t}^{j}.

Choose N(s) as in lemma 3.4 for the triple F_{w}, n − h, t. Then the action of (1 + w^{s}O_{D_{Fw,n−h}}) on Σ_{F_{w},n−h} lifts to an action on Σ̃_{F_{w},n−h} ×_{R_{Fw,n−h,t}} R_{F_{w},n−h,t}/𝔪_{R_{Fw,n−h,t}}^{N(s)}. Moreover for any positive integer a ≥ t the action of

(1 + w^{s−t+a}O_{D_{Fw,n−h}}) on Σ̃_{F_{w},n−h}[w^{a}] ×_{R_{Fw,n−h,t}} R_{F_{w},n−h,t}/𝔪_{R_{Fw,n−h,t}}^{N(s)} is trivial. Thus (1 + w^{s}O_{D_{Fw,n−h}})/(1 + w^{s+a−t}O_{D_{Fw,n−h}}) acts diagonally on

Σ̃_{F_{w},n−h,t}[w^{a}] × Y_{s+a−t}^{∧}{N(s)}.

As Y_{s+a−t}^{∧}{N(s)}/Y_{s}^{∧}{N(s)} is finite, etale and Galois with Galois group

(1 + w^{s}O_{D_{Fw,n−h}})/(1 + w^{s+a−t}O_{D_{Fw,n−h}}),

the quotient is a finite, flat group scheme H_{s}^{0}[w^{a}]/Y_{s}^{∧}{N(s)}. The direct system of the H^{0}[w^{a}] define a formal Barsotti-Tate O_{F,w}-module H_{s}^{0}/Y_{s}^{∧}{N(s)}. Note that H_{s}^{0}[w^{t}] ≅ Σ̃_{F_{w},n−h,t}[w^{t}], and so H_{s}^{0} inherits a Drinfeld w^{t}-structure. Also note that over W_{s+a−t} we have an isomorphism

H_{s}^{0}[w^{a}] ≅ Σ_{F_{w},n−h}[w^{a}] →^{∼} G^{0}[w^{a}].

The composite isomorphism descends to an isomorphism over W_{s}. Thus we see that over W_{s} we get a canonical isomorphism between G^{0} and H_{s}^{0}. More generally for s′ > s we get a canonical isomorphism between the pull back of H_{s}^{0} to Y_{s′}^{∧}{N(s)} and the restriction of H_{s′}^{0} to Y_{s′}^{∧}{N(s)}.

Let H_{s}^{et}/Y_{s}^{∧} denote the unique lifting of G^{et}/W_{s} to an ind-etale Barsotti-Tate O_{F,w}-module. By corollary 2.10 we can recursively find an extension of Barsotti-Tate O_{F,w}-modules

(0) ⟶ H_{s}^{0} ⟶ H_{s} ⟶ H_{s}^{et} ⟶ (0)

over Y_{s}^{∧}{N(s)} which restricts over Y_{s}^{∧}{N(s − 1)} to the pull back

(0) ⟶ H_{s−1}^{0} ⟶ H_{s−1} ⟶ H_{s−1}^{et} ⟶ (0).

We will simply write H for any H_{s}.

From the universal property of (I_{U^{p},m}^{(h)})^{∧}(t) we obtain a unique morphism of formal schemes over Spf OF̂_{w}^{nr},

κ_{s}^{∗} : Y_{s}^{∧}{N(s)} ⟶ (I_{U^{p},m}^{(h)})^{∧}(t) × Spf OF̂_{w}^{nr}

which extends the natural map W_{s} → I_{U^{p},m}^{(h)} × Spec k(w)^{ac} and such that G pulls back to H and the universal Drinfeld w^{t}-structure on G^{0} pulls back to the Drinfeld w^{t}-structure we have just defined on H^{0}. From the definition of W_{s}^{∧} and lemma 6.2 we see that κ_{s}^{∗} lifts to a unique morphism

κ_{s}^{∗} : Y_{s}^{∧}{N(s)} ⟶ W_{s}^{∧}

which extends the identity map on W_{s} and such that G pulls back to H and the universal Drinfeld w^{t}-structure on G^{0} pulls back to the Drinfeld w^{t}-structure we have just defined on H^{0}. Note that

Y_{s}^{∧}{N(s)} ⟶^{κ_{s}^{∗}} W_{s}^{∧}
↑  ↑
Y_{s+1}^{∧}{N(s)} ⟶^{κ_{s+1}^{∗}} W_{s+1}^{∧}

commutes. Note also that if x is a closed point of W_{s} and if x_{∞} is a closed point of J_{U^{p},m,∞}^{(h)} lying above x then

(Y_{s}^{∧}{N(s)})_{x}^{∧}  ⟶^{κ_{s}^{∗}}  (W_{s}^{∧})_{x}^{∧}
↓  ↓
Spf R_{F_{w},n−h,t}/𝔪_{R_{Fw,n−h,t}}^{N(s)} ↪ Spf R_{F_{w},n−h,t}

commutes, where the right hand vertical arrow is j_{x_{∞}}^{∗}.

The rest of the argument is entirely due to Vladimir Berkovich, to whom we are extremely grateful. By his lemma II.5 we see that for every positive integer N we can find a positive integer s such that N(s) ≥ N and a morphism

κ(N)^{∗} : Y_{s}^{∧} ⟶ W_{s}^{∧}

such that for each closed point x of W_{s} and each closed point x_{∞} of J_{U^{p},m,∞}^{(h)} lying above x

(Y_{s}^{∧}{N})_{x}^{∧}  ⟶^{κ(N)^{∗}}  (W_{s}^{∧})_{x}^{∧}
↓  ↓
Spf R_{F_{w},n−h,t}/𝔪_{R_{Fw,n−h,t}}^{N} ↪ Spf R_{F_{w},n−h,t}

commutes, where the right hand vertical arrow is j_{x_{∞}}^{∗}.

We will let Ψ^{j}(ℤ/l^{r}ℤ) denote the j^{th} vanishing cycle sheaf constructed for the constant sheaf ℤ/l^{r}ℤ on Spf R_{F_{w},n−h,t} and Φ^{j}(ℤ/l^{r}ℤ) the j^{th} vanishing cycle sheaf constructed for the constant sheaf ℤ/l^{r}ℤ on W_{s}^{∧}. Thus

Ψ_{F_{w},l,n−h,t}^{j} ≅ (lim_{←r} Ψ^{j}(ℤ/l^{r}ℤ)) ⊗_{ℤ_{l}} ℚ_{l}^{ac}

and

Φ^{j}(t) ≅ (lim_{←r} Φ^{j}(ℤ/l^{r}ℤ)) ⊗_{ℤ_{l}} ℚ_{l}^{ac}.

By theorem 4.1 of [Berk3], for any positive integer r we may choose a positive integer N such that any two morphisms of formal schemes over Of̂_{w}^{nr},

Spf R_{F_{w},n−h,t}[[X_{1}, ..., X_{h}]] ⟶ Spf R_{F_{w},n−h,t}

which agree on Spf (R_{F_{w},n−h,t}/𝔪_{R_{Fw,n−h,t}}^{N})[[X_{1}, ..., X_{h}]] induce the same map on vanishing cycles

Ψ^{j}(ℤ/l^{r}ℤ) ⟵ Ψ^{j}(ℤ/l^{r}ℤ).

As the formal completion of any W_{s}^{∧} or Y_{s}^{∧} at a closed point is isomorphic to Spf R_{F_{w},n−h,t}[[X_{1}, ..., X_{h}]] we see that we can find a positive integer s (still perhaps depending on r) and a morphism of constructible sheaves over W_{s},

κ : Φ^{j}(ℤ/l^{r}ℤ) ⟶ Ψ^{j}(ℤ/l^{r}ℤ),

such that for any closed point x of W_{s} and any closed point x_{∞} of J_{U^{p},m,∞}^{(h)} above x the morphism

κ : Φ^{j}(ℤ/l^{r}ℤ)_{x} ⟶ Ψ^{j}(ℤ/l^{r}ℤ)

coincides with the inverse of (j_{x_{∞}}^{∗})^{∗}. In particular we see that κ is an isomorphism.

Moreover if δ ∈ O_{D_{Fw,n−h}}^{×} we see that the natural map δ^{∗}Φ^{j}(ℤ/l^{r}ℤ) →^{∼} Φ^{j}(ℤ/l^{r}ℤ) (which arises as Φ^{j}(ℤ/l^{r}ℤ) is a pull back from W) corresponds under κ to the composite of the natural map δ^{∗}Ψ^{j}(ℤl^{r}ℤ) →^{∼} Ψ^{j}(ℤ/l^{r}ℤ) (which arises as Φ^{j}(ℤ/l^{r}ℤ) is a pull back from Spec k(w)^{ac}) with the automorphism δ of Ψ^{j}(ℤ/l^{r}ℤ) (which arises from the action of O_{D_{Fw,n−h}}^{×} on Spf R_{F_{w},n−h,t}). (This can be checked by working on stalks and using the commutativity of the diagram

Φ^{j}(ℤ/l^{r}ℤ)_{δx} ⟶^{(j_{δx∞}^{∗})^{∗}} Ψ^{j}(ℤ/l^{r}ℤ)
↓  ↓
Φ^{j}(ℤ/l^{r}ℤ)_{x} ⟶^{(j_{x∞}^{∗})^{∗}} Ψ^{j}(ℤ/l^{r}ℤ),

where the right hand vertical arrow is δ.)

We will let F(Ψ^{j}(ℤ/l^{r}ℤ)) denote the etale sheaf over W obtained by descending Ψ^{j}(ℤ/l^{r}ℤ)/W_{s} by the diagonal action of (O_{D_{Fw,n−h}}/w^{s})^{×}. Thus

F(Ψ_{F_{w},l,n−h,t}^{j}) ≅ (lim_{←r} F(Ψ^{j}(ℤ/l^{r}ℤ))) ⊗_{ℤ_{l}} ℚ_{l}^{ac}.

We see that κ descends to an isomorphism

κ : Φ^{j}(ℤ/l^{r}ℤ) ⟶ F(Ψ^{j}(ℤ/l^{r}ℤ))

over W such that for any closed point x of W and any closed point x_{∞} of J_{U^{p},m,∞}^{(h)} above x the morphism

κ : Φ^{j}(ℤ/l^{r}ℤ)_{x} ⟶ F(Ψ^{j}(ℤ/l^{r}ℤ))_{x}

coincides with the inverse of

F(Ψ^{j}(ℤ/l^{r}ℤ))_{x} →^{∼} F(Ψ^{j}(ℤ/l^{r}ℤ))_{x_{s}} ≅ Ψ^{j}(ℤ/l^{r}ℤ) ⟶^{(j_{x∞}^{∗})^{∗}} Φ_{x}^{j},

where x_{s} is the image in W_{s} of x_{∞}. By looking at stalks we see that the morphisms κ are compatible as r varies and hence we can glue the morphisms κ to give the map whose existence is asserted in the proposition. □

The next two corollaries are checked by working on stalks.

Corollary 7.3 The homomorphism κ in the proposition is an isomorphism.

Corollary 7.4 Suppose that g = (g^{p}, g_{p,0}, g_{w}^{0}, g_{w}^{et}, g_{w_{i}}) is an element of

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×},

and that

[g] : I_{U^{p},m}^{(h)} ⟶ I_{(U^{p})′,m′}^{(h)}.

Suppose also that (g_{w}^{0}, δ, σ) ∈ A_{F_{w},n−h} and that t >> t′. Then

([g] × (Frob_{w}^{w(σ)})^{∗})^{∗}Φ^{j}(t′)  ⟶^{g×σ}  Φ^{j}(t)
↓  ↓
([g] × (Frob_{w}^{w(det g_{w}^{0})−w(det δ)})^{∗})^{∗}F(Ψ_{F_{w},l,n−h,t′}^{j}) ⟶^{[g]×(g_{w}^{0},δ,σ)} F(Ψ_{F_{w},l,n−h,t})

commutes (where the vertical arrows are induced by κ).

The next three corollaries follow easily from the previous one.

Corollary 7.5 Under κ the homomorphism F̃r_{ϖ}^{∗}^{f_{1}(n−h)} × 1 from

H_{c}^{i}(I_{U^{p},m}^{(h)} × Spec k(w)^{ac}, Φ^{j}(t) ⊗ L_{ξ})

to

H_{c}^{i}(I_{U^{p},m}^{(h)} × Spec k(w)^{ac}, Φ^{j}(t) ⊗ L_{ξ})

corresponds to the homomorphism 1 × (ϖ^{−1}, ϖ^{−1}, 1) from

H_{c}^{i}(I_{U^{p},m}^{(h)} × Spec k(w)^{ac}, F(Ψ_{F_{w},l,n−h,t}^{j}) ⊗ L_{ξ})

to

H_{c}^{i}(I_{U^{p},m}^{(h)} × Spec k(w)^{ac}, F(Ψ_{F_{w},l,n−h,t}^{j}) ⊗ L_{ξ}).

Hence lemma 6.6 follows. Corollary 7.6 κ induces an isomorphism

H_{c}^{i}(I^{(h)}, Φ^{j} ⊗ L_{ξ}) →^{∼} H_{c}^{i}(I^{(h)}, F(Ψ^{j}) ⊗ L_{ξ}).

Moreover if g = (g^{p}, g_{p,0}, g_{w}^{0}, g_{w}^{et}, g_{w_{i}}) is an element of

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}

and if σ ∈ W_{F_{w}}, then the action of g × σ on the left hand side corresponds to the action of [g] × g_{w}^{0} × σ on the right hand side. Corollary 7.7 The action of

G(𝔸^{∞,p}) × ℚ_{p}^{×} × (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))^{+} × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}

on H_{c}^{i}(I^{(h)}, Φ^{j} ⊗ L_{ξ}) extends uniquely to an action of

G(𝔸^{∞,p}) × ℚ_{p}^{×} × GL_{n−h}(F_{w}) × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}.

Combining these with lemma 7.1 we get the following corollary. Corollary 7.8 We have an isomorphism

H_{c}^{i}(I^{(h)}, Φ^{j} ⊗ L_{ξ})^{n−h} →^{∼} ⨁_{ρ}(H_{c}^{i}(I^{(h)}, F_{ρ} ⊗ L_{ξ}) ⊗ Ψ_{F_{w},l,n−h}^{j}(ρ))^{(n−h)/e[ρ]},

where ρ runs over irreducible admissible representations of D_{F_{w},n−h}^{×} up to inertial equivalence. Moreover if (g^{p}, g_{p,0}, g_{w}^{0}, g_{w}^{et}, g_{w_{i}}, σ) is an element of

G(𝔸^{∞,p}) × ℚ_{p}^{×} × GL_{n−h}(F_{w}) × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×} × W_{F_{w}},

then the action of (g^{p}, g_{p,0}, g_{w}^{0}, g_{w}^{et}, g_{w_{i}}, σ) on the left hand side corresponds to the action of

(g^{p}, g_{p,0}p^{−f_{1}w(σ)}, w(δ), g_{w}^{et}, g_{w_{i}}) × (g_{w}^{0}, σ)

on H_{c}^{i}(I^{(h)}, F_{ρ} ⊗ L_{ξ}) ⊗ Ψ_{F_{w},l,n−h}^{j}(ρ), where δ ∈ D_{F_{w},n−h}^{×} satisfies

w(det δ) = w(det g_{w}^{0}) − w(σ).

We will let d_{h} denote the homomorphism

\begin{matrix}GL_{n−h}(F_{w}) × W_{F_{w}} & ⟶ & G^{(h)}(𝔸^{∞})/O_{D_{Fw,n−h}}^{×} \\ (g_{w}^{0}, σ) & ⟼ & (1, p^{−f_{1}w(σ)}, δ, 1, 1), \end{matrix}

where

w(det δ) = w(det g_{w}^{0}) − w(σ).

If we set

[H_{c}(I^{(h)}, F_{ρ} ⊗ L_{ξ})] = ∑_{i}(−1)^{h−i}[H_{c}^{i}(I^{(h)}, F_{ρ} ⊗ L_{ξ})],

then we can combine corollaries 6.8 and 7.8 to obtain the following theorem.

Theorem 7.9 For h = 0, ..., n − 1 choose homomorphisms j_{h}^{0} : Λ_{11} → → O_{F,w}^{n−h} and j_{h}^{et} : Λ_{11} → → O_{F,w}^{h} such that j_{h}^{0} ⊕ j_{h}^{et} is an isomorphism. Let M_{h} = ker j_{h}^{et}. Then we have an equality of virtual G(𝔸^{∞}) × W_{F_{w}}-modules

[H(X, L_{ξ})^{ℤ_{p}^{×}}] = ∑_{h=0}^{n−1} ∑_{ρ} e[ρ]^{−1}Ind _{P_{M_{h}}(F_{w})}^{(B_{w}^{op})^{×}}([H_{c}(I^{(h)}, F_{ρ} ⊗ L_{ξ})] ∗_{d_{h}} [Ψ_{F_{w},l,n−h}(ρ)]),

where ρ runs over representatives of the inertial equivalence classes of irreducible admissible representations of D_{F_{w},n−h}^{×}

(See the discussion before lemma 3.7 for the definition of d_{n}. See appendix I for the definition of ∗_{d_{n}}.)

8 k(w)^{ac} points.

The purpose of this section is to give a reasonably explicit description of the inverse system of sets J_{U^{p},m,s}^{(h)}(k(w)^{ac}) together with its action of G^{(h)}(𝔸^{∞}). This we will do by combining the theory of Honda and Tate (which describes simple abelian varieties over k(w)^{ac} up to isogeny - see [Tat1]) with some Galois cohomology calculations which closely follow work of Kottwitz (and which allows us to describe the possible polarisations on these abelian varieties - see [Ko3] and [Ko4]).

If U^{p} is an open compact subgroup of G(𝔸^{∞,p}), if m ∈ ℤ_{≥0}^{r} and if s ∈ ℤ_{≥0} then set

U^{p}(m, s) = U^{p}×ℤ_{p}^{×}×(1+w^{s}O_{D_{Fw,n−h}})×(1+w^{m_{1}}M_{h}(O_{F,w}))×∏_{i=2}^{r}(1+w_{i}^{m_{i}}O_{B,w_{i}}^{op}),

an open compact subgroup of G^{(h)}(𝔸^{∞}).

Also set

J^{(h)}(k(w)^{ac}) = lim_{⟵U^{p},m,s} J_{U^{p},m,s}^{(h)}(k(w)^{ac}).

This set has a natural right action of G^{(h)}(𝔸^{∞})^{+} and we see that

J_{U^{p},m,s}^{(h)}(k(w)^{ac}) = J^{(h)}(k(w)^{ac})/U^{p}(m, s).

Also if x ∈ J^{(h)}(k(w)^{ac}) then we define the stalk of F_{ρ} ⊗ L_{ξ} to be the direct limit of its stalks at the images of x in each J_{U^{p},m,s}^{(h)}. For each such x there is a canonical isomorphism

(F_{ρ} ⊗ L_{ξ})_{x} →^{∼} ρ ⊗ ξ

such that for any g ∈ G^{(h)}(𝔸^{∞})^{+} we have a commutative diagram

(F_{ρ} ⊗ L_{ξ})_{xg}  ⟶^{g} (F_{ρ} ⊗ L_{ξ})_{x}
↓  ↓
ρ ⊗ ξ  ⟶^{ρ(g_{w}^{0})⊗ξ(g_{l})} ρ ⊗ ξ.

As a first step we will see that when considering k(w)^{ac}-points one can (as in characteristic zero) work with abelian varieties up to isogeny, rather than up to prime to p-isogeny. To this end we have the following lemma, whose proof is straight forward. (The main point being that if x is a closed geometric point of J_{U^{p},m,s}^{(h)} then G_{x} ≅ G_{x}^{0} × G_{x}^{et}.)

Lemma 8.1 1. J^{(h)}(k(w)^{ac}) is in bijection with equivalence classes of (r +

6)-tuples

(A, λ, i, η^{p}, η_{p,0}, η_{w}^{0}, η_{w}^{e}, η_{w_{i}}),

where • A/k(w)^{ac} is an abelian variety of dimension dn^{2};

• λ : A → A^{∨} is a polarisation;

• i : B ↪ End (A) ⊗_{ℤ} ℚ such that – for all b ∈ B we have λ ∘ i(b) = i(b^{∗})^{∨} ∘ λ,

– for i > 1 the ring O_{F,w_{i}} acts trivially on Lie A,

– Lie A ⊗_{O_{F,p}} O_{F,w} is a one-dimensional k(w)^{ac}-vector space;

• η^{p} : V ⊗_{ℚ}𝔸^{∞,p} →^{∼} V^{p}A is an isomorphism of B⊗_{ℚ}𝔸^{∞,p}-modules under which the standard pairing, ( , ), on V corresponds to a (𝔸^{∞,p})^{×} multiple of the λ-Weil pairing on V^{p}A;

• η_{p,0} : ℚ_{p} →^{∼} ℚ_{p}(1);

• η_{w}^{0} : Σ_{n−h} ⟶ εA[w^{∞}]^{0} is an isogeny;

• η_{w}^{e} : F_{w}^{h} →^{∼} εV_{w}A;

• for i > 1, η_{w_{i}} : Λ_{i} ⊗_{ℤ_{p}} ℚ_{p} →^{∼} V_{w_{i}}A is an isomorphism of B_{w_{i}}-

modules. Here we call two such (r + 6)-tuples, (A, λ, i, η^{p}, η_{p,0}, η_{w}^{0}, η_{w}^{e}, η_{w_{i}}) and (A′, λ′, i′, (η^{p})′, η′_{p,0}, (η_{w}^{0})′, (η_{w}^{e})′, η′_{w_{i}}), equivalent if there is an isogeny α : A → A′, γ ∈ ℚ^{×} and a ∈ ℤ_{p}^{×} such that

• γλ = α^{∨} ∘ λ′ ∘ α,

• α ∘ i(b) = i′(b) ∘ α for all b ∈ B,

• (η^{p})′ ∘ α = η^{p},

• γaη_{p,0} = η′_{p,0},

• (η_{w}^{0})′ ∘ α = η_{w}^{0},

• (η_{w}^{e})′ ∘ α = η_{w}^{e},

• (η_{w_{i}})′ ∘ α = η_{w_{i}}.

2. Under this bijection the action of

(g^{p}, g_{p,0}, g_{w}^{0}, g_{w}^{e}, g_{w_{i}}) ∈ G^{(h)}(𝔸^{∞})^{+}

on J^{(h)}(k(w)^{ac}) corresponds to the map which sends

(A, λ, i, η^{p}, η_{p,0}, η_{w}^{0}, η_{w}^{e}, η_{w_{i}})

to

(A, λ, i, η^{p} ∘ g^{p}, η_{p,0} ∘ g_{p,0}, η_{w}^{0} ∘ g_{w}^{0}, η_{w}^{e} ∘ g_{w}^{e}, η_{w_{i}} ∘ g_{w_{i}}).

3. In particular the action of G^{(h)}(𝔸^{∞})^{+} on J^{(h)}(k(w)^{ac}) extends to an action of G^{(h)}(𝔸^{∞}).

In the rest of this section we will give a group theoretic description of J^{(h)}(k(w)^{ac}). We start with an application of the theory of Honda and Tate (see [Tat1]).

Lemma 8.2 1. There is a bijection between isogeny classes of pairs (A, i) where

• A/k(w)^{ac} is an abelian variety of dimension dn^{2};

• i : B ↪ End (A) ⊗_{ℤ} ℚ such that

– for i > 1 the ring O_{F,w_{i}} acts trivially on Lie A,

– Lie A ⊗_{O_{F,p}} O_{F,w} is a one-dimensional k(w)^{ac}-vector space;

• εA[w^{∞}]^{e} has height h;

and pairs (M, w̃) where

• M/F is a CM field extension which is embeddable into B over F,

• w̃ is a place of M above w such that

[Mw̃ : F_{w}]n = [M : F](n − h),

• there is no intermediate field M ⊃ N ⊃ F such that w̃|_{N} is inert in M.

2. Suppose that (A, i) and (M, w̃) correspond. Then C = End _{B}^{0}(A) is the division algebra with centre M and invariants as follows.

• If x is a place of M not dividing ww^{c} then inv_{x}(C) = inv_{x}(B^{op} ⊗_{F} M).

• If x is a place of M which divides ww^{c} but not w̃w̃^{c} then inv_{x}(C) = 0.

• inv w̃(C) = [M : F]/n.

• inv _{}_{w̃^{c}}(C) = −[M : F]/n.

Moreover

dim_{M}(End_{B}(A) ⊗_{ℤ} ℚ) = (n/[M : F])^{2}.

Proof: In the proof of this lemma we will use results from appendix III without comment.

We will first show that any A as in the first part of the lemma is a simple object in the category of abelian varieties up to isogeny with B-action. To this end choose a simple factor A′ of A in the category of abelian varieties up to isogeny with B-action. Suppose that A′ corresponds to a p-adic type over F with minimal representative (M, η). Choose a place x of F such that B_{x} is a division algebra and such that x |̸u^{c}. Let y be a place of M above x. If x|p then (as x|u but x ≠ w) A[y^{∞}] is etale and hence η_{y} = 0. In any case we see that, if C′ = End _{B}^{0}(A′), then

inv_{y}(C′) = −[M_{y} : F_{w}]inv_{x}(B),

and so

[C′ : M]^{1/2} ≥ n/[M : F].

Thus

dim A′ ≥ dn^{2},

which implies A = A′, as desired.

Thus let A correspond to a p-adic type over F with minimal representative (M, η). Note that if y is a place of M dividing u but not dividing w then A[y^{∞}] is etale and hence η_{y} = 0. Moreover as A[w^{∞}]^{0} is a simple object in the category of Barsotti-Tate groups up to isogeny with B_{w}-action we see that there is a unique place w̃ of M above w with ηw̃ ≠ 0. Then A[w^{∞}]^{0} = A[w̃^{∞}] and as this Barsotti-Tate group has height [F_{w} : ℚ_{p}]n(n − h) we see that

[Mw̃ : F_{w}][C : M]^{1/2} = n − h.

Also as the Newton polygon of A[w^{∞}]^{0} has Newton polygon which is pure of slope 1/([F_{w} : ℚ_{p}](n − h)) we see that

ηw̃ = e_{w̃/w}/((n − h)f_{w/p}).

Hence we see that

• inv w̃C = [Mw̃ : F_{w}]/(n − h) = [C : M]^{−1/2},

• inv _{}_{w̃^{c}}C = −[Mw̃ : F_{w}]/(n − h) = −[C : M]^{−1/2}, • and for any other place x of M we have

inv_{x}C = −[M_{x} : F_{x}]inv_{x}(B).

As

dn^{2} = dim A = [M : ℚ]n[C : M]^{1/2}/2

we see that

n = [M : F][C : M]^{1/2}

and hence that

(n − h)[M : F] = n[Mw̃ : F_{w}].

Because we can find an extension N : M such that N splits B and [N : F]^{2} = [B : F], we see that M embeds in B over F. Finally as (M, η) was chosen minimal we see that there is no intermediate field M ⊃ N ⊃ F such that w̃|_{N} is inert in M.

Conversely if (M, w̃) is as in the theorem we consider the p-adic type over F, (M, η) where

• ηw̃ = e_{w̃/w}/((n − h)f_{w/p}),

• η_{x} = 0 for any other place x|u of M,

• η_{}_{w̃^{c}} = e_{w̃/p}(1 − 1/((n − h)[F_{w} : ℚ_{p}])),

• η_{x} = e_{x/p} for any other place x|u^{c} of M.

Let (A, i) be the corresponding abelian variety with B-action. Again using the results listed in appendix III we see that (A, i) has the properties listed in the lemma. □

We now want to add polarisations to the picture. To this end we follow the approach of Kottwitz, [Ko3]. The next lemma is presumably the analogue in our language of “the vanishing of the Kottwitz invariant”.

Lemma 8.3 Keep the notation of the last lemma and suppose that (A, i) and (M, w̃) correspond. Then we can find

• a polarisation λ_{0} : A → A^{∨} such that the λ_{0}-Rosati involution preserves B ⊗ M and induces the involution ∗ ⊗ c on it,

• and a finitely generated B ⊗ M-module W_{0} together with an alternating

pairing

⟨ , ⟩_{0} : W_{0} × W_{0} ⟶ ℚ

which is ∗ ⊗ c-Hermitian such that

1. there is an isomorphism of B ⊗ M ⊗ 𝔸^{∞,p}-modules

W_{0} ⊗ 𝔸^{∞,p} →^{∼} V^{p}A

which takes ⟨ , ⟩_{0} to an (𝔸^{∞,p})^{×}-multiple of the λ_{0}-Weil pairing on V^{p}A,

2. and there is an isomorphism of B ⊗ ℝ-modules

W_{0} ⊗ ℝ →^{∼} V ⊗ ℝ

which takes ⟨ , ⟩_{0} to a ℝ^{×} multiple of our standard pairing ( , ) on V ⊗ ℝ.

Proof: By lemma 9.2 of [Ko3] there is a polarisation λ_{0} : A → A^{∨} such that the λ_{0}-Rosati involution preserves B ⊗_{F} M and acts on it as ∗ ⊗ c. Let C = End_{B}(A) ⊗_{ℤ} ℚ and let ‡_{0} denote the λ_{0}-Rosati involution restricted to C. We see from lemma 8.2 that C is a division algebra with centre M and that ‡_{0}|_{M} = c.

The first step of the proof will be to show that, up to isogeny, we can lift A with its B ⊗_{F} M-action and its polarisation over O_{F_{w}^{ac}}.

Let M^{+} denote the maximal totally real subfield of M. For a|[C : M]^{1/2} we will let X_{a} ⊂ C denote the locally closed M^{+}-subvariety of semi-simple elements δ such that δ^{‡^{0}} = δ and the characteristic polynomial of δ over M is an a^{th}, but no higher, power. (Note that if a polynomial over F^{+} is a a^{th} power over (M^{+})^{ac} it is already one over M^{+}.) Then

C^{‡_{0}=1} = ∐ X_{a}(M^{+}).

Computing over (M^{+})^{ac} we see that

dim X_{a} = [C : M] + (a − [C : M]/a).

As C^{‡_{0}=1} is an affine space over M^{+} of dimension [C : M] and as for a ≠ [C : M]^{1/2} we have dim X_{a} < [C : M], we see that we can find an element

δ ∈ C^{‡_{0}=1} − ∐_{a≠[C:M]^{1/2}} X_{a}(M^{+}).

Set N = M(δ). Then N is a maximal subfield of C, which is preserved by ‡_{0}, which is a CM-field and which satisfies ‡_{0}|_{N} = c. Set N^{+} = M^{+}(δ) the maximal totally real subfield of N.

We see that N also splits B (as it does so locally at all places of F by lemma 8.2). Choose an isomorphism α : B ⊗_{F} N →^{∼} M_{n}(N) and let ∗_{α} denote the involution α ∘ (∗ ⊗ c) ∘ α^{−1} on M_{n}(N). If x ∈ M_{n}(N) we will define x′ ∈ M_{n}(M) by (x′)_{ij} = x_{ji}^{c}. Then we have x^{∗_{α}} = ax′a^{−1} for some a ∈ GL_{n}(N) with a′a^{−1} ∈ N^{×}. We see that (a′a^{−1})(a′a^{−1})^{c} = (a′)^{−1}(a′a^{−1})a = 1 and so by Hilbert’s theorem 90 a′a^{−1} = γ/γ^{c} for some γ ∈ N^{×}. Replacing a by γa we see that we may suppose that a′ = a. If we compose α with conjugation by b in GL_{n}(N) then we change a to bab′. Thus by suitable choice of α we may suppose that

α(x^{∗⊗c}) = aα(x)′a^{−1}

for some diagonal matrix a ∈ GL_{n}(N^{+}), with diagonal entries a_{1}, ..., a_{n} say. As ∗ ⊗ c is a positive involution on M_{n}(N) we see that a_{i}a_{j} is totally positive for all i and j. Thus multiplying a by a scalar we may suppose that each a_{i} is totally positive.

Considering ε ∈ M_{n}(N), let A_{1} = εA and let j : A_{1} ↪ A denote the tautological inclusion. Let i_{1} : N ↪ End^{0}(A_{1}) be the map induced by i and let λ_{1} = j^{∨} ∘ λ_{0} ∘ j, a polarisation on A_{1}. If for i = 1, ..., n we let ε_{i} denotes the element of M_{n}(N) which has a 1 in the i^{th} entry in the first column and zeroes elsewhere, then we get an isogeny

(ε_{1} ∘ j) + ... + (ε_{n} ∘ j) : A_{1}^{n} ⟶ A.

Then the diagram

A ⟶^{λ} A^{∨}
↑  ↓
A_{1}^{n} ⟶^{⊕λ_{1}∘(a_{i}/a_{1})} A_{1}^{n}

commutes. Note that, as the centraliser of B ⊗_{F} N in B ⊗_{F} End _{B}^{0}(A) is N, the centraliser of i_{1}(N) in End^{0}(A_{1}) is just i_{1}(N) itself.

Now (up to isogeny) we can lift A_{1} to an abelian scheme Ã_{1}/O_{F_{w}^{ac}} in such a way that the action i_{1} of N lifts to an action ĩ_{1} of N on Ã_{1}. (See [Tat1].) Choose a polarisation μ : Ã_{1} ×F_{w}^{ac} → Ã_{1}^{∨} ×F_{w}^{ac} for which the Rosati involution induces c on ĩ_{1}(N) (see lemma 9.2 of [Ko3]). It extends to a homomorphism μ : Ã_{1} → Ã_{1}^{∨}, which is again a polarisation (see [Ko3], page 392). Let \overline{μ} denote the pull back of μ to A_{1}. As λ_{1} and \overline{μ} both induce c on N and as i_{1}(N) is its own centraliser in End^{0}(A_{1}) we see that λ_{1} = \overline{μ}∘x for some x ∈ N. As λ_{1} and \overline{μ} are both polarisations, x must in fact be a totally positive element of N^{+}. Then λ̃_{1} = μ ∘ x is a polarisation of Ã_{1} which reduces to λ_{1}. Set Ã = (Ã_{1})^{n}, ĩ = M_{n}(ĩ_{1} ∘ α) : B ⊗_{F} M → End^{0}(Ã) and λ̃_{0} = ⊕_{j=1}^{n}λ̃_{1} ∘ ĩ(a_{j}/a_{1}).

Because Lie Ã_{1} ⊗_{O_{F{ac \atop w}}} F_{w}^{ac} is a F ⊗ F_{w}^{ac} ≅ (F_{w}^{ac})^{Hom (F,F_{w}^{ac})}-module, we get a decomposition

Lie Ã_{1} ⊗_{O_{F{ac \atop w}}} F_{w}^{ac} ≅ ⨁_{σ∈Hom (F,F_{w}^{ac})}(Lie Ã_{1})_{σ}.

We also have a decomposition

Lie Ã_{1} ⊗_{O_{F{ac \atop w}}} F_{w}^{ac} ≅ Lie Ã_{1}[p^{∞}] ⊗_{O_{F{ac \atop w}}} F_{w}^{ac} ≅ ⨁_{x} Lie Ã_{1}[x^{∞}] ⊗_{O_{F{ac \atop w}}} F_{w}^{ac},

where x runs over places of F above p. Then

Lie Ã_{1}[x^{∞}] ⊗_{O_{F{ac \atop w}}} F_{w}^{ac} ≅ ⨁_{σ}(Lie Ã_{1})_{σ},

where now σ runs over embeddings σ : F ↪ F_{w}^{ac} such that x is the induced place of F. Let Hom (F, F_{w}^{ac})^{+} denote those embeddings which induce the place u on E, so that Hom (F, F_{w}^{ac}) = Hom (F, F_{w}^{ac})^{+} ∐ Hom (F, F_{w}^{ac})^{+} ∘ c. Then we deduce that there is a unique σ_{0} ∈ Hom (F, F_{w}^{ac})^{+} such that (Lie Ã_{1})_{σ_{0}} ≠ (0). Moreover σ_{0} induces w on F and (Lie Ã_{1})_{σ_{0}} ≅ F_{w}^{ac}.

We can find an embedding κ : F_{w}^{ac} ↪ ℂ such that κ ∘ σ_{0} = τ_{0}, our distinguished embedding F ↪ ℂ. Set

W_{0} = H_{1}((Ã ×_{Spec O_{F{ac \atop w},κ}} Spec ℂ)(ℂ), ℚ).

This is a B ⊗_{F} M-module with an alternating pairing (coming from λ̃_{0}) which is ∗ ⊗ c-Hermitian for the action of B ⊗_{F} M. We see at once that W_{0} ⊗ 𝔸^{∞,p} is equivalent to V^{p}A as a B ⊗_{F} M ⊗_{ℚ} 𝔸^{∞,p}-module with ∗ ⊗ c-Hermitian 𝔸^{∞,p}-alternating pairing.

It remains to show that W_{0}⊗ℝ has invariants (see section 1) (n−1, 1) at τ_{0} and (n, 0) at any other embedding F^{+} ↪ ℝ. Note first that the invariants of W_{0} ⊗ ℝ are the same as the invariants of the F ⊗_{ℚ}ℝ-module with c-Hermitian alternating pairing ⟨ , ⟩_{1} on W_{1} ⊗_{ℚ} ℝ, where

W_{1} = H_{1}((Ã_{1} ×_{Spec O_{F{ac \atop w},κ}} Spec ℂ)(ℂ), ℚ),

and ⟨ , ⟩_{1} comes from λ̃_{1}. As an F -module we see that W_{1} ≅ F^{n}. Also

W_{1} ⊗_{ℚ} ℝ ≅ (Lie Ã_{1}) ⊗_{O_{F{ac \atop w}},κ} ℂ,

and so it is an F ⊗_{ℚ} ℂ-module, not simply an F ⊗_{ℚ} ℝ-module. Corresponding to F ⊗_{ℚ} ℂ ≅ ℂ^{Hom (F,ℂ)} we get a decomposition

W_{1} ⊗_{ℚ} ℝ ≅ ⨁_{τ∈Hom (F,ℂ)}(W_{1} ⊗_{ℚ} ℝ)_{τ}.

As W_{1} ⊗_{ℚ} ℝ ≅ (F ⊗_{ℚ} ℝ)^{n} we see that for all τ,

dim_{ℂ}(W_{1} ⊗_{ℚ} ℝ)_{τ} + dim_{ℂ}(W_{1} ⊗_{ℚ} ℝ)_{τ∘c} = n.

On the other hand

• (W_{1} ⊗_{ℚ} ℝ)_{κ∘σ_{0}} ≅ ℂ,

• but if τ ≠ κ ∘ σ_{0} while τ|_{E} = (κ ∘ σ_{0}|_{E} then (W_{1} ⊗_{ℚ} ℝ)_{τ} = (0).

As the alternating form on (W_{1} ⊗_{ℚ} ℝ) is c ⊗ c-Hermitian for the action of F ⊗_{ℚ} ℂ we see that

W_{1} ⊗_{ℚ} ℝ = ⨁_{τ}(W_{1} ⊗_{ℚ} ℝ)_{τ}

is an orthogonal direct sum for this alternating form.

For a suitable choice of i ∈ ℂ a square root of −1 the c-symmetric form (W_{1} ⊗_{ℚ} ℝ) × (W_{1} ⊗_{ℚ} ℝ) → ℂ given by

x × y ⟼ ⟨ix, y⟩_{1} + i⟨x, y⟩_{1}

is positive definite. Choose \sqrt{−1} ∈ E ⊗_{ℚ} ℝ such that κ ∘ σ_{0}(\sqrt{−1}) = −i. If τ|_{E} = c ∘ κ ∘ σ_{0}|_{E} then

x × y ⟼ ⟨\sqrt{−1}x, y⟩_{1} + \sqrt{−1}⟨x, y⟩_{1}

is positive definite on (W_{1} ⊗_{ℚ} ℝ)_{τ}, while if τ|_{E} = κ ∘ σ_{0}|_{E} then it is negative definite on (W_{1} ⊗_{ℚ} ℝ)_{τ}. It follows that the invariants of W_{0} ⊗_{ℚ} ℝ coincide with those of V ⊗_{ℚ} ℝ and so these are equivalent as B ⊗_{ℚ} ℝ-modules with 8-Hermitian ℝ-alternating pairings up to ℝ^{×}-multiples. □

Keep the notation of the last two lemmas. Let ‡_{0} denote the λ_{0}-Rosati involution on C = End _{B}^{0}(A). Let H_{0}^{AV}/ℚ denote the reductive algebraic group such that for any ℚ-algebra R the R-points of H_{0}^{AV} are the set of g ∈ C ⊗_{ℚ} R such that g^{‡^{0}}g ∈ R^{×}. Also let D = End_{B⊗M}(W_{0}), so that D is isomorphic to the centraliser of M in B^{op}, and let †_{0} denote the involution on B^{op} = End_{B}(W_{0}) and on D induced by ⟨ , ⟩_{0}. Let H_{0}^{LA}/ℚ (resp. G_{0}/ℚ) denote the reductive algebraic group such that for any ℚ-algebra R the R-points of H_{0}^{LA} are the set of g ∈ D ⊗_{ℚ} R such that g^{†^{0}}g ∈ R^{×} (resp. g ∈ B^{op} ⊗_{ℚ} R such that g^{†^{0}}g ∈ R^{×}). Then

• H_{0}^{LA} ⊂ G_{0},

• G and G_{0} are inner forms of each other, • and H_{0}^{AV} and H_{0}^{LA} are inner forms of each other. (If Isom (H_{0}^{AV}, H_{0}^{LA})

denotes the variety of isomorphisms between these algebraic groups, we need to show that Isom (H_{0}^{AV}, H_{0}^{LA}) has a connected component which is geometrically connected. Thus it suffices to show that there is an isomorphism H_{0}^{LA} ×_{ℚ} \overline{𝔸}^{∞,p} →^{∼} H_{0}^{AV} ×_{ℚ} \overline{𝔸}^{∞,p} which is conjugate to all its Gal (ℚ^{ac}/ℚ) transforms. The existence of such a isomorphism follows from the equivalence of W_{0} ⊗_{ℚ} \overline{𝔸}^{∞,p} and V^{p}A ⊗_{𝔸^{∞,p}} \overline{𝔸}^{∞,p} as B ⊗_{F} M ⊗_{ℚ} 𝔸^{∞,p}-modules with ∗ ⊗ c-Hermitian 𝔸^{∞,p}-alternating pairing.)

Moreover we have a natural isomorphism

H_{0}^{AV}(𝔸^{∞,p}) ≅ H_{0}^{LA}(𝔸^{∞,p}).

Also let ϕ_{0} denote the class in H^{1}(ℚ, G_{0}) which represents the difference between (V, ( , )) and (W_{0}, ⟨ , ⟩_{0}).

We see that there are natural bijections between the following sets.

1. G(ℚ)-conjugacy classes of F -embeddings j : M ↪ B^{op} such that #∘j = j ∘ c.

2. Equivalence classes of B ⊗_{F} M-modules together with a ∗ ⊗ c-Hermitian ℚ-alternating form on that module which are equivalent to V as Bmodules with ∗-Hermitian ℚ-alternating pairing.

3. The preimage of ϕ_{0} under H^{1}(ℚ, H_{0}^{LA}) ⟶ H^{1}(ℚ, G_{0}).

(The map from the first to the second set sends j to (V, ( , )) considered as a B ⊗_{F} M-module via Id ⊗ j : B ⊗_{F} M → B ⊗_{F} B^{op}.)

Lemma 8.4 This bijection induces a bijection between the following sets.

1. G(𝔸^{∞,p})-conjugacy classes of F -embeddings j : M ↪ B^{op} such that # ∘ j = j ∘ c.

2. The preimage of ϕ_{0} under H^{1}(ℚ, H_{0}^{LA}(\overline{𝔸}^{∞,p})) ⟶ H^{1}(ℚ, G_{0}(\overline{𝔸}^{∞,p})).

Proof: We see that G(𝔸^{∞,p})-conjugacy classes of F -embeddings j : M ↪ B^{op} such that # ∘ j = j ∘ c are in bijection with classes x ∈ H^{1}(ℚ, H_{0}^{LA}(\overline{𝔸}^{∞,p})) which lift to a class y ∈ H^{1}(ℚ, H_{0}^{LA}) mapping to ϕ_{0} ∈ H^{1}(ℚ, G_{0}). Any such x certainly maps to ϕ_{0} in H^{1}(ℚ, G_{0}(\overline{𝔸}^{∞,p})). So we must show that if x ∈ H^{1}(ℚ, H_{0}^{LA}(\overline{𝔸}^{∞,p})) maps to ϕ_{0} in H^{1}(ℚ, G_{0}(\overline{𝔸}^{∞,p})), then we can lift x to such a y.

Note that ϕ_{0} maps to zero in H^{1}(ℝ, G_{0}) (by lemma 8.3) and in

H^{1}(ℚ_{p}, G_{0}) = H^{1}(ℚ_{p}, (B_{u}^{op} ⊗_{ℚ_{p}} ℚ_{p}^{ac})^{×} × (ℚ_{p}^{ac})^{×}) = (0).

Thus x ∈ H^{1}(ℚ, H_{0}^{LA}(\overline{𝔸}^{∞,p})) ⊂ H^{1}(ℚ, H_{0}^{LA}(\overline{𝔸})) maps to ϕ_{0} ∈ H^{1}(ℚ, G_{0}(\overline{𝔸})).

Let A(G_{0}) and A(H_{0}^{LA}) be the groups defined in section 2.1 of [Ko2]. Then according to proposition 2.6 of [Ko2] we have a commutative diagram with exact rows as follows.

(0) → ker^{1}(ℚ, H_{0}^{LA}) → H^{1}(ℚ, H_{0}^{LA}) → H^{1}(ℚ, H_{0}^{LA}(\overline{𝔸})) → A(H_{0}^{LA})
↓ ↓ ↓ ↓
(0) → ker^{1}(ℚ, G_{0}) → H^{1}(ℚ, G_{0}) → H^{1}(ℚ, G_{0}(\overline{𝔸})) → A(G_{0}).

The lemma now follows from a diagram chase, because of the following two observations.

1. ker^{1}(ℚ, H_{0}^{LA}) → → ker^{1}(ℚ, G_{0}).

2. A(H_{0}^{LA}) ↪ A(G_{0}).

The first of these follows because letting Z_{0} denote the centre of G_{0} the composite homomorphism

ker^{1}(ℚ, Z_{0}) ⟶ ker^{1}(ℚ, H_{0}^{LA}) ⟶ ker^{1}(ℚ, G_{0})

is an isomorphism by the argument on pages 393 and 394 of [Ko3]. The second follows by direct computation from the definitions. In fact, if [F^{+} : ℚ][B : F]^{1/2} = [M^{+} : ℚ][D : M]^{1/2} is odd then A(G_{0}) = (0) and A(H_{0}^{LA}) = (0). If on the other hand [F^{+} : ℚ][B : F]^{1/2} = [M^{+} : ℚ][D : M]^{1/2} is even then the natural homomorphism A(H_{0}^{LA}) → A(G_{0}) is the unique isomorphism

ℤ/2ℤ ⟶^{∼} ℤ/2ℤ.

□

We also see that there are bijections between the following sets.

1. Equivalence classes of polarisations λ : A → A^{∨} such that the λ-Rosati involution takes B ⊗_{F} M to itself and acts on it as ∗ ⊗ c, and where two polarisations λ and λ′ are equivalent if there exists δ ∈ End _{B}^{0}(A) such that λ′ is a ℚ^{×}-multiple of δ^{∨}λδ.

2. Equivalence classes of non-zero elements γ ∈ C^{‡_{0}=1} such that γ = δ^{‡^{0}}δ for some δ ∈ C ⊗_{ℚ} ℝ, and where γ and γ′ are equivalent if there exists δ ∈ ℂ^{×} and μ ∈ ℚ_{>0}^{×} such that γ′ = μδ^{‡^{0}}γδ.

3. Equivalence classes of non-zero elements γ ∈ C^{‡_{0}=1} such that γ = μδ^{‡^{0}}δ for some δ ∈ C ⊗_{ℚ}ℝ and some μ ∈ ℝ^{×}, and where γ and γ′ are equivalent if there exists δ ∈ ℂ^{×} and μ ∈ ℚ^{×} such that γ′ = μδ^{‡^{0}}γδ.

4. ker(H^{1}(E/ℚ, H_{0}^{AV}(E)) ⟶ H^{1}(ℂ/ℝ, H_{0}^{AV}(ℂ))).

5. ker(H^{1}(ℚ, H_{0}^{AV}) ⟶ H^{1}(ℝ, H_{0}^{AV})).

(The equivalence between the first two parts sends γ to λ_{0}∘γ. The equivalence between the last two parts results because

H^{1}(E, H_{0}^{AV}) = H^{1}(E, (C ⊗_{E} E^{ac})^{×} × (E^{ac})^{×}) = (0).)

We deduce that we have a bijection between the following two sets.

1. Equivalence classes of polarisations λ : A → A^{∨} such that

• the λ-Rosati involution takes B ⊗_{F} M to itself and acts on it as ∗ ⊗ c,

• there is an equivalence of B⊗𝔸^{∞,p}-modules with ∗-Hermitian 𝔸^{∞,p}- alternating pairings between V^{p}A with the λ-Weil pairing and V ⊗ 𝔸^{∞,p} with our standard pairing ( , );

and where two polarisations λ and λ′ are equivalent if there exists δ ∈ End _{B}^{0}(A) such that λ′ is a ℚ^{×}-multiple of δ^{∨}λδ.

2. Those elements of ker(H^{1}(ℚ, H_{0}^{AV}) ⟶ H^{1}(ℝ, H_{0}^{AV})) which map to ϕ_{0} in H^{1}(ℚ, G_{0}(\overline{𝔸}^{∞,p})).

We will call two polarisations λ, λ′ : A → A^{∨} nearly equivalent if V^{p}A with its λ-Weil pairing is equivalent to V^{p}A with its λ′-Weil pairing as a B ⊗_{F} M ⊗_{ℚ} 𝔸^{∞,p}-module with ∗ ⊗ c-Hermitian 𝔸^{∞,p}-alternating pairing. This is a strictly coarser equivalence relation than our previous notion of equivalent. We have the following lemma.

Lemma 8.5 There are bijections between the following sets.

1. Near equivalence classes of polarisations λ : A → A^{∨} such that

• the λ-Rosati involution takes B ⊗_{F} M to itself and acts on it as ∗ ⊗ c,

• there is an equivalence of B ⊗𝔸^{∞,p}-modules with ∗-Hermitian 𝔸^{∞,p}- alternating pairing between V^{p}A with its λ-Weil pairing and V ⊗ 𝔸^{∞,p} with our standard pairing ( , ).

2. The preimage of ϕ_{0} under

H^{1}(ℚ, H_{0}^{AV}(\overline{𝔸}^{∞,p})) ≅ H^{1}(ℚ, H_{0}^{LA}(\overline{𝔸}^{∞,p})) ⟶ H^{1}(ℚ, G_{0}(\overline{𝔸}^{∞,p})).

3. G(𝔸^{∞,p})-conjugacy classes of F -embeddings j : M ↪ B^{op} such that # ∘ j = j ∘ c.

This bijection can be arranged so that λ and j correspond if and only if V ⊗𝔸^{∞,p} with its standard pairing ( , ) is equivalent to V^{p}A with its λ-Weil pairing as B ⊗_{F} M ⊗_{ℚ} 𝔸^{∞,p}-modules with ∗ ⊗ c-Hermitian 𝔸^{∞,p}-alternating pairings.

Proof: We have to show that the preimage of ϕ_{0} under the homomorphism

H^{1}(ℚ, H_{0}^{AV}(\overline{𝔸}^{∞,p})) ≅ H^{1}(ℚ, H_{0}^{LA}(\overline{𝔸}^{∞,p})) ⟶ H^{1}(ℚ, G_{0}(\overline{𝔸}^{∞,p}))

is contained in the image of ker(H^{1}(ℚ, H_{0}^{AV}) → H^{1}(ℝ, H_{0}^{AV})). Suppose that x ∈ H^{1}(ℚ, H_{0}^{AV}(\overline{𝔸}^{∞,p})) maps to ϕ_{0} ∈ H^{1}(ℚ, G_{0}(\overline{𝔸}^{∞,p})). Again by proposition 2.6 of [Ko3] we have an exact sequence

H^{1}(ℚ, H_{0}^{AV}) ⟶ H^{1}(ℚ, H_{0}^{AV}(\overline{𝔸}^{∞,p})) ⊕ H^{1}(ℝ, H_{0}^{AV}) ⟶ A(H_{0}^{AV}),

and so it suffices to check that x maps to zero under the map

H^{1}(ℚ, H_{0}^{AV}(\overline{𝔸}^{∞,p})) ⟶ A(H_{0}^{AV}).

By lemma 2.8 of [Ko3] we have a commutative diagram

H^{1}(ℚ, H_{0}^{AV}(\overline{𝔸}^{∞,p})) ⟶^{∼} H^{1}(ℚ, H_{0}^{LA}(\overline{𝔸}^{∞,p})) ⟶ H^{1}(ℚ, G_{0}(\overline{𝔸}^{∞,p}))
↓  ↓  ↓
A(H_{0}^{AV})  =  A(H_{0}^{LA})  ↪  A(G_{0}).

(The injectivity of the map A(H_{0}^{LA}) → A(G_{0}) was explained in the proof of lemma 8.4.) Thus it suffices to show that ϕ_{0} ∈ H^{1}(ℚ, G_{0}(\overline{𝔸}^{∞,p})) maps to zero in A(G_{0}). But ϕ_{0} ∈ ker(H^{1}(ℚ, G_{0}) → H^{1}(ℝ, G_{0})) by lemma 8.3 and H^{1}(ℚ_{p}, G_{0}) = (0). Thus ϕ_{0} ∈ H^{1}(ℚ, G_{0}(\overline{𝔸}^{∞,p})) has the same image in A(G_{0}) as ϕ_{0} ∈ H^{1}(ℚ, G_{0}(\overline{𝔸})), i.e. 0 (by proposition 2.6 of [Ko3]). □

Now let PHT^{(h)} denote the set of triples (M, w̃, [j]) where

• M is a CM-field extension of F,

• w̃ is a place of M above w such that

[Mw̃ : F_{w}]n = [M : F](n − h),

• there is no intermediate field M ⊃ N ⊃ F such that w̃|_{N} is inert in M,

• [j] is a G(𝔸^{∞,p})-conjugacy class of F -embeddings

j : M ↪ B^{op}

such that # ∘ j = j ∘ c.

Combining lemmas 8.1 and 8.5 we see that there is a surjective map

P : J^{(h)}(k(w)^{ac}) ⟶ PHT^{(h)}

which sends

(A, λ, i, η^{p}, η_{p,0}, η_{w}^{0}, η_{w}^{e}, η_{w_{i}}),

to the triple (M, w̃, [j]) associated to the near equivalence class of (A, i, λ) by lemma 8.5. Note that the action of G^{(h)}(𝔸^{∞}) on J^{(h)}(k(w)^{ac}) preserves the fibres of P.

Let z = (M, w̃, [j]) ∈ PHT^{(h)}. Suppose that j ∈ [j]. We make the following definitions.

1. D_{j} will denote the centraliser in B^{op} of jM and †_{j} will denote the restriction of # to D_{j}.

2. V_{j} will denote the B ⊗_{F} M-module V with its B ⊗_{F} M action via Id⊗j : B ⊗_{F} M → B ⊗_{F} B^{op}. If j′ ∈ [j] then the B ⊗_{F} M ⊗_{ℚ}𝔸^{∞,p}-modules with ∗⊗c-Hermitian 𝔸^{∞,p}-alternating pairings V_{j} ⊗_{ℚ}𝔸^{∞,p} and V_{j′} ⊗_{ℚ}𝔸^{∞,p} are equivalent. This equivalence can be realised by an element of G(𝔸^{∞,p}).

3. H_{j}/ℚ will denote the reductive algebraic group coming from the B⊗_{F} Mautomorphisms of V_{j} which preserve the standard alternating pairing ( , ) up to a scalar multiple. This group comes with a natural embedding ι_{j} : H_{j} ↪ G. Moreover, if j′ ∈ [j] then we have an isomorphism

H_{j} ×_{ℚ} 𝔸^{∞,p} →^{∼} H_{j′} ×_{ℚ} 𝔸^{∞,p},

which is canonical up to conjugation in H_{j′}(𝔸^{∞,p}). This isomorphism is achieved via the maps ι_{j} and ι_{j′} and conjugation in G(𝔸^{∞,p}).

4. Unless F^{+} = ℚ and n = 2, there is a distinguished extension τ̃_{0}(j) : M ↪ ℂ of τ_{0} : F ↪ ℂ. It is defined as follows. The equivalence class of V_{j} as a B ⊗_{F} M ⊗_{ℚ}ℝ-module with ∗⊗c-Hermitian ℝ-alternating pairing is classified by a collection of pairs (a_{τ}, b_{τ}) for τ : M^{+} ↪ ℝ. In fact we have (a_{τ}, b_{τ}) = (0, n/[M : F]) for all but one embedding τ_{1} for which (a_{τ_{1}(j)}, b_{τ_{1}}(j)) = (1, n/[M : F]−1). Moreover τ_{1}|_{F^{+}} = τ_{0}|_{F^{+}}. We see that τ_{1} is uniquely determined except in the exceptional case excluded at the start of this paragraph. We define τ̃_{0}(j) to be the unique embedding M ↪ ℂ extending τ_{1} on M^{+} and τ_{0} on F.

If j, j′ ∈ [j] and τ̃_{0}(j) = τ̃_{0}(j′) then we have an equivalence of B ⊗_{F} M ⊗_{ℚ} 𝔸-modules with ∗ ⊗ c-Hermitian 𝔸-alternating pairings between V_{j} ⊗_{ℚ}𝔸 and V_{j′} ⊗_{ℚ} 𝔸. This equivalence can be realised by an element of G(𝔸). Thus we have an isomorphism

H_{j} ×_{ℚ} 𝔸 →^{∼} H_{j′} ×_{ℚ} 𝔸,

which is canonical up to conjugation in H_{j′}(𝔸). (If F^{+} = ℚ and n = 2 we may suppress the requirement that τ̃_{0}(j) = τ̃_{0}(j′).)

Lemma 8.6 Keep the above notation.

1. For any embedding τ̃_{0} : M ↪ ℂ extending τ_{0} we can find a j′ ∈ [j] with τ̃_{0}(j′) = τ̃_{0}.

2. If j, j′ ∈ [j] and τ̃_{0}(j) = τ̃_{0}(j′) then we can find an isomorphism H_{j} →^{∼} H_{j′} compatible with one of our canonical isomorphisms H_{j}×_{ℚ}𝔸 →^{∼} H_{j′}×_{ℚ} 𝔸. (If F^{+} = ℚ and n = 2 we may suppress the requirement that τ̃_{0}(j) = τ̃_{0}(j′).)

Proof: We first look at part one. Choose j_{0} ∈ [j]. As in the discussion before lemma 8.4 we see that G(ℚ)-conjugacy classes of elements of [j] correspond to elements of

ker(H^{1}(ℚ, H_{j_{0}}) ⟶ H^{1}(ℚ, H_{j_{0}}(\overline{𝔸}^{∞})) ⊕ H^{1}(ℚ, G)).

Thus the possibilities for V_{j} ⊗_{ℚ} ℝ correspond to the image of

ker(H^{1}(ℚ, H_{j_{0}}) ⟶ H^{1}(ℚ, H_{j_{0}}(\overline{𝔸}^{∞})) ⊕ H^{1}(ℚ, G))

in H^{1}(ℝ, H_{j_{0}}). As in lemma 8.4 we have a commutative diagram with exact rows

(0) → ker^{1}(ℚ, H_{j_{0}}) → H^{1}(ℚ, H_{j_{0}}) → H^{1}(ℚ, H_{j_{0}}(\overline{𝔸})) → A(H_{j_{0}})
↓ ↓ ↓ ↓
(0) → ker^{1}(ℚ, G) → H^{1}(ℚ, G) → H^{1}(ℚ, G(\overline{𝔸})) → A(G).

where • ker^{1}(ℚ, H_{j_{0}}) → → ker^{1}(ℚ, G)

• and A(H_{j_{0}}) ↪ A(G).

Then a diagram chase shows that the possibilities for V_{j} ⊗_{ℚ} ℝ correspond to

ker(H^{1}(ℝ, H_{j_{0}}) ⟶ H^{1}(ℝ, G)),

i.e. to equivalence classes of B ⊗_{F} M ⊗_{ℚ} ℝ-modules with a ∗ ⊗ c-Hermitian ℝ-alternating form which are equivalent as B ⊗_{ℚ} ℝ-module with ∗-Hermitian ℝ-alternating form to V ⊗_{ℚ} ℝ with its standard pairing ( , ). The first part of the lemma follows.

We now turn to the second part of the lemma. Let Z_{j} denote the centre of H_{j}. The key point will be that

ker^{1}(ℚ, Z_{j}) ⟶^{∼} ker^{1}(ℚ, H_{j}).

This is proved as on pages 393 and 394 of [Ko3]. As V_{j} ⊗ 𝔸 is equivalent to V_{j′} ⊗ 𝔸 the difference between V_{j} and V_{j′} corresponds to an element of

ker(ker^{1}(ℚ, H_{j}) ⟶ H^{1}(ℚ, G)),

i.e. to an element of

ker(ker^{1}(ℚ, Z_{j}) ⟶ H^{1}(ℚ, G)).

Let this element of ker^{1}(ℚ, Z_{j}) be represented by a cocycle σ ↦ z_{σ}. Then we can find α ∈ G(ℚ^{ac}) and β ∈ Z_{j}(\overline{𝔸}) with

z_{σ} = α^{−1}σ(α) = β^{−1}σ(β)

for all σ ∈ Gal (ℚ^{ac}/ℚ). If h ∈ H_{j} then αhα^{−1} ∈ H_{j′}. Moreover if σ ∈ Gal (ℚ^{ac}/ℚ) then

σ(αhα^{−1}) = αz_{σ}σ(h)z_{σ}^{−1}α^{−1} = ασ(h)α^{−1}.

Thus conjugation by α gives an isomorphism H_{j} →^{∼} H_{j′} over ℚ.

Suppose that g ∈ G(𝔸) gives an isomorphism V_{j} ⊗_{ℚ} 𝔸 →^{∼} V_{j′} ⊗_{ℚ} 𝔸. Then βα^{−1}g ∈ H_{j}(\overline{𝔸}). For all σ ∈ Gal (ℚ^{ac}/ℚ) we see that σ(βα^{−1}g) = βα^{−1}g, and so βα^{−1}g ∈ H_{j}(𝔸). Altering g on the right by an element of H_{j}(𝔸) we see that we may suppose that

g = αβ^{−1}.

As conjugation by β^{−1} is trivial on H_{j} we see that g and α induce the same isomorphism

H_{j} ×_{ℚ} 𝔸 →^{∼} H_{j′} ×_{ℚ} 𝔸,

as desired. □

We will write D_{z} for any of the M-algebras D_{j} for any j ∈ [j]. (They are all isomorphic as their invariants are given by

inv_{x}(D_{j}) = [M_{x} : F_{x}]inv_{x}(B^{op}).)

Note that up to conjugacy we have a unique embedding D_{z} ↪ B^{op}. If τ̃_{0} : M ↪ ℂ extends τ_{0} : F ↪ ℂ then we will write H_{z,τ̃_{0}}^{LA}/ℚ for any one of the algebraic groups H_{j} for j ∈ [j] with τ̃_{0}(j) = τ̃_{0}. It comes with an embedding ι_{z,τ̃_{0}}^{LA} : H_{z,τ̃_{0}}^{LA} ↪ G which is well defined up to G(𝔸)-conjugacy.

Now suppose that (A, i, λ) is a triple associated to z. We will make the following definitions.

1. We will let C_{(A,i,λ)} denote End _{B}^{0}(A) and we will let ‡_{(A,i,λ)} denote the λ-Rosati involution on C_{(A,i,λ)}.

2. We will let H_{(A,i,λ)}/ℚ denote the algebraic group such that for any ℚalgebra R, H_{(A,i,λ)}(R) is the group of elements g ∈ C_{(A,i,λ)} ⊗_{ℚ} R with g^{‡^{(A,i,λ)}}g ∈ R^{×}.

If j ∈ [j] then we get an equivalence of B ⊗_{F} M ⊗_{ℚ} 𝔸^{∞,p}-modules with ∗ ⊗ c-Hermitian 𝔸^{∞,p}-alternating pairings between V_{j} ⊗_{ℚ} 𝔸^{∞,p} and V^{p}A. Thus we get an isomorphism

H_{j} ×_{ℚ} 𝔸^{∞,p} ⟶^{∼} H_{(A,i,λ)} ×_{ℚ} 𝔸^{∞,p},

which is well defined up to H_{(A,i,λ)}(𝔸^{∞,p})-conjugacy. We also then get an isomorphism

H_{j} ×_{ℚ} ℚ^{ac} ⟶^{∼} H_{(A,i,λ)} ×_{ℚ} ℚ^{ac},

which is specified up to H_{(A,i,λ)}(ℚ^{ac})-conjugacy by the condition it is compatible with the isomorphism

V_{j} ⊗_{ℚ} \overline{𝔸}^{∞,p} →^{∼} V^{p}A ⊗_{𝔸^{∞,p}} \overline{𝔸}^{∞,p}

(itself defined up to H_{(A,i,λ)}(𝔸^{∞,p})-conjugacy). Thus H_{(A,i,λ)} is an inner form of H_{j} (cf the discussion after lemma 8.3). Since the λ-Rosati involution is positive, H_{(A,i,λ)}(ℝ) is compact mod centre.

Lemma 8.7 Suppose that (A, i, λ) and (A′, i′, λ′) are both associated to z. Suppose also that j ∈ [j]. Then we can find an isomorphism

H_{(A,i,λ)} ⟶^{∼} H_{(A′,i′,λ′)}

compatible with some choices of the canonical isomorphisms H_{j} ×_{ℚ} 𝔸^{∞,p} ⟶^{∼}

H_{(A,i,λ)} ×_{ℚ} 𝔸^{∞,p} and H_{j} ×_{ℚ} 𝔸^{∞,p} ⟶^{∼} H_{(A′,i′,λ′)} ×_{ℚ} 𝔸^{∞,p}.

Proof: We can find an isogeny

α : A ⟶ A′

such that

• i′(x) = αi(x)α^{−1} for all x ∈ B ⊗_{F} M,

• and λ′ = α^{−∨}λγα^{−1} for some γ ∈ C_{(A,i,λ)}^{‡_{(A,i,λ)}=1} with γ = δ^{‡^{(A,i,λ)}}δ for some δ ∈ C_{(A,i,λ)} ⊗_{ℚ} ℝ.

Then (γ, 1) ∈ C_{(A,i,λ)}^{×} × E^{×} represents an element of

ker(H^{1}(E/ℚ, H_{(A,i,λ)}(E)) ⟶ H^{1}(ℚ, H_{(A,i,λ)}(\overline{𝔸}))).

Let Z_{(A,i,λ)} denote the centre of H_{(A,i,λ)}. As

• H^{1}(E, H_{(A,i,λ)}) = (0),

• H^{1}(E, Z_{(A,i,λ)}) = (0),

• and ker^{1}(ℚ, Z_{(A,i,λ)}) →^{∼} ker^{1}(ℚ, H_{(A,i,λ)}) (again by the same argument used on pages 393 and 394 of [Ko3]),

we deduce that

ker(H^{1}(E/ℚ, H_{(A,i,λ)}(E)) ⟶ H^{1}(ℚ, H_{(A,i,λ)}(\overline{𝔸})))

equals

ker(H^{1}(E/ℚ, Z_{(A,i,λ)}(E)) ⟶ H^{1}(E/ℚ, Z_{(A,i,λ)}(𝔸_{E}))).

Thus we can find (δ, μ) ∈ C_{(A,i,λ)}^{×} × E^{×} such that

(μ^{−c}δ^{‡^{(A,i,λ)}}γδ, μ/μ^{c}) ∈ M^{×} × E^{×}.

Note that we may take μ = 1. Then replacing α by αδ we see that without loss of generality we may suppose that γ ∈ M^{×} (and hence that γ is a totally positive element of M^{+}).

It follows (from γ ∈ M) that α induces an isomorphism C_{(A,i,λ)} →^{∼} C_{(A′,i′,λ′)} (x ↦ αxα^{−1}) which takes ‡_{(A,i,λ)} to ‡_{(A′,i′,λ′)} and hence induces an isomorphism

H_{(A,i,λ)} →^{∼} H_{(A′,i′,λ′)}.

We may choose μ ∈ (𝔸^{∞,p})^{×} and δ ∈ (𝔸_{M}^{∞,p})^{×} such that γ = μ^{−1}δ^{c}δ. Suppose that β : V^{p}A →^{∼} V^{p}A′ is an isomorphism of B ⊗_{F} M ⊗_{ℚ}𝔸^{∞,p}-modules with alternating pairings up to (𝔸^{∞,p})^{×}-multiples. Then we see that

β^{−1}(V^{p}α)δ^{−1} ∈ H_{(A,i,λ)}(𝔸^{∞,p}).

Altering β on the right by an element of H_{(A,i,λ)}(𝔸^{∞,p}), we may suppose that β = (V^{p}α)δ^{−1} and hence that β and V^{p}α induce the same isomorphism

H_{(A,i,λ)} ×_{ℚ} 𝔸^{∞,p} →^{∼} H_{(A′,i′,λ′)} ×_{ℚ} 𝔸^{∞,p}.

the second part of the lemma follows. □

We will write C_{z} for any of the M-algebras C_{(A,i,λ)}. (They are all isomorphic as they have the same invariants by lemma 8.2.) Also from the invariants we see that for any place x ≠ w̃ or w̃^{c} of M we have C_{z,x} ≅ D_{z,x}. We will write H_{z}^{AV} for any of the algebraic groups H_{(A,i,λ)} for (A, i, λ) associated to z. This is well defined up to inner automorphism. If τ̃_{0} : M ↪ ℂ is any extension of τ_{0} : F ↪ ℂ then there is an isomorphism

H_{z,τ̃_{0}}^{LA} ×_{ℚ} 𝔸^{∞,p} ⟶^{∼} H_{z}^{AV} ×_{ℚ} 𝔸^{∞,p},

which is canonical up to H_{z}^{AV}(𝔸^{∞,p})-conjugacy. In particular we have an embedding

ι_{z}^{AV} : H_{z}^{AV}(𝔸^{∞,p}) ↪ G(𝔸^{∞,p})

well defined up to G(𝔸^{∞,p})-conjugacy. We also have isomorphisms

H_{z}^{AV}(ℚ_{p}) ≅ ℚ_{p}^{×} × C_{z,u}^{×} ≅ ℚ_{p}^{×} × C_{z,w̃}^{×} × ∏_{x|w,x≠w̃} GL_{n/[M:F]}(M_{x}) × ∏_{i>1} ∏_{x|w_{i}} C_{z,x}^{×}.

But there exist embeddings (unique up to conjugation)

• C_{z,w̃} ↪ D_{F_{w},n−h} over F_{w} (firstly M_{w} ↪ D_{F_{w},n−h} because [Mw̃ : F_{w}]|(n − h), and secondly we find that the centraliser of M_{w} in D_{F_{w},n−h} has invariant [Mw̃ : F_{w}]/(n − h) = [M : F]/n = inv w̃C_{z}),

• ∏_{x|w,x≠w̃} M_{n/[M:F]}(M_{x}) ↪ M_{h}(F_{w}) over F_{w} (because

(n/[M : F]) ∑_{x|w,x≠w̃}[M_{x} : F_{w}] = n − n[Mw̃ : F_{w}]/[M : F] = h),

• and for each i > 1, ∏_{x|w_{i}} C_{z,x} ≅ D_{z,w_{i}} ↪ B_{w_{i}}^{op} over F_{w_{i}}.

Thus we may extend ι_{z}^{AV} to an embedding

ι_{z}^{AV} : H_{z}^{AV}(𝔸^{∞}) ↪ G^{(h)}(𝔸^{∞})

well defined up to G^{(h)}(𝔸^{∞})-conjugacy.

We finish this section with the promised description of J^{(h)}(k(w)^{ac}).

Proposition 8.8 The map

P : J^{(h)}(k(w)^{ac}) ⟶ PHT^{(h)}

is a surjection with fibres preserved by G^{(h)}(𝔸^{∞}). If z ∈ PHT^{(h)} then P^{−1}(z) is isomorphic as a right G^{(h)}(𝔸^{∞})-set to # ker^{1}(ℚ, H_{z}^{AV}) copies of

ι_{z}^{AV}(H_{z}^{AV}(ℚ))\G^{(h)}(𝔸^{∞})/ℤ_{p}^{×}.

This identification is canonical up to replacing ι_{z}^{AV} by y ↦ gι_{z}^{AV}(xyx^{−1})g^{−1} for g ∈ G^{(h)}(𝔸^{∞}) and x ∈ H_{z}^{AV}(ℚ) and simultaneously left translating the isomorphism by g.

Proof: We have already proved the first sentence, so fix z ∈ PHT^{(h)}. Then z corresponds to a near equivalence class of triples (A, i, λ). This near equivalence class consists of ker^{1}(ℚ, H_{z}^{AV}) equivalence classes. Fix one such equivalence class C = [(A, i, λ)] and let C̃ denote the set of points

(A′, λ′, i′, η^{p}, η_{p,0}, η_{w}^{0}, η_{w}^{e}, η_{w_{i}})

of J^{(h)}(k(w)^{ac}) with (A′, i′, λ′) equivalent to (A, i, λ). We will show that C̃ is isomorphic as a right G^{(h)}(𝔸^{∞})-set to

ι_{(A,i,λ)}(H_{(A,i,λ)}(ℚ))\G^{(h)}(𝔸^{∞})/ℤ_{p}^{×}.

The proposition will then follow.

First we see that any point of C̃ can be written in the form

(A, λ, i, η^{p}, η_{p,0}, η_{w}^{0}, η_{w}^{e}, η_{w_{i}}).

Thus G^{(h)}(𝔸^{∞}) acts transitively on C̃. Finally the stabiliser in G^{(h)}(𝔸^{∞}) of x_{0} is just ι_{(A,i,λ)}(H_{(A,i,λ)})ℤ_{p}^{×}. □

9 An application of Fujiwara’s trace formula.

Let φ ∈ C_{c}^{∞}(G^{(h)}(𝔸^{∞})^{+}/ℤ_{p}^{×} × O_{D_{Fw} ,n−h}^{×}). In this section we will find (subject to some restrictions) a group theoretic description for

tr (φ|H_{c}(I^{(h)}, F_{ρ} ⊗ L_{ξ})).

Our main tool will be Fujiwara’s trace formula (see [Fu]). This is a form of the Lefschetz trace formula for the cohomology with compact supports of smooth but not necessarily proper varieties over finite fields. This formula was conjectured by Deligne and had been proved modulo some sort of resolution of singularities by Pink (see [P]).

Any such φ can be written as a finite sum of the form

φ = ∑_{g} a_{g}char_{U^{p}(m,0)gU^{p}(m,0)}

for some fixed U^{p} and m (depending on φ). As always we can and will assume that U^{p} is sufficiently small.

By a fixed point of [U^{p}(m, 0)gU^{p}(m, 0)] we will mean a point

x ∈ J^{(h)}(k(w)^{ac})/(U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1})

such that x = xg ∈ J^{(h)}(k(w)^{ac})/U^{p}(m, 0). This set appears to depend on g not just on U^{p}(m, 0)gU^{p}(m, 0), but if we replace g by u_{1}gu_{2} (with u_{1}, u_{2} ∈ U^{p}(m, 0)) the two sets are in natural bijection via x ↦ xu_{1}^{−1}. (If u_{1}gu_{2} = u′_{1}gu′_{2}, with u′_{1}, u′_{2} ∈ U^{p}(m, 0) as well, then x ↦ x(u′_{1})^{−1} gives the same map.) We will denote this set defined up to canonical bijection Fix([U^{p}(m, 0)gU^{p}(m, 0)]).

Suppose x is such a fixed point. Choose g ∈ U^{p}(m, 0)gU^{p}(m, 0) and a point x̃ ∈ J^{(h)}(k(w)^{ac}) above x ∈ J^{(h)}(k(w)^{ac})/U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1}. Then we see that

x̃g = x̃u

for some u ∈ U^{p}(m, 0). We will set

tr [U^{p}(m, 0)gU^{p}(m, 0)]|(F_{ρ} ⊗ L_{ξ})_{x} = tr (π ⊗ ξ)(gu^{−1}).

We will check that this is independent of the various choices. First if we replace x̃ by x̃v for some v ∈ U^{p}(m, 0)∩gU^{p}(m, 0)g^{−1} then gu^{−1} is replaced by v^{−1}gu^{−1}v and so the value of the trace is unchanged. Secondly if we replace g by u_{1}gu_{2} and x̃ by x̃(u_{1})^{−1} then gu^{−1} is replaced by u_{1}gu^{−1}u_{1}^{−1} and again the value of the trace is unchanged. Thus tr [U^{p}(m, 0)gU^{p}(m, 0)]|(F_{ρ} ⊗ L_{ξ})_{x} is a well-defined function on Fix([U^{p}(m, 0)gU^{p}(m, 0)]).

Again suppose that x ∈ Fix([U^{p}(m, 0)gU^{p}(m, 0)]) and again choose g ∈ U^{p}(m, 0)gU^{p}(m, 0) and x̃ ∈ J^{(h)}(k(w)^{ac}) above x ∈ J^{(h)}(k(w)^{ac})/U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1}. Let z = P(x̃). Then we can represent x̃ by an element y ∈ G^{(h)}(𝔸^{∞}), and we see that

yg = ι_{z}^{AV}(a)yu

for some a ∈ H_{z}^{AV}(ℚ) and some u ∈ U^{p}(m, 0). We will show that the conjugacy class [a] of a in H_{z}^{AV}(ℚ) depends only on x. We have to check independence of the following choices.

• We could postmultiply a by an element of

(ι_{z}^{AV})^{−1}(yU^{p}(m, 0)y^{−1}) ∩ H_{z}^{AV}(ℚ).

But as H_{z}^{AV}(ℝ) is compact modulo the centre this intersection is a finite group and so as U^{p}(m, 0) is sufficiently small we see that

(ι_{z}^{AV})^{−1}(yU^{p}(m, 0)y^{−1}) ∩ H_{z}^{AV}(ℚ) = {1}.

• We could replace y by ι_{z}^{AV}(b)yv with b ∈ H_{z}^{AV}(ℚ) and v ∈ U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1}. In this case a is replaced by bab^{−1} and u is replaced by v^{−1}u(g^{−1}vg).

• We could replace g by u_{1}gu_{2} and y by yu_{1}^{−1} with u_{1}, u_{2} ∈ U^{P} (m, 0). Then a remains unchanged and u is replaced by uu_{2}.

• We could preconjugate ι_{z}^{AV} by b ∈ H_{z}^{AV}(ℚ) and postconjugate by g′ ∈ G^{(h)}(𝔸^{∞}) while replacing y by g′y. Then u is unchanged and a is replaced by b^{−1}ab.

Thus we may write [a(x)] for this conjugacy class. Notice that

tr [U^{p}(m, 0)gU^{p}(m, 0)]|(F_{ρ} ⊗ L_{ξ}) = tr (ρ ⊗ ξ)(ι_{z}^{AV}(a(x))),

because gu^{−1} = y^{−1}ι_{z}^{AV}(a)y.

Now we ask the converse question: given a ∈ H_{z}^{AV}(ℚ) how many points x ∈ Fix([U^{p}(m, 0)gU^{p}(m, 0)]) are there with [a(x)] = [a]? One may check that the answer is the cardinality of the double coset space

#(ι_{z}^{AV}(H_{z}^{AV}(ℚ))\X/U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1})

where

X = {y ∈ G^{(h)}(𝔸^{∞}) : y^{−1}ι_{z}^{AV}([a])y ∩ gU^{p}(m, 0) ≠ ∅}.

If a, b ∈ H_{z}^{AV}(ℚ) and if both y^{−1}ι_{z}^{AV}(a)y and y^{−1}ι_{z}^{AV}(b)y ∈ gU^{p}(m, 0) then ι_{z}^{AV}(a^{−1}b) ∈ y^{−1}gUg^{−1}y and so (because U^{p} is sufficiently small and H_{z}^{AV}(ℝ) is compact modulo its centre) we see that a = b. We deduce that the number of x ∈ Fix([U^{p}(m, 0)gU^{p}(m, 0)]) with [a(x)] = [a] is also given by

#(ι_{z}^{AV}(Z_{H_{z}^{AV}}(a)(ℚ))\X′/U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1})

where

X′ = {y ∈ G^{(h)}(𝔸^{∞}) : y^{−1}ι_{z}^{AV}(a)y ∈ gU^{p}(m, 0)}.

A similar argument shows that for any y ∈ G^{(h)}(𝔸^{∞}) we have

ι_{z}^{AV}(Z_{H_{z}^{AV}}(a)(ℚ))y(U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1}) =
= ∐_{b∈Z_{H{AV \atop z}} (a)(ℚ)} ι_{z}^{AV}(b)y(U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1}),

and so the number of x ∈ Fix([U^{p}(m, 0)gU^{p}(m, 0)]) with [a(x)] = [a] is also given by

vol (U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1})^{−1}
vol ({y ∈ ι_{z}^{AV}(Z_{H_{z}^{AV}}(a)(ℚ))\G^{(h)}(𝔸^{∞}) : y^{−1}ι_{z}^{AV}(a)y ∈ gU^{p}(m, 0)}),

where we use any Haar measure on G^{(h)}(𝔸^{∞}) and where we use a Haar measure on ι_{z}^{AV}(Z_{H_{z}^{AV}}(a)(ℚ)) which gives each point volume 1. This can be rewritten

vol (U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1})^{−1}
vol (ι_{z}^{AV}(Z_{H_{z}^{AV}}(a)(ℚ))\Z_{G^{(h)}(𝔸^{∞})}(a))O_{ι_{z}^{AV}(a)}^{G^{(h)}(𝔸^{∞})}(char_{gU^{p}(m,0)}),

where again the measure on ι_{z}^{AV}(Z_{H_{z}^{AV}}(a)(ℚ)) gives each point volume 1 and where the Haar measures on each other groups are arbitrary as long as they are chosen consistently for each occurrence of a given group. This appears to depend on the choice of g ∈ [U^{p}(m, 0)gU^{p}(m, 0)]. Adding the formulas for g running over a set of representatives for U^{p}(m, 0)gU^{p}(m, 0)/U^{p}(m, 0) and dividing by

#(U^{p}(m, 0)gU^{p}(m, 0)/U^{p}(m, 0)) = [U^{p}(m, 0) : U^{p}(m, 0) ∩ gU^{p}(m, 0)g^{−1}],

we see that the number of x ∈ Fix([U^{p}(m, 0)gU^{p}(m, 0)]) with [a(x)] = [a] is also given by

vol (U^{p}(m, 0))^{−1}
vol (ι_{z}^{AV}(Z_{H_{z}^{AV}}(a)(ℚ))\Z_{G^{(h)}(𝔸^{∞})}(a))O_{ι_{z}^{AV}(a)}^{G^{(h)}(𝔸^{∞})}(char_{U^{p}(m,0)gU^{p}(m,0)}).

We remark that this number may be infinite.

We will say that φ ∈ C_{c}^{∞}(G^{(h)}(𝔸^{∞})^{+}/ℤ_{p}^{×} × O_{D_{Fw} ,n−h}^{×}) is acceptable if it can be written as a finite sum

φ = ∑_{i} α_{i}char_{U^{p}(m,0)g_{i}U^{p}(m,0)}

with U^{p} sufficiently small and where

1. for y ∈ ⋃_{i} U^{p}(m, 0)g_{i}U^{p}(m, 0) the p-adic valuation of every eigenvalue of y_{w}^{0} is strictly less than the p-adic valuation of every eigenvalue of y_{w}^{e};

2. each Fix([U^{p}(m, 0)g_{i}U^{p}(m, 0)]) is a finite set;

3. and, for each i,

∑_{i}(−1)^{i}tr ([U^{p}(m, 0)g_{i}U^{p}(m, 0)]|H_{c}^{i}(I_{U^{p},m}^{(h)} × Spec k(w)^{ac}, F_{ρ} ⊗ L_{ξ})) = = ∑_{x∈Fix([U^{p}(m,0)g_{i}U^{p}(m,0)])} tr ([U^{p}(m, 0)g_{i}U^{p}(m, 0)]|(F_{ρ} ⊗ L_{ξ})_{x}).

Here [U^{p}(m, 0)g_{i}U^{p}(m, 0)] defines a correspondence on I_{U^{p},m}^{(h)} which lifts to a natural cohomological correspondence on F_{ρ} ⊗ L_{ξ} over (I_{U^{p},m}^{(h)} × Spec k(w)^{ac}). It therefore acts on H_{c}^{i}(I_{U^{p},m}^{(h)} × Spec k(w)^{ac}, F_{ρ} ⊗ L_{ξ}). (See [Fu].)

This definition is only useful if we have a good supply of acceptable functions φ. This is provided by the following lemma, whose key ingredient is Fujiwara’s trace formula.

Lemma 9.1 Suppose that φ ∈ C_{c}^{∞}(G^{(h)}(𝔸^{∞})^{+}/ℤ_{p}^{×} × O_{D_{Fw} ,n−h}^{×}). Fix δ ∈ D_{F_{w},n−h}^{×} with w(det δ) = 1. Then for N >> 0 the function φ|(1, p^{f_{1}N}, δ^{N}, 1, 1) defined by

(φ|(1, p^{f_{1}N}, δ^{N}, 1, 1))(y) = φ(y(1, p^{f_{1}N}, δ^{N}, 1, 1))

is acceptable.

Proof: The first condition is easily checked. The latter two conditions follow from corollary 5.4.5 of [Fu] because

[U^{p}(m, 0)g_{i}(1, p^{−f_{1}N}, δ^{−N}, 1, 1)U^{p}(m, 0)] = (Fr^{∗})^{f_{1}N}.[U^{p}(m, 0)g_{i}U^{p}(m, 0)]

in the notation of [Fu] (see the third paragraph of section 7). □

We now show how we can calculate the trace of an acceptable function on the cohomology of Igusa varieties of the first kind.

Lemma 9.2 Suppose that φ ∈ C_{c}^{∞}(G^{(h)}(𝔸^{∞})^{+}/ℤ_{p}^{×} × O_{D_{Fw} ,n−h}^{×}) is acceptable, then

tr (φ|H_{c}(I^{(h)}, F_{ρ} ⊗ L_{ξ})) = (−1)^{h} ∑_{z∈PHT^{(h)}} ∑_{[a]⊂H_{z}^{AV}(ℚ)}(# ker^{1}(ℚ, H_{z}^{AV}))

vol (Z_{H_{z}^{AV}}(a)(ℚ)\Z_{H_{z}^{AV}}(a)(𝔸^{∞}))O_{ι_{z}^{AV}(a)}^{G^{(h)}(𝔸^{∞})}(φ)tr (ρ ⊗ ξ)(ι_{z}^{AV}(a)).

This sum is finite and all the terms occurring are finite numbers. For each non-zero term we have F (a) ⊃ M and

Z_{G^{(h)}(𝔸^{∞})}(a) = ι_{z}^{AV}(Z_{H_{z}^{AV}}(a)(𝔸^{∞})).

We may make any consistent choices for the Haar measures implicit in this formula as long as the Haar measure on Z_{H_{z}^{AV}}(a)(ℚ) gives points measure 1 and as long as the measures used on Z_{G^{(h)}(𝔸^{∞})}(a) and Z_{H_{z}^{AV}}(a)(𝔸^{∞}) correspond under ι_{z}^{AV}.

Proof: We first check that if O_{ι_{z}^{AV}(a)}^{G^{(h)}(𝔸^{∞})}(φ) ≠ 0 then

Z_{G^{(h)}(𝔸^{∞})}(a) = ι_{z}^{AV}(Z_{H_{z}^{AV}}(a)(𝔸^{∞})).

If O_{ι_{z}^{AV}(a)}^{G^{(h)}(𝔸^{∞})}(φ) ≠ 0 then the p-adic valuation of every eigenvalue of ι_{z}^{AV}(a)_{w}^{0} is strictly less than the p-adic valuation of every eigenvalue of ι_{z}^{AV}(a)_{w}^{e} (because φ is acceptable). Thus there is a constant κ such that a place x of M(a) ⊂ C_{z} above w divides w̃ if and only if

|a|_{x}^{1/[M(a)_{x}:F_{w}]} > κ.

Hence if x is a place of M(a) above w̃ then any other place of M(a) above x|_{F (a)} also lies above w̃. Let N/(F (a) ∩ M) denote the normal closure of M(a)/F (a) ∩ M, then N has the same property that two places x and x′ of N above the same place of F (a) either both lie above w̃ or neither lie above w̃. Fix a place x of N above w̃ and let Δ denote the decomposition group for x in Gal (N/F (a) ∩ M). Let σ ∈ Gal (N/F (a) ∩ M). Then some (resp. all) places of N above the place σ(x)|_{F (a)} lie above w̃ if and only if

(Gal (N/F (a))σΔ) ∩ (Gal (N/M)Δ) ≠ ∅

(resp.

(Gal (N/F (a))σΔ) ⊂ (Gal (N/M)Δ) ).

Thus we see that Gal (N/M)Δ is a union of double cosets of the form

Gal (N/F (a))σΔ,

i.e.

Gal (N/F (a))Gal (N/M)Δ = Gal (N/M)Δ.

As Gal (N/(F (a) ∩ M)) is generated by Gal (N/F (a)) and Gal (N/M) we see that

Gal (N/(F (a) ∩ M)) = Gal (N/(F (a) ∩ M))Gal (N/M)Δ = Gal (N/M)Δ.

This translates into w̃|_{F (a)∩M} being inert in M. By the minimality of (M, w̃) we conclude that F (a) ⊃ M and so

Z_{G^{(h)}(𝔸^{∞})}(a) ⊂ Z_{G^{(h)}(𝔸^{∞})}(M) = H_{z}^{AV}(𝔸^{∞}).

This establishes the desired equality.

If

φ = ∑_{i} α_{i}char_{U^{p}(m,0)g_{i}U^{p}(m,0)}

then

tr (φ|H_{c}(I^{(h)}, F_{ρ} ⊗ L_{ξ})) = vol (U^{p}(m, 0)) ∑_{i,j} α_{i}(−1)^{h−j}
tr ([U^{p}(m, 0)g_{i}U^{p}(m, 0)]|H_{c}^{j}(I_{U^{p},m}^{(h)} × Spec k(w)^{ac}, F_{ρ} ⊗ L_{ξ})),

where we use the same Haar measure to compute tr φ as we do to compute vol (U^{p}(m, 0)). The lemma now follows from the definition of acceptable and from our previous calculations. □

Although it appears that we have a purely group theoretic expression for the trace this is illusory as the definition of H_{z}^{AV} involves abelian varieties. Thus we must further massage the formula.

Let ν_{z}^{AV} : H_{z}^{AV} ⟶ 𝔾_{m} denote the multiplier character (so that ν_{z}^{AV}(x) = x^{‡}x). Also let Z_{H_{z}^{AV}}(a)(𝔸)^{1} (resp. Z_{H_{z}^{AV}}(a)(ℝ)^{1}) denote the kernel of

|ν| : Z_{H_{z}^{AV}}(a)(𝔸) ⟶ ℝ_{>0}^{×}

(resp. the kernel of

|ν| : Z_{H_{z}^{AV}}(a)(ℝ) ⟶ ℝ_{>0}^{×}).

We have an exact sequence

{1} → Z_{H_{z}^{AV}}(a)(ℝ)^{1} → Z_{H_{z}^{AV}}(a)(ℚ)\Z_{H_{z}^{AV}}(a)(𝔸)^{1} →
→ Z_{H_{z}^{AV}}(a)(ℚ)\Z_{H_{z}^{AV}}(a)(𝔸^{∞}) → {1},

and hence vol (Z_{H_{z}^{AV}}(a)(ℚ)\Z_{H_{z}^{AV}}(a)(𝔸^{∞})) equals

vol (Z_{H_{z}^{AV}}(a)(ℚ)\Z_{H_{z}^{AV}}(a)(𝔸))vol (Z_{H_{z}^{AV}}(a)(ℝ)^{1})^{−1}.

Moreover if we use Tamagawa measure on Z_{H_{z}^{AV}}(a)(𝔸)^{1} then the main theorem of [Ko5] tells us that

vol (Z_{H_{z}^{AV}}(a)(ℚ)\Z_{H_{z}^{AV}}(a)(𝔸^{∞}))
= #A(Z_{H_{z}^{AV}}(a)(ℚ))(# ker^{1}(ℚ, Z_{H_{z}^{AV}}(a))^{−1}vol (Z_{H_{z}^{AV}}(a)(ℝ)^{1})^{−1}
= κ_{B}(# ker^{1}(ℚ, Z_{H_{z}^{AV}}(a)(ℚ)))^{−1}vol (Z_{H_{z}^{AV}}(a)(ℝ)^{1})^{−1},

where κ_{B} = 2 if [B : ℚ]/2 is even and κ_{B} = 1 otherwise. (See the introduction to [Ko5], formula 4.2.2 of [Ko1], compute A(Z_{H_{z}^{AV}}(a)(ℚ)) directly from the definition and note that [B : ℚ]/2 is even if and only if [Z_{C_{z}}(a) : ℚ]/2 is.) Thus the right hand side of the main equality of lemma 9.2 can be rewritten

(−1)^{h}κ_{B} ∑_{z∈PHT^{(h)}} ∑_{[a]⊂H_{z}^{AV}(ℚ)}(# ker^{1}(ℚ, H_{z}^{AV})/# ker^{1}(ℚ, Z_{H_{z}^{AV}}(a)))
O_{ι_{z}^{AV}(a)}^{G^{(h)}(𝔸^{∞})}(φ)vol (Z_{H_{z}^{AV}}(a)(ℝ)^{1})^{−1}tr (ρ ⊗ ξ)(ι_{z}^{AV}(a)),

where we use measures on Z_{H_{z}^{AV}}(a)(ℝ)^{1} and Z_{G^{(h)}(𝔸^{∞})}(a) compatible with Tamagawa measure on Z_{H_{z}^{AV}}(a)(𝔸)^{1} and the exact sequence

{1} → Z_{H_{z}^{AV}}(a)(ℝ)^{1} → Z_{H_{z}^{AV}}(a)(𝔸)^{1} → Z_{G^{(h)}(𝔸^{∞})}(ι_{z}^{AV}(a)) → {1}.

Now suppose that a and a′ ∈ H_{z}^{AV}(ℚ) are conjugate under H_{z}^{AV}(𝔸). Then Z_{H_{z}^{AV}}(a) and Z_{H_{Z}^{AV}}(a′) are inner forms of each other which become isomorphic over 𝔸. Moreover the Tamagawa measures on Z_{H_{z}^{AV}}(a)(𝔸)^{1} and Z_{H_{z}^{AV}}(a′)(𝔸)^{1} agree under this isomorphism (use the definition of Tamagawa measure and the discussion in paragraph two on page 631 of [Ko3]). Thus

O_{ι_{z}^{AV}(a)}^{G^{(h)}(𝔸^{∞})}(φ)vol (Z_{H_{z}^{AV}}(a)(ℝ)^{1})^{−1} = O_{ι_{z}^{AV}(a′)}^{G^{(h)}(𝔸^{∞})}(φ)vol (Z_{H_{z}^{AV}}(a′)(ℝ)^{1})^{−1}.

The number of H_{z}^{AV}(ℚ)-conjugacy classes in the H_{z}^{AV}(𝔸)-conjugacy class of a in H_{z}^{AV}(ℚ) is

# ker(ker^{1}(ℚ, Z_{H_{z}^{AV}}(a)) ⟶ ker^{1}(ℚ, H_{z}^{AV})).

For the moment let Z (resp. Z′) denote the centre of H_{z}^{AV} (resp. Z_{H_{z}^{AV}}(a)). Then we have homomorphisms

ker^{1}(ℚ, Z) ⟶ ker^{1}(ℚ, Z′) ⟶ ker^{1}(ℚ, Z_{H_{z}^{AV}}(a)) ⟶ ker^{1}(ℚ, H_{z}^{AV}),

where as on pages 393 and 394 of [Ko3] we see that

ker^{1}(ℚ, Z′) ⟶^{∼} ker^{1}(ℚ, Z_{H_{z}^{AV}}(a))

and

ker^{1}(ℚ, Z) ⟶^{∼} ker^{1}(ℚ, H_{z}^{AV}).

Thus

# ker(ker^{1}(ℚ, Z_{H_{z}^{AV}}(a)) ⟶ ker^{1}(ℚ, H_{z}^{AV}))

equals

# ker^{1}(ℚ, Z_{H_{z}^{AV}}(a))/# ker^{1}(ℚ, H_{z}^{AV}).

In particular this number only depends on a up to H_{z}^{AV}(𝔸)-conjugacy. Thus we may again rewrite the right hand side of the main equality of lemma 9.2 as

(−1)^{h}κ_{B} ∑_{z∈PHT^{(h)}} ∑_{a} O_{ι_{z}^{AV}(a)}^{G^{(h)}(𝔸^{∞})}(φ)vol (Z_{H_{z}^{AV}}(a)(ℝ)^{1})^{−1}tr (ρ ⊗ ξ)(ι_{z}^{AV}(a)),

where the second sum runs over representatives of H_{z}^{AV}(𝔸)-conjugacy classes of elements a ∈ H_{z}^{AV}(ℚ) such that F (a) ⊃ M (if z = (M, w̃, [j])) and where we use Haar measures on Z_{H_{z}^{AV}}(a)(ℝ)^{1} and Z_{G^{(h)}(𝔸^{∞})}(a) compatible with Tamagawa measure on Z_{H_{z}^{AV}}(a)(𝔸)^{1} and the exact sequence

{1} → Z_{H_{z}^{AV}}(a)(ℝ)^{1} → Z_{H_{z}^{AV}}(a)(𝔸)^{1} → Z_{G^{(h)}(𝔸^{∞})}(ι_{z}^{AV}(a)) → {1}.

Consider the following three sets.

1. The set FP_{AV}^{(h)} of pairs (z, [a]) where z = (M, w̃, [j]) ∈ PHT^{(h)} and [a] is a H_{z}^{AV}(𝔸)-conjugacy class in H_{z}^{AV}(ℚ) such that F (a) ⊃ M.

2. The set FP_{LA}^{(h)} of triples (z, τ̃_{0}, [a]) where z = (M, w̃, [j]) ∈ PHT^{(h)}, τ̃_{0} : M ↪ ℂ extends τ_{0} and [a] is a H_{z,τ̃_{0}}^{LA}(𝔸) conjugacy class of elements a ∈ H_{z,τ̃_{0}}^{LA}(ℚ) such that F (a) ⊃ M and with a is elliptic in both H_{z,τ̃_{0}}^{LA}(ℝ) and D_{z,w̃}^{×}. (If F^{+} = ℚ and n = 2 drop the τ̃_{0}, so that FP_{LA}^{(h)} simply consists of pairs (z, [a]).))

3. The set FP^{(h)} of equivalence classes of pairs (a, w̃) where a ∈ G(ℚ) is an element which is elliptic in G(ℝ) and where w̃ is a place of the field F (a) above w such that

(n − h)[F (a) : F] = n[F (a)w̃ : F_{w}].

We consider two pairs (a, w̃) and (a′, w̃′) equivalent if a and a′ are conjugate by an element of G(𝔸) which induces an isomorphism F (a)_{w} →^{∼} F (a′)_{w} taking w̃ to w̃′.

If (a, w̃) ∈ FP^{(h)} then we define an element ι(a, w̃) ∈ G^{(h)}(𝔸^{∞}) as follows. We set ι(a, w̃)^{p} = a ∈ G(𝔸^{∞,p}). We set ι(a, w̃)_{p,0} = ν(a) ∈ ℚ_{p}^{×}. If x|u is a place of F other than w then we set ι(a, w̃)_{x} = a ∈ (B_{x}^{op})^{×}. We set ι(a, w̃)_{w}^{0} =

a ∈ F (a)× ; w̃ ↪ D_{F_{w},n−h}^{×}. Finally we set ι(a, w̃)_{w}^{e} = a ∈ ∏_{x} F (a)_{x}^{×} ⊂ GL_{h}(F_{w}),

where the product is over places x ≠ w̃ of F (a) which divide w. Note that ι(a, w̃) is well defined up to G^{(h)}(𝔸^{∞})-conjugacy.

Our expression for tr (φ|H_{c}(I^{(h)}, F_{ρ} ⊗ L_{ξ})) is in terms of a sum over FP_{AV}^{(h)}. We wish to turn this into a sum over FP^{(h)}. To this end we will consider two maps.

1. The map ϕ : FP_{LA}^{(h)} → FP_{AV}^{(h)} which sends (z, τ̃_{0}, [a]) to the pair (z, [a′]) where a′ is an element of H_{z}^{AV}(ℚ) conjugate to a in H_{z}^{LA}(𝔸^{∞,p}) ≅ H_{z}^{AV}(𝔸^{∞,p}). We will see below that such an a′ exists and is unique up to H_{z}^{AV}(𝔸)-conjugacy.

2. The map ψ : FP_{LA}^{(h)} → FP^{(h)} which sends (z, τ̃_{0}, [a]) (with z = (M, w̃, [j]) to (ι_{z}^{LA}(a), w̃′) where w̃′ is the unique place of M(a) above the place w̃ of M.

Lemma 9.3 The map ϕ is well defined. It is surjective and the fibre of (z, [a]) has [F (a) : F] elements, unless F^{+} = ℚ, n = 2 and [F (a) : F] = 2 in which case it has 1 element.

Proof: Fix z = (M, w̃, [j]) ∈ PHT^{(h)} and τ̃_{0} : M ↪ ℂ extending τ_{0}. We will give a natural map from conjugacy classes in H_{z,τ̃_{0}}^{LA}(𝔸), which are elliptic in H_{z,τ̃_{0}}^{LA}(ℝ) and D_{z,w̃}^{×}, to conjugacy classes in H_{z}^{AV}(𝔸) as follows.

1. We use the isomorphism H_{z,τ̃_{0}}^{LA}(𝔸^{∞,p}) ≅ H_{z}^{AV}(𝔸^{∞,p}), which is canonical up to conjugacy, to associate conjugacy classes in these two groups.

2. We use the isomorphism

ℚ_{p}^{×} × ∏_{x|u,x≠w̃} D_{z,x}^{×} ≅ ℚ_{p}^{×} × ∏_{x|u,x≠w̃} C_{z,x}^{×},

which is canonical up to conjugacy, to associate conjugacy classes in these two groups.

3. We use the natural identification with elliptic conjugacy classes in D_{z,w̃}^{×} ≅ GL_{n/[M:F]}(Mw̃) with conjugacy classes in C_{z,w̃}^{×}. (C_{z,w̃} is a division algebra with centre Mw̃ and dimension (n/[M : F])^{2}.)

4. We use the natural map from elliptic conjugacy classes in H_{z,τ̃_{0}}^{LA}(ℝ) to conjugacy classes in its compact mod centre inner form, H_{z}^{AV}(ℝ). This map is surjective. (Elliptic conjugacy classes transfer to any inner form and in H_{z}^{AV}(ℝ) stable conjugacy and conjugacy coincide because H_{z}^{AV}(ℝ) is compact mod centre.)

We claim that under this map the preimage of [a] has [(M ⊗_{ℚ} ℝ)(a_{∞}) : M ⊗_{ℚ}ℝ] elements. (Unless M^{+} = ℚ and n = 2 when it has only 1 element.) If [a′] is one such preimage then the other preimages are just the [(a′)^{∞}×a′′_{∞}] with a′′_{∞} stably conjugate to a′_{∞} in H_{z,τ̃_{0}}^{LA}(ℝ). Thus the preimages are in bijection with

ker(H^{1}(ℝ, Z_{H_{z,τ̃_{0}}^{LA}}(a′_{∞})) ⟶ H^{1}(ℝ, H_{z,τ̃_{0}}^{LA})).

Elements of this set also parametrise equivalence classes of B ⊗_{F} ((M ⊗_{ℚ} ℝ)(a′_{∞}))-modules with a ∗⊗c-Hermitian ℝ-alternating pairing which are equivalent as B ⊗_{F} M ⊗_{ℚ} ℝ-modules with ∗ ⊗ c-Hermitian ℝ-alternating pairing to V ⊗ ℝ with its standard pairing, ( , ). These in turn are parametrised by sequences of pairs of integers (a_{τ}, b_{τ}) for τ : (M ⊗_{ℚ} ℝ)(a′_{∞}) → ℝ where (a_{τ}, b_{τ}) = (n/[(M ⊗_{ℚ} ℝ)(a′_{∞}) : F ⊗_{ℚ} ℝ], 0) for all but one such embedding τ_{1} for which τ_{1}|_{M} = τ̃_{0} and (a_{τ_{1}}, b_{τ_{1}}) = (n/[(M ⊗_{ℚ}ℝ)(a′_{∞}) : F ⊗_{ℚ}ℝ] − 1, 1). Thus the preimage of [a] ⊂ H_{z}^{AV}(𝔸) does have [(M ⊗_{ℚ} ℝ)(a_{∞}) : M ⊗_{ℚ} ℝ] elements, unless M^{+} = ℚ and n = 2 when it has only 1 element.

To complete the proof of the lemma it suffices to show that [a] ⊂ H_{z,τ̃_{0}}^{LA}(𝔸) maps to [a′] ⊂ H_{z}^{AV}(𝔸) then [a] contains an element of H_{z,τ̃_{0}}^{LA}(ℚ) if and only if [a′] contains an element of H_{z}^{AV}(ℚ). This follows from theorem 6.6 of [Ko2] if we can verify that the group ℜ(I/ℚ) of that paper vanishes for every centraliser I of a semi-simple element in H(ℚ) where H/ℚ is a quasi-split inner form of H_{z}^{AV} (and hence of H_{z,τ̃_{0}}^{LA}). This can be verified as in lemma 2 of [Ko4]. Alternatively one can note that it fits in an exact sequence

ker^{1}(ℚ, I) → ker^{1}(ℚ, H) → Hom (ℜ(I/ℚ), ℚ/ℤ) → A(I) → A(H) → (0)

(combine section 4.6 of [Ko2] with the definition of A(H) and A(I) and the isomorphism 4.2.2 of [Ko1]). If [B : ℚ]/2 is odd then a direct calculation shows that A(H) = A(I) = (0), while if [B : ℚ]/2 is even then the morphism A(I) → A(H) is the unique isomorphism ℤ/2ℤ →^{∼} ℤ/2ℤ. On the other hand the map ker^{1}(ℚ, I) ⟶ ker^{1}(ℚ, H) is surjective because (arguing as on pages 393 and 394 of [Ko3] we see that) the composite

ker^{1}(ℚ, Z(H)) ⟶ ker^{1}(ℚ, Z(I)) ⟶ ker^{1}(ℚ, I) ⟶ ker^{1}(ℚ, H)

(where Z(H) (resp. Z(I)) denotes the centre of H (resp. I)) is an isomorphism. Thus ℜ(I/ℚ) = (0). □

Lemma 9.4 The map ψ is a bijection

ψ : FP_{LA}^{(h)} ⟶^{∼} FP^{(h)}.

Proof: Suppose (a, w̃) represents an element of FP^{(h)}. Let M denote the minimal subfield of F (a) which contains F and for which w̃|_{M} is inert in F (a). (It is an exercise in the splitting of primes in number fields to show such a unique minimal subfield exists.) Let j : M ↪ B^{op} be the tautological embedding. Then the equivalence class of V ⊗_{ℚ} ℝ as a B ⊗_{F} M ⊗_{ℚ} ℝ-module (via j) with ∗ ⊗ c-Hermitian ℝ-alternating pairing has invariants (a_{τ}, b_{τ}) (for τ : M^{+} ↪ ℝ) with (a_{τ}, b_{τ}) = (n/[M : F], 0) for all τ except one for which we get (n/[M : F] − 1, 1). Call this exceptional embedding τ̃_{0}. Its restriction to F^{+} is τ_{0}. Then ((M, w̃|_{M}, [j]), τ̃_{0}, [a]) is a point of FP_{LA}^{(h)} mapping to the class of (a, w̃) and it is unique. The lemma follows. (If F^{+} = ℚ and n = 2 drop τ̃_{0}, which is no longer well defined.) □

Putting these results together we get the following formula for the trace of φ on H_{c}(I^{(h)}, F_{ρ} ⊗ L_{ξ}).

Proposition 9.5 Suppose that φ ∈ C_{c}^{∞}(G^{(h)}(𝔸^{∞})^{+}/ℤ_{p}^{×}×O_{D_{Fw} ,n−h}^{×}) is acceptable, then

tr (φ|H_{c}(I^{(h)}, F_{ρ} ⊗ L_{ξ})) = (−1)^{h}κ_{B} ∑_{[(a,w̃)]∈FP^{(h)}}
[F (a) : F]^{−1}vol (Z_{G}(a)(ℝ)_{0}^{1})^{−1}O_{ι(a)}^{G^{(h)}(𝔸^{∞})}(φ)tr (ρ ⊗ ξ)(ι(a)),

where κ_{B} = 2 if [B : ℚ]/2 is even and κ_{B} = 1 otherwise. (If F^{+} = ℚ and n = 2 we must drop the [F (a) : F]^{−1} term.)

This sum is finite and all the terms occurring are finite numbers.

We choose measures on Z_{G^{(h)}(𝔸^{∞})}(a) and on Z_{G}(a)(ℝ)_{0}^{1} compatible with

• Tamagawa measure on Z_{G}(a)(𝔸)^{1},

• the exact sequence

{1} ⟶ Z_{G}(a)(ℝ)^{1} ⟶ Z_{G}(a)(𝔸)^{1} ⟶ ℤ_{G}(a)(𝔸^{∞}) ⟶ {1},

• the association of measures on Z_{G}(a)(ℝ)_{0}^{1} and Z_{G}(a)(ℝ)^{1} (see page 631 of [Ko5]),

• the association of measures on

Z_{(B_{w}^{op})^{×}}(a) ≅ ∏_{x|w} GL_{n/[F (a):F]}(F (a)_{x})

and on

\begin{matrix}Z_{D_{Fw,n−h}^{×}×GL_{h}(F_{w})}(ι(a)) ≅ \\ D_{F (a)w̃,(n−h)/[F (a)w̃:F_{w}]}^{×} × ∏_{x|w,x≠w̃} GL_{h/([F (a):F]−[F (a)w̃:F_{w}])}(F (a)_{x}) \end{matrix}

(see page 631 of [Ko5]),

• and the isomorphism

\begin{matrix}Z_{G}(a)(𝔸^{∞}) ≅ \\ Z_{G}(a)(𝔸^{∞,p}) × ℚ_{p}^{×} × Z_{D_{Fw,n−h}^{×}×GL_{h}(F_{w})}(ι(a)) × ∏_{i=2}^{r} Z_{(B_{w_{i}}^{op})^{×}}(a). \end{matrix}

10 The cohomology of Igusa varieties

In this section we compute H_{c}(I^{(h)}, F_{ρ} ⊗ L_{ξ}) in terms of H(X, L_{ξ}). We will use results and notation from appendix IV without comment.

For h = 0, ..., n − 1 let P_{h} ⊂ GL_{n} denote the parabolic subgroup consisting of block upper triangular matrices with an (n − h) × (n − h)-block in the top left hand corner and an (h × h)-block in the bottom right hand corner. We will let P_{h}^{op} denote the opposite parabolic (lower triangular matrices with the same block structure). Also let N_{h} (resp. N_{h}^{op}) denote the unipotent radical of P_{h} (resp. P_{h}^{op}) and L_{h} denote the Levi component P_{h} ∩P_{h}^{op} (≅ GL_{n−h} ×GL_{h}).

Fix an irreducible admissible representation ρ of D_{F_{w},n−h}^{×}. We will define a homomorphism

Red_{ρ}^{(h)} : Groth (GL_{n}(F_{w})) ⟶ Groth (D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×} × GL_{h}(F_{w}))

as follows.

First we have a homomorphism

Groth (GL_{n}(F_{w})) ⟶ Groth (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))

which sends [π] to [J_{N_{h}^{op}}(π) ⊗ δ_{P_{h}}^{1/2}] (see appendix I). Secondly we have a homomorphism

Groth (GL_{n−h}(F_{w}) × GL_{h}(F_{w})) ⟶ Groth (D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×} × GL_{h}(F_{w}))

which sends [α ⊗ β] to

∑_{ψ} vol (D_{F_{w},n−h}^{×}/F_{w}^{×})^{−1}tr α(φ_{JL (ρ^{∨}⊗ψ)})[ψ ⊗ β],

where ψ runs over characters of D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×} so that α and ρ^{∨}⊗ψ have the same central character, and where we use associated measures on GL_{n−h}(F_{w}) and D_{F_{w},n−h}^{×}. (That this second map is well defined follows from corollary IV.5.) Red_{ρ}^{(h)} will denote the composite.

The homomorphism Red_{ρ}^{(h)} extends naturally to a homomorphism

Red_{ρ}^{(h)} : Groth (G(𝔸^{∞})) ⟶ Groth (G^{(h)}(𝔸^{∞})).

We will need a couple of lemmas in local harmonic analysis. It is convenient to fix

• the Haar measure on F_{w}^{×} giving O_{F,w}^{×} volume 1,

• the Haar measure on D_{F_{w},n−h}^{×} giving O_{D_{F_{W} ,n−h}}^{×} volume 1,

• and the Haar measure on GL_{n−h}(F_{w}) associated to our choice of Haar measure on D_{F_{w},n−h}^{×}.

Lemma 10.1 Suppose that φ^{0} ∈ C_{c}^{∞}(D_{F_{w},n−h}^{×}/O_{F_{w},n−h}^{×}). Then we can find an element PC_{ρ}(φ^{0}) ∈ C_{c}^{∞}(GL_{n−h}(F_{w})) with the following properties.

1. If π is an irreducible admissible representation of GL_{n−h}(F_{w}) then

tr π(PC_{ρ}(φ^{0})) = ∑_{ψ} tr ψ(φ^{0})vol (D_{F_{w},n−h}^{×}/F_{w}^{×})^{−1}tr π(φ_{JL (ρ^{∨}⊗ψ)})

where ψ runs over characters of D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×} so that α and ρ^{∨} ⊗ ψ have the same central character.

2. If g ∈ GL_{n−h}(F_{w}) is a non-elliptic semi-simple element then

O_{g}^{GL_{n−h}(F_{w})}(PC_{ρ}(φ^{0})) = 0.

3. If g ∈ GL_{n−h}(F_{w}) is an elliptic semi-simple element and if g̃ ∈ D_{F_{w},n−h}^{×} is an element with the same characteristic polynomial then

O_{g}^{GL_{n−h}(F_{w})}(PC_{ρ}(φ^{0})) = (−1)^{(n−h)(1−[F_{w}(g):F_{w}]^{−1})}OD_{Fw,n−h}^{×} ; g̃(φ^{0})tr ρ(g̃).

4. If g ∈ GL_{n−h}(F_{w}) is in the support of PC_{ρ}(φ^{0}) and if λ is an eigenvalue of g then (n − h)w(λ) is in the image under w ∘ det of the support of φ^{0}.

Proof: Via w ∘ det : D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×} →^{∼} ℤ we may and we shall think of φ^{0} ∈ C_{c}^{∞}(ℤ). We will let A^{0} denote the set of elements of GL_{n−h}(F_{w}) all whose eigenvalues have w-valuation in the image under (n − h)^{−1}w ∘ det of the support of φ^{0}. We will let A_{m} ⊂ GL_{n−h}(F_{w}) denote (w ∘ det)^{−1}({m}). We then set

PC_{ρ}(φ^{0}) = (φ^{0} ∘ w ∘ det)φ_{JL (ρ^{∨})}char_{A^{0}},

and

φ̃ = (φ^{0} ∘ w ∘ det)φ_{JL (ρ^{∨})}.

Note that by definition PC_{ρ}(φ^{0}) has property 4 of the lemma.

It follows from lemma IV.1 that for a non-elliptic regular semi-simple element g ∈ GL_{n−h}(F_{w}) we have

O_{g}^{GL_{n−h}(F_{w})}(φ̃) = 0,

while if g ∈ GL_{n−h}(F_{w}) is a elliptic regular semi-simple element and if g̃ ∈ D_{F_{w},n−h}^{×} is an element with the same characteristic polynomial then

O_{g}^{GL_{n−h}(F_{w})}(φ̃) = (−1)^{n−h−1}vol (D_{F_{w},n−h}^{×}/Z_{D_{Fw,n−h}^{×}}(g̃))φ^{0}(g̃)tr ρ(g̃).

Parts 2 and 3 of the lemma follow (because the valuation of all eigenvalues of an elliptic element of GL_{n}(F_{w}) are equal).

It also follows that at all regular semi-simple elements of GL_{n−h}(F_{w}) the functions φ̃ and PC_{ρ}(φ^{0}) have the same orbital integrals. Hence by theorem 2f of appendix 1 of [DKV] we see that for any admissible representation π of GL_{n−h}(F_{w}) we have

tr π(φ̃) = tr π(PC_{ρ}(φ^{0})).

Thus it suffices to prove the first part of the lemma with φ̃ replacing PC_{ρ}(φ^{0}).

Suppose that π is an irreducible admissible representation of GL_{n−h}(F_{w}) with central character ψ_{π}. Also let ψ_{ρ} denote the central character of ρ. If ψ_{π}|_{O_{Fw}^{×}} ≠ ψ_{ρ}|_{O_{Fw}^{×}}^{−1} then tr π(φ̃) = 0 as desired. Thus suppose that ψ_{π}|_{O_{Fw}^{×}} =

ψ_{ρ}|_{O_{Fw}^{×}}. Then

tr π(φ̃)
= ∑_{x∈ℤ} φ^{0}(x)tr π(φ_{JL (ρ^{∨})}char_{A_{x}})
= ∑_{i=0}^{n−h−1} ∑_{x≡imodn−h} φ^{0}(x)(ψ_{π}ψ_{ρ})((x − i)/(n − h))tr π(φ_{JL (ρ^{∨})}char_{A_{i}})
= ∑_{i=0}^{n−h−1} ∑_{x∈ℤ}(n − h)^{−1} ∑_{ψ^{n−h}=ψ_{π}ψ_{ρ}} φ^{0}(x)ψ(x − i)tr π(φ_{JL (ρ^{∨})}char_{A_{i}})
= (n − h)^{−1} ∑_{ψ^{n−h}=ψ_{π}ψ_{ρ}} ∑_{x∈ℤ} φ^{0}(x)ψ(x) ∑_{i=0}^{n−h−1} tr π(φ_{JL (ρ^{∨})⊗ψ}char_{A_{i}})
= ∑_{ψ^{n−h}=ψ_{π}ψ_{ρ}} tr ψ(φ^{0})(n − h)^{−1}vol (O_{D_{Fw,n−h}}^{×})^{−1}tr π(φ_{JL (ρ^{∨})⊗ψ})
= ∑_{ψ^{n−h}=ψ_{π}ψ_{ρ}} tr ψ(φ^{0})vol (D_{F_{w},n−h}^{×}/F_{w}^{×})^{−1}tr π(φ_{JL (ρ^{∨})⊗ψ}).

The lemma follows. □

Lemma 10.2 Let ρ be an irreducible admissible representation of D_{F_{w},n−h}^{×}. Suppose that φ^{0} ∈ C_{c}^{∞}(D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×}) and that φ^{e} ∈ C_{c}^{∞}(GL_{h}(F_{w})). Suppose moreover that if g^{0} (resp. g^{e}) is in the support of φ^{0} (resp. φ^{e}) then the padic valuation of every eigenvalue of g^{0} is strictly less than the p-adic valuation of every eigenvalue of g^{e}. Also fix Haar measures μ_{n} and μ_{h} on GL_{n}(F_{w}) and GL_{h}(F_{w}). Then we can find a function IPC_{ρ}(φ^{0}, φ^{e}; μ_{n}, μ_{h}) ∈ C_{c}^{∞}(GL_{n}(F_{w})) with the following properties.

1. Let g ∈ GL_{n}(F_{w}) be a semi-simple element. Then

\begin{matrix}O_{g}^{GL_{n}(F_{w})}(IPC_{ρ}(φ^{0}, φ^{e}; μ_{n}, μ_{h})) = \\ (−1)^{n−h−1} ∑_{(g^{0},g^{e})} O_{g^{e}}^{GL_{h}(F_{w})}(φ^{e})O_{g^{0}}^{D_{Fw,n−h}^{×}}(φ^{0})tr ρ(g^{0}), \end{matrix}

where (g^{0}, g^{e}) runs over a set of representatives for conjugacy classes in D_{F_{w},n−h}^{×} × GL_{h}(F_{w}) such that, if g̃^{0} denotes a semi-simple element of GL_{n−h}(F_{w}) with the same characteristic polynomial as g^{0}, then (g̃^{0}, g^{e}) is conjugate to g in GL_{n}(F_{w}). Whenever (tr ρ(g^{0}))O_{g^{e}}^{GL_{h}(F_{w})}(φ^{e}) ≠ 0 we see that Z_{GL_{n−h}(F_{w})×GL_{h}(F_{w})}(g̃^{0} × g^{e}) = Z_{GL_{n}(F_{w})}(g̃^{0}, g^{e}) and we take corresponding measures on these two groups in the two sides of this equality of orbital integrals.

2. If π is an irreducible admissible representation of GL_{n}(F_{w}) and if

[J_{N_{h}^{op}}(π) ⊗ δ_{P_{h}}^{1/2}] = ∑_{α,β} m_{α,β}[α ⊗ β]

with α (resp. β) running over irreducible admissible representations of GL_{n−h}(F_{w}) (resp. GL_{h}(F_{w})) and with m_{α,β} ∈ ℤ (and almost all 0), then

tr π(IPC_{ρ}(φ^{0}, φ^{e}; μ_{n}, μ_{h})) =
= ∑_{α,β,ψ} tr ψ(φ^{0})m_{α,β}vol (D_{F_{w},n−h}^{×}/F_{w}^{×})^{−1}tr α(φ_{JL (ρ^{∨}⊗ψ)})tr β(φ^{e})

with α (resp. β, resp. ψ) running over irreducible admissible representations of GL_{n−h}(F_{w}) (resp. GL_{h}(F_{w}), resp. D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×}).

Proof: The choice of measure μ_{h} on GL_{h}(F_{w}) and our fixed choice of Haar measure on GL_{n−h}(F_{w}) determine a Haar measure on L_{h}(F_{w}).

Let 𝔈^{0} (resp. 𝔈^{e}) denote the set of p-adic valuations of elements in the support of φ^{0} (resp. φ^{e}). By our assumptions 𝔈^{0} and 𝔈^{e} are disjoint finite sets. Let 𝔖 denote the set of elements g^{0} × g^{e} ∈ L_{h}(F_{w}) such that the p-adic valuations all the eigenvalues of g^{0} lie in 𝔈^{0} and the p-adic valuations of all the eigenvalues of g^{e} lie in 𝔈^{e}. If g ∈ 𝔖 then Z_{GL_{n}(F_{w})}(g) = Z_{L_{h}(F_{w})}(g). By lemma 10.1 𝔖 contains the support of PC_{ρ}(φ^{0}) × φ^{e}.

Define a function on GL_{n}(F_{w}) by

W (g) = ∑_{g′} O_{g′}^{L_{h}(F_{w})}(PC_{ρ}(φ^{0}) × φ^{e}),

where the sum is over sets of representatives of L_{h}(F_{w})-conjugacy classes contained in the GL_{n}(F_{w}) conjugacy class of g. We will show that this function satisfies the hypotheses of theorem B of section 1.n. of [V1]. It will then follow from that theorem that there is a function

IPC(φ^{0}, φ^{e}; μ_{n}, μ_{h}) ∈ C_{c}^{∞}(GL_{n}(F_{w}))

such that for all g ∈ GL_{n}(F_{w}) we have

O_{g}^{GL_{n}(F_{w})}(IPC(φ^{0}, φ^{e}; μ_{n}, μ_{h})) = ∑_{g′} O_{g′}^{L_{h}(F_{w})}(PC_{ρ}(φ^{0}) × φ^{e}),

where the sum is over sets of representatives of L_{h}(F_{w})-conjugacy classes contained in the GL_{n}(F_{w}) conjugacy class of g, where we use our fixed measures on GL_{n}(F_{w}) and L_{h}(F_{w}), and where whenever O_{g′}^{L_{h}(F_{w})}(PC_{ρ}(φ^{0}) × φ^{e}) ≠ 0 we use conjugate measures on the conjugate groups Z_{GL_{n}}(g)(F_{w}) and Z_{L_{h}}(g′)(F_{w}). The first part of the lemma will then follow from this and lemma 10.1.

First note that W is clearly invariant under conjugation.

Now suppose that (T, u) is a standard pair (for GL_{n}(F_{w})) in the sense of section 1.m of [V1]. If t ∈ T we will let 𝔈(t) denote the multiset of p-adic valuations of eigenvalues of t. We will also let 𝔈^{0}(t) (resp. 𝔈^{e}(t)) denote the submultiset of 𝔈(t) consisting of elements which also lie in 𝔈^{0} (resp. 𝔈^{e}). Note that tu is conjugate to an element of 𝔖 if and only if t is conjugate to an element of 𝔖 if and only if #𝔈^{0} = n − h and #𝔈^{e} = h. In particular the support of W in T u is contained in the set tu such that the p-adic valuation of all the eigenvalues of t lie in the finite set 𝔈^{0} ∪ 𝔈^{e}. Hence W has compact support in T u.

We will let 𝔖(T) denote the set of elements t ∈ T which are conjugate to an element of 𝔖. If s ∈ 𝔖(T) and gsg^{−1} ∈ 𝔖 then

gT ug^{−1} ⊂ Z_{GL_{n}(F_{w})}(gtg^{−1}) ⊂ L_{h}(F_{w}).

If t ∈ T ∩ g^{−1}𝔖g then

W (tu) = O_{gtug^{−1}}^{L_{h}(F_{w})}(PC_{ρ}(φ^{0}) × φ^{e}).

In particular if s is a regular element of T (in the sense of section 1.l. of [V1]) then we see that W is constant in some neighbourhood of s (by, for instance, theorem A of section 1.n. of [V1]).

Suppose s ∈ T. Let u_{1} = u, u_{2}, ..., u_{m} be unipotent elements of Z_{GL_{n}(F_{w})}(s) such that

• for all i the GL_{n}(F_{w})-orbit of su_{i} is open in the union of the GL_{n}(F_{w})- orbits of su_{j} for j ≤ i;

• and for any unipotent element v ∈ Z_{GL_{n}(F_{w})}(s), su is in the closure of the GL_{n}(F_{w})-orbit of sv if and only if sv is GL_{n}(F_{w}) conjugate to some su_{j}.

(See section 1.j. of [V1].) Fix a Haar measure on Z_{GL_{n}}(s)(F_{w}). This determines a canonical Haar measure on each Z_{GL_{n}}(su_{j})(F_{w}) (see section 1.d. of [V1]). Then we can choose functions f_{1}^{GL_{n}(F_{w})}, ..., f_{m}^{GL_{n}(F_{w})} ∈ C_{c}^{∞}(GL_{n}(F_{w})) such that

• for j < i the support of f_{i}^{GL_{n}(F_{w})} does not meet the GL_{n}(F_{w})-orbit of su_{j},

• if j ≠ i then O_{su_{j}}^{GL_{n}(F_{w})}(f_{i}^{GL_{n}(F_{w})}) = 0,

• and O_{su_{i}}^{GL_{n}(F_{w})}(f_{i}^{GL_{n}(F_{w})}) = 1.

(See section 1.k. of [V1].) To verify the hypotheses of theorem B of section 1.n. of [V1] it remains to show that there is a neighbourhood V of s in T such that for all t ∈ V which are regular in T (in the sense of section 1.l. of [V1]) we have the equality

W (tu) = ∑_{i} W (su_{i})O_{tu}^{GL_{n}(F_{w})}(f_{i}^{GL_{n}(F_{w})}).

(This equality is independent of the choice of Haar measure on Z_{GL_{n}}(tu)(F_{w}) as long as in the case W (tu) ≠ 0 we choose the conjugate Haar measure on Z_{L_{h}}(atua^{−1})(F_{w}) = aZ_{GL_{n}}(tu)(F_{w})a^{−1}, where atua^{−1} ∈ 𝔖.)

First suppose that s ∉ 𝔖(T). Then each W (su_{i}) = 0. On the other hand we can find a neighbourhood V of s in T such that if t ∈ V then 𝔈(t) = 𝔈(s). Then V ∩ 𝔖(T) = ∅ and so for any t ∈ V we have W (tu) = 0 and the desired equality holds.

Now suppose that s ∈ 𝔖(T) and that gsg^{−1} ∈ 𝔖. Replacing T by gT g^{−1}, s by gsg^{−1} and u by gug^{−1} we may suppose that T ⊂ L_{h}, s ∈ 𝔖 and u ∈ L_{h}(F_{w}). Because s ∈ 𝔖 we see • that for each i, u_{i} ∈ L_{h}(F_{w});

• that for all i the L_{h}(F_{w})-orbit of su_{i} is open in the union of the L_{h}(F_{w})- orbits of the su_{j} for j ≤ i;

• and that for any unipotent element v ∈ Z_{GL_{n}(F_{w})}(s), su is in the closure of the L_{h}(F_{w})-orbit of sv if and only if sv is L_{h}(F_{w})-conjugate to some su_{j}.

It follows from lemma 2.5. of [V1] that we can find a neighbourhood V of s in T such that

• V ⊂ 𝔖

• and for any compact set A ⊂ GL_{n}(F_{w}) one can find a compact set C ⊂

L_{h}(F_{w})\GL_{n}(F_{w}) such that, if L_{h}(F_{w})g ∉ C then for each i = 1, ..., m

g^{−1}V u_{i}g ∩ A = ∅.

As in section 2.5. of [V1] it follows that we can find h ∈ C_{c}^{∞}(GL_{n}(F_{w})) such that

∫_{L_{h}(F_{w})} h(xg)dx

is 1 if L_{h}(F_{w})g ∈ C and 0 otherwise. If we set

f_{i}^{L_{h}(F_{w})}(g) = ∫_{GL_{n}(F_{w})} h(x)f_{i}^{GL_{n}(F_{w})}(x^{−1}gx)dx

then we see that • f_{i}^{L_{h}(F_{w})} ∈ C_{c}^{∞}(L_{h}(F_{w})),

• for j < i the support of f_{i}^{L_{h}(F_{w})} does not meet the L_{h}(F_{w})-orbit of su_{j},

• for t ∈ V and for any i, j = 1, ..., m we have

O_{tu_{j}}^{GL_{n}(F_{w})}(f_{i}^{GL_{n}(F_{w})}) = O_{tu_{j}}^{L_{h}(F_{w})}(f_{i}^{L_{h}(F_{w})})

(argue as page 954 of [V1]),

• if j ≠ i then O_{su_{j}}^{L_{h}(F_{w})}(f_{i}^{L_{h}(F_{w})}) = 0,

• and O_{su_{i}}^{L_{h}(F_{w})}(f_{i}^{L_{h}(F_{w})}) = 1.

Using theorem A of section 1.n. of [V1] we see that for t ∈ V which is also regular as an element of T we have

\begin{aligned}W (tu) & = & ∑_{i} O_{su_{i}}^{L_{h}(F_{w})}(PC_{ρ}(φ^{0}) × φ^{e})O_{tu}^{L_{h}(F_{w})}(f_{i}^{L_{h}(F_{w})}) \\ & = & ∑_{i} W (su_{i})O_{tu}^{GL_{n}(F_{w})}(f_{i}^{GL_{n}(F_{w})}). \end{aligned}

This completes the proof of the first part of the lemma.

For the second part, the Weyl integration formula tells us that

tr π(IPC_{ρ}(φ^{0}, φ^{e}; μ_{n}, μ_{h})) = ∑_{T} (#W_{G}(T))^{−1}
∫_{T^{reg}} D_{G}(t)O_{t}^{GL_{n}(F_{w})}(IPC_{ρ}(φ^{0}, φ^{e}; μ_{n}, μ_{h}))χ_{π}(t)dt,

where

• T runs over GL_{n}(F_{w})-conjugacy classes of maximal tori in GL_{n}(F_{w}),

• W_{G}(T) denotes the normaliser of T in GL_{n}(F_{w}) modulo T,

• T^{reg} denotes the subset of regular elements of T,

• D_{G}(t) = | det((ad (t) − 1)|_{Lie G/Lie T})|.

(See for instance section A.3.f of [DKV] and note that χ_{π} is locally integrable.) By the first part of the lemma this can be rewritten

tr π(IPC_{ρ}(φ^{0}, φ^{e}; μ_{n}, μ_{h})) = ∑_{T} (#W_{L_{h}(F_{w})}(T))^{−1}
∫_{T^{reg}} D_{G}(t^{0} × t^{e})O_{t^{0}}^{GL_{n−h}(F_{w})}(PC_{ρ}(φ^{0}))O_{t^{e}}^{GL_{h}(F_{w})}(φ^{e})χ_{ρ}(t)dt^{0}dt^{e},

where now

• T runs over L_{h}(F_{w})-conjugacy classes of maximal tori in L_{h}(F_{w}),

• and W_{L_{h}(F_{w})}(T) denotes the normaliser of T in L_{h}(F_{w}) modulo T.

Let P_{t^{0}×t^{e}} denote the parabolic associated to t^{0} ×t^{e} as in section 2 of [Cas]. If P_{t^{0}×t^{e}} is not a subset of P_{h}^{op} then by the assumption on the supports of φ^{0} and φ^{e} and by lemma 10.1 we see that

O_{t^{0}}^{GL_{n−h}(F_{w})}(PC_{ρ}(φ^{0}))O_{t^{e}}^{GL_{h}(F_{w})}(φ^{e}) = 0.

If on the other hand P_{t^{0}×t^{e}} ⊂ P_{h}^{op} then

• D_{G}(t^{0} × t^{e}) = D_{L_{h}}(t^{0} × t^{e})δ_{P_{h}}(t^{0} × t^{e}) (from the definitions)

• and χ_{π}(t^{0} × t^{e}) = χ_{π_{N{op \atop h}}}(t^{0} × t^{e}) (by theorem 5.2 of [Cas]).

Thus we obtain

tr π(IPC_{ρ}(φ^{0}, φ^{e}; μ_{n}, μ_{h}))

= ∑_{T} (#W_{L_{h}(F_{w})}(T))^{−1}

∫_{T^{reg}} D_{L_{h}(F_{w})}(t^{0} × t^{e})O_{t^{0}}^{GL_{n−h}(F_{w})}(PC_{ρ}(φ^{0}))O_{t^{e}}^{GL_{h}(F_{w})}(φ^{e})χ_{π_{N{op \atop h}}}(t)δ_{P_{h}}(t)dt

= tr (J_{N_{h}^{op}}(π) ⊗ δ_{P_{h}}^{1/2})(PC_{ρ}(φ^{0}) × φ^{e}),

and the second part of the lemma follows. □

Corollary 10.3 If π is an admissible representation of GL_{n}(F_{w}) and if φ^{0}, φ^{e}, μ_{n} and μ_{h} are as in the lemma then

tr Red_{ρ}^{(h)}(π)(φ^{0} × φ^{e}) = tr π(IPC_{ρ}(φ^{0}, φ^{e}; μ_{n}, μ_{h})).

We can now prove our second main theorem. Theorem 10.4

Red_{ρ}^{(h)}[H(X, L_{ξ})^{ℤ_{p}^{×}}] = n[H_{c}(I^{(h)}, F_{ρ} ⊗ L_{ξ})]

in Groth (G^{(h)}(𝔸^{∞})).

Proof: By lemma 9.1, it suffices to check that for any

• φ^{p} ∈ C_{c}^{∞}(G(𝔸^{∞,p}) × (ℚ_{p}^{×}/ℤ_{p}^{×}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}),

• φ^{0} ∈ C_{c}^{∞}(D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×}),

• and φ^{e} ∈ C_{c}^{∞}(GL_{h}(F_{w})),

such that φ = φ^{w} × φ^{0} × φ^{e} is acceptable, we have

tr (Red_{ρ}^{(h)}(H(X, L_{ξ})))(φ) = ntr H_{c}(I^{(h)}, L_{ξ} ⊗ F_{ρ})(φ).

(Note that, if C ∈ ℝ and ψ_{0}, ..., ψ_{m} are characters of ℤ we can a function φ′ ∈ C_{c}^{∞}(ℤ_{>C}) such that ψ_{0}(φ′) = 1 but ψ_{i}(φ′) = 0 for i = 1, ..., m.)

But by corollary 10.3, proposition VII.1 and lemma 10.2 we have

tr (Red_{ρ}^{(h)}(H(X, L_{ξ})))(φ)
= tr H(X, L_{ξ})(φ^{w} × IPC_{ρ}(φ^{0}, φ^{e}; μ_{n}, μ_{h}))
= (−1)^{n}κ_{B}n ∑_{a}[F (a) : F]^{−1}(−1)^{n/[F (a):F]}vol (Z_{G}(a)(ℝ)_{0}^{1})^{−1}
O_{a}^{G(𝔸^{∞})}(φ^{w} × IPC_{ρ}(φ^{0}, φ^{e}; μ_{n}, μ_{h}))
= (−1)^{h}κ_{B}n ∑_{[(a,w̃)]∈FP^{(h)}}[F (a) : F]^{−1}(−1)^{n/[F (a):F]−(n−h)/[F (a)w̃:F_{w}]}
vol (Z_{G}(a)(ℝ)_{0}^{1})^{−1}O_{ι(a)}^{G^{(h)}(𝔸^{∞})}(φ)tr (ρ ⊗ ξ)(ι(a)),

where we drop the term [F (a) : F]^{−1} if n = 2 and F^{+} = ℚ. Here κ_{B} = 2 if [B : ℚ]/2 is even and = 1 otherwise and the choices of measures are as in proposition 9.5. Comparing this formula with proposition 9.5, the trace identity and hence the theorem follows. □

11 The main theorems

To state our third main theorem let us establish a little more notation. Let

red_{ρ}^{(h)} : Groth (GL_{n}(F_{w})) ⟶ Groth (GL_{h}(F_{w}))

to be the composite of the map

Groth (GL_{n}(F_{w})) ⟶ Groth (GL_{n−h}(F_{w}) × GL_{h}(F_{w}))
[π] ⟼ [J_{N_{h}^{op}}(π) ⊗ δ_{P_{h}}^{1/2}]

and the map

Groth (GL_{n−h}(F_{w}) × GL_{h}(F_{w})) ⟶ Groth (GL_{h}(F_{w}))

which sends [α ⊗ β] to

vol (D_{F_{w},n−h}^{×}/F_{w}^{×})^{−1}tr α(φ_{JL (ρ^{∨})})[β]

if the product of the central characters of α and ρ is 1 and sends [α ⊗ β] to 0 otherwise. Then red_{ρ}^{(h)} extends to a homomorphism

Groth (G(𝔸^{∞})) ⟶ Groth (G(𝔸^{∞,p}) × ℚ_{p}^{×} × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×}).

Note that

Red_{ρ}^{(h)}(π) = ∑_{ψ} ψ ⊗ red_{ρ}^{(h)}(π),

where ψ runs over characters of D_{F_{w},n−h}^{×}/O_{D_{Fw,n−h}}^{×}.

Combining theorems 7.9 and 10.4 we at once obtain the following result.

Theorem 11.1 In Groth_{l}(G(𝔸^{∞}) × W_{F_{w}}) we have

n[H(X, L_{ξ})^{ℤ_{p}^{×}}] = ∑_{h=0}^{n−1} ∑_{ρ} Ind _{P_{h}(F_{w})}^{GL_{n}(F_{w})}red_{ρ}^{(h)}[H(X, L_{ξ})^{ℤ_{p}^{×}}] ∗_{|Art _{K}^{−1}|} [Ψ_{F_{w},l,n−h}(ρ)]

where ρ runs over irreducible admissible representations of D_{F_{w},n−h}^{×} and where |Art _{K}^{−1}| : W_{F_{w}} → p^{ℤ} ⊂ ℚ_{p}^{×}.

Let us rephrase this theorem another way. Define

n-red_{ρ}^{(h)} : Groth (G(𝔸^{∞})) ⟶ Groth (G(𝔸^{∞,p}) × ℚ_{p}^{×} × GL_{h}(F_{w}) × ∏_{i=2}^{r}(B_{w_{i}}^{op})^{×})

in the same manner we defined red_{ρ}^{(h)} except that we replace J_{N_{h}^{op}}(π) ⊗ δ_{P_{h}}^{1/2} by simply J_{N_{h}^{op}}(π). We may decompose

[H(X, L_{ξ})] = ∑_{π}[π][R_{ξ}(π)]

where π runs over irreducible admissible representations of G(𝔸^{∞}) and where [R_{ξ}(π)] ∈ Groth (Gal (F^{ac}/F)). We will need the following lemma, which follows from lemma 4.1 and corollary VI.2. Lemma 11.2 Suppose that π and π′ are irreducible admissible representations of G(𝔸^{∞}) such that

• π^{p} ≅ (π′)^{p},

• [R_{ξ}(π)] ≠ 0,

• and [R_{ξ}(π′)] ≠ 0. Then π_{p} ≅ π′_{p}.

Then we have the following reformulation of theorem 11.1. Theorem 11.3 Suppose that π = π^{p} × π_{p,0} × π_{w_{1}} × ... × π_{w_{r}} is an irreducible admissible representation of G(𝔸^{∞}) such that π_{p,0}|_{ℤ_{p}^{×}} = 1. Then in Groth_{l}(GL_{n}(F_{w}) × W_{F_{w}}) we have

n[π_{w}][R_{ξ}(π)|_{W_{Fw}}] = (dim[R_{ξ}(π)]) ∑_{h=0}^{n−1} ∑_{ρ} n-Ind _{P_{h}(F_{w})}^{GL_{n}(F_{w})}((n-red_{ρ}^{(h)}[π_{w}])
[Ψ_{F_{w},l,n−h}(ρ) ⊗ ((π_{p,0}^{−1} ⊗ | |_{p}^{−h/2}) ∘ Art _{ℚ_{p}}^{−1})|_{W_{Fw}}])

where ρ runs over irreducible admissible representations of D_{F_{w},n−h}^{×}.

As a special case we get the following consequence. Corollary 11.4 Suppose that π = π^{p} × π_{p,0} × π_{w_{1}} × ... × π_{w_{r}} is an irreducible admissible representation of G(𝔸^{∞}) such that π_{p,0}|_{ℤ_{p}^{×}} = 1 and π_{w} is supercuspidal. Then in Groth_{l}(GL_{n}(F_{w}) × W_{F_{w}}) we have n[π_{w}][R_{ξ}(π)|_{W_{Fw}}] = (dim[R_{ξ}(π)])[Ψ_{F_{w},l,n}(JL^{−1}(π_{w})^{∨}) ⊗ (π_{p,0}^{−1} ∘ Art _{ℚ_{p}}^{−1})|_{W_{Fw}}].

Proof: Use the following facts.

• If π_{w} is supercuspidal and if N is the unipotent radical of a proper parabolic subgroup of GL_{n}(F_{w}) then J_{N}(π_{w}) = (0).

• If π_{w} is a supercuspidal representation of GL_{n}(F_{w}) and if π′_{w} is a square integrable representation of GL_{n}(F_{w}) with the same central character, then it follows from the results listed in appendix IV that tr π_{w}(φ_{π′_{w}}) = 0 unless π′_{w} ≅ π_{w} in which case it equals vol (D_{F_{w},n}^{×}/F_{w}^{×}).

□

Theorem 11.5 Suppose that K is a p-adic field and l ≠ p is a prime. Suppose that π is a supercuspidal representation of GL_{g}(K). Then there is a continuous semi-simple representation

r_{l}(π) : W_{K} ⟶ GL_{g}(ℚ_{l}^{ac})

such that

[Ψ_{K,l,g}(JL^{−1}(π^{∨}))] = [π ⊗ r_{l}(π)].

Proof: Choose (E, F^{+}, w, B, ∗, τ_{0}, β, Λ_{i}, ξ) as in section 1 and such that

• F_{w}^{+} ≅ K,

• [B : F] = g^{2},

• and ξ is sufficiently regular that H^{i}(X, L_{ξ}) = (0) for i ≠ g − 1 (see the last paragraph of section 1 of [Ko4] for the existence of such a ξ).

By corollary VII.2 we may choose an irreducible admissible representation π̃ of G(𝔸^{∞}) such that

• [R_{ξ}(π̃)] ≠ 0,

• π̃_{w} ≅ π ⊗ (ψ ∘ det) for some unramified character ψ of K^{×}/O_{K}^{×},

• and π_{p,0}|_{ℤ_{p}^{×}} = 1.

By the previous corollary we see that in Groth_{l}(GL_{g}(K) × W_{K}) we have

g[π̃_{w} ⊗ R_{ξ}(π̃)|_{W_{K}}] = (dim[R_{ξ}(π̃)])[Ψ_{K,l,g}(JL^{−1}(π̃_{w}^{∨})) ⊗ (π̃_{p,0}^{−1} ∘ Art ℚ_{p}^{1})|_{W_{K}}].

Thus, if we set

[r_{l}(π)] = g(dim[R_{ξ}(π̃)])^{−1}[R_{ξ}(π̃)|_{W_{K}} ⊗ (π̃_{p,0} ∘ Art _{ℚ_{p}}^{−1})|_{W_{K}} ⊗ (ψ ∘ Art _{K}^{−1})],

we see that

[r_{l}(π)] ∈ Groth_{l}(W_{K})

and

[Ψ_{K,l,g}(JL^{−1}(π^{∨}))] = [π][r_{l}(π)].

Clearly dim[r_{l}(π)] = g. Finally by our assumption on ξ, we see that R_{ξ}(π̃) is a true representation and hence [r_{l}(π)] can also be represented by a true representation r_{l}(π). □

The following lemma follows from the definitions, from lemma 3.7 and from lemma 3.8.

Lemma 11.6 Suppose that K and K′ are p-adic fields and l ≠ p is a prime.

1. If σ : K →^{∼} K′ is an isomorphism of ℚ_{p}-algebras then for any irreducible supercuspidal representation π′ of GL_{g}(K′) we have

r_{l}(π′ ∘ σ) = r_{l}(π′)^{σ},

where, if we fix an extension σ̃ of σ to an isomorphism σ̃ : K^{ac} →^{∼} (K′)^{ac}, then r_{l}(π′)^{σ}(x) = r_{l}(π′)(σ̃xσ̃^{−1}).

2. If σ ∈ Aut (ℚ_{l}^{ac}) and if π is an irreducible supercuspidal representation

of GL_{g}(K) then

r_{l}(σ(π)) = σ(r_{l}(π)).

3. If π is an irreducible supercuspidal representation of GL_{g}(K) and if ψ is a character of K^{×}/O_{K}^{×} then

r_{l}(π ⊗ (ψ ∘ det)) = r_{l}(π) ⊗ (ψ^{−1} ∘ Art _{K}^{−1}).

4. If π is a character of K^{×} then r_{l}(π) = π^{−1} ∘ Art _{K}^{−1}.

For the rest of this section we will use without comment the notation established in appendix IV.

Theorem 11.7 Suppose that K is a p-adic field and l ≠ p is a prime. Suppose also that s and g are positive integers and that π is an irreducible supercuspidal representation of GL_{g}(K). Then

[Ψ_{K,l,gs}(JL^{−1}(Sp_{s}(π)^{∨}))] =
∑_{j=1}^{s}(−1)^{s−j}[Sp_{j}(π) ⊞ (π ⊗ | det |^{j}) ⊞ ... ⊞ (π ⊗ | det |^{s−1})]
[r_{l}(π ⊗ | det |^{j−1}) ⊗ |Art _{K}^{−1}|^{g(1−s)/2}].

Proof: We will argue by induction on s. The case s = 1 is just the previous theorem. Thus suppose the theorem is proved for all s′ < s.

Choose (E, F^{+}, w, B, ∗, τ_{0}, β, Λ_{i}, ξ) as in section 1 and such that • F_{w}^{+} ≅ K,

• and [B : F] = g^{2}s^{2}. By corollary VII.2 we may choose an irreducible admissible representation π̃ of G(𝔸^{∞}) such that

• dim[R_{ξ}(π̃)] ≠ 0,

• π̃_{w} ≅ Sp_{s}(π) ⊗ (ψ ⊗ det) for some character ψ of K^{×}/O_{K}^{×},

• and π_{p,0}|_{ℤ_{p}^{×}} = 1. If we write π′ for π ⊗ (ψ ∘ det), then by theorem 11.3 and lemma IV.3 we

see that gs[Sp_{s}(π′)][R_{ξ}(π̃)] = (dim[R_{ξ}(π̃)]) ∑_{h′=0}^{s−1} n-Ind _{P_{gh′}(K)}^{GL_{sg}(K)}[Sp_{h′}(π′ ⊗ | det |^{s−h′})× Ψ_{K,l,g(s−h′)}(JL^{−1}(Sp_{s−h′}(π′)^{∨})) ⊗ ((π̃_{p,0}^{−1} ⊗ | |_{p}^{−gh′/2}) ∘ Art _{ℚ_{p}}^{−1})|_{W_{K}}].

By the inductive hypothesis we may rewrite this gs[Sp_{s}(π′)][R_{ξ}(π̃) ⊗ ((π̃_{p,0} ⊗ | |_{p}^{g(s−1)/2}) ∘ Art _{ℚ_{p}}^{−1})|_{W_{K}}] = (dim[R_{ξ}(π̃)])[Ψ_{K,l,gs}(JL^{−1}(Sp_{s}(π′)^{∨})) ⊗ |Art _{K}^{−1}|^{g(s−1)/2}]+ (dim[R_{ξ}(π̃)]) ∑_{h′=1}^{s−1} ∑_{j=1}^{s−h′}(−1)^{s−h′−j}n-Ind _{P_{h′g}(K)}^{GL_{gs}(K)}[(Sp_{j}(π′) ⊞ (π′ ⊗ | det |^{j})⊞ ... ⊞ (π′ ⊗ | det |^{s−h′−1})) × Sp_{h′}(π′ ⊗ | det |^{s−h′})][r_{l}(π′ ⊗ | det |^{j−1})].

By lemma IV.2 this can be rewritten

[Ψ_{K,l,gs}(JL^{−1}(Sp_{s}(π′)^{∨})) ⊗ |Art _{K}^{−1}|^{g(s−1)/2}] =

gs(dim[R_{ξ}(π̃)])^{−1}[Sp_{s}(π′)][R_{ξ}(π̃) ⊗ ((π̃_{p,0} ⊗ | |_{p}^{g(s−1)/2}) ∘ Art _{ℚ_{p}}^{−1})|_{W_{K}}]− ∑_{h′=1}^{s−1}[Sp_{s}(π′)][r_{l}(π′ ⊗ | det |^{s−1−h′})]− ∑_{h′=1}^{s−1} ∑_{j=1}^{s−1−h′}(−1)^{s−h′−j}[ω(Γ⃗_{j,s−1−h′−j,h′+1}(π′))][r_{l}(π′ ⊗ | det |^{j−1})]− ∑_{h′=1}^{s−1} ∑_{j=1}^{s−h′}(−1)^{s−h′−j}[ω(Γ⃗_{j,s−h′−j,h′}(π′))][r_{l}(π′ ⊗ | det |^{j−1})] =

gs(dim[R_{ξ}(π̃)])^{−1}[Sp_{s}(π′)][R_{ξ}(π̃) ⊗ ((π̃_{p,0} ⊗ | |_{p}^{g(s−1)/2}) ∘ Art _{ℚ_{p}}^{−1})|_{W_{K}}]− ∑_{h′=1}^{s−1}[Sp_{s}(π′)][r_{l}(π′ ⊗ | det |^{s−1−h′})]+ ∑_{j=1}^{s−2} ∑_{h′′=2}^{s−j}(−1)^{s−h′′−j}[ω(Γ⃗_{j,s−h′−j,h′}(π′))][r_{l}(π′ ⊗ | det |^{j−1})]− ∑_{j=1}^{s−1} ∑_{h′=1}^{s−j}(−1)^{s−h′−j}[ω(Γ⃗_{j,s−h′−j,h′}(π′))][r_{l}(π′ ⊗ | det |^{j−1})] =

gs(dim[R_{ξ}(π̃)])^{−1}[Sp_{s}(π′)][R_{ξ}(π̃) ⊗ ((π̃_{p,0} ⊗ | |_{p}^{g(s−1)/2}) ∘ Art _{ℚ_{p}}^{−1})_{W_{K}}]− ∑_{h′=1}^{s−1}[Sp_{s}(π′)][r_{l}(π′ ⊗ | det |^{s−1−h′})]+ ∑_{j=1}^{s−1}(−1)^{s−j}[ω(Γ⃗_{j,s−1−j,1}(π′))][r_{l}(π′ ⊗ | det |^{j−1})] =

∑_{j=1}^{s}[Sp_{j}(π′) ⊞ (π′ ⊗ | det |^{j}) ⊞ ... ⊞ (π′ ⊗ | det |^{s−1})][r_{l}(π′ ⊗ | det |^{j−1})]+ [Sp_{s}(π′)][A′],

for some [A′] ∈ Groth (W_{K}). Then by lemma 3.7 we see that

[Ψ_{K,l,gs}(JL^{−1}(Sp_{s}(π)^{∨}))] =
∑_{j=1}^{s}[Sp_{j}(π) ⊞ (π ⊗ | det |^{j}) ⊞ ... ⊞ (π ⊗ | det |^{s−1})]
[r_{l}(π′ ⊗ | det |^{j−1}) ⊗ |Art _{K}^{−1}|^{g(1−s)/2}] + [Sp_{s}(π)][A],

for some [A] ∈ Groth (W_{K}).

It remains to show that [A] = 0. By lemma VII.4 we may choose an irreducible admissible representation π̃′ of G(𝔸^{∞}) such that

• dim[R_{ξ}(π̃′)] ≠ 0,

• π̃′_{w} ≅ (π ⊞ ... ⊞ (π ⊗ | det |^{s−1})) ⊗ (ψ′ ∘ det)^{−1} for some character ψ′ of K^{×}/O_{K}^{×},

• and π̃′_{p,0}|_{ℤ_{p}^{×}} = 1.

Now let π′ denote π ⊗ (ψ′ ∘ det)^{−1}. By theorem 11.3 and lemma IV.3 we also see that

gs[π′ ⊞ ... ⊞ (π′ ⊗ | det |^{s−1})][R_{ξ}(π̃′)] =

(dim[R_{ξ}(π̃′)]) ∑_{h′=0}^{s−1}(−1)^{s−1−h′}n-Ind _{P_{gh′}(K)}^{GL_{gs}(K)}[(π′ ⊞ ... ⊞ (π′ ⊗ | det |^{h′−1}))× Ψ_{K,l,g(s−h′)}(JL^{−1}(Sp_{s−h′}(π′ ⊗ | det |^{h′})^{∨}))⊗ (((π̃′_{p,0})^{−1} ⊗ | |_{p}^{−gh′/2}) ∘ Art _{ℚ_{p}}^{−1})|_{W_{K}}].

By the inductive hypothesis and what we have already proved, this can be rewritten

gs[π′ ⊞ ... ⊞ (π′ ⊗ | det |^{s−1})][R_{ξ}(π̃′)|_{W_{K}} ⊗ (π̃′_{p,o} ∘ Art _{ℚ_{p}}^{−1})|_{W_{K}}] =

(−1)^{s−1}(dim[R_{ξ}(π̃′)])[Sp_{s}(π′)][A ⊗ (ψ′ ∘ det)]+ (dim[R_{ξ}(π̃′)]) ∑_{h′=0}^{s−1} ∑_{j=1}^{s−h′}(−1)^{j−1}n-Ind _{P_{gh′}(K)}^{GL_{gs}(K)}[(π′ ⊞ ... ⊞ (π′ ⊗ | det |^{h′−1})) ×(Sp_{j}(π′ ⊗ | det |^{h′}) ⊞ (π′ ⊗ | det |^{h′+j}) ⊞ ... ⊞ (π′ ⊗ | det |^{s−1}))] [r_{l}(π′ ⊗ | det |^{h′+j−1}) ⊗ |Art _{K}^{−1}|^{g(1−s)/2}].

Again using lemma IV.2 this becomes

gs[π′ ⊞ ... ⊞ (π′ ⊗ | det |^{s−1})][R_{ξ}(π̃′)|_{W_{K}} ⊗ (π̃′_{p,o} ∘ Art _{ℚ_{p}}^{−1})|_{W_{K}}] =

(−1)^{s−1}(dim[R_{ξ}(π̃′)])[Sp_{s}(π′)][A ⊗ (ψ′ ∘ det)]+ (dim[R_{ξ}(π̃′)]) ∑_{h′=0}^{s−1} ∑_{j=1}^{s−h′}(−1)^{j−1}[ω(Γ⃗′_{h′,j,s−1−h′−j}(π′))] [r_{l}(π′ ⊗ | det |^{h′+j−1}) ⊗ |Art _{K}^{−1}|^{g(1−s)/2}]− ∑_{h′′=0}^{s−2} ∑_{j′=2}^{s−h′′}(−1)^{j′−1}[ω(Γ⃗′_{h′′,j′,s−1−h′′−j′}(π′))] [r_{l}(π′ ⊗ | det |^{h′+j−1}) ⊗ |Art _{K}^{−1}|^{g(1−s)/2}] =

(−1)^{s−1}(dim[R_{ξ}(π̃′)])[Sp_{s}(π′)][A ⊗ (ψ′ ∘ det)]+ (dim[R_{ξ}(π̃′)]) ∑_{h′=0}^{s−1}[π′ ⊞ ... ⊞ (π′ ⊗ | det |^{s−1})] [r_{l}(π′ ⊗ | det |^{h′}) ⊗ |Art _{K}^{−1}|^{g(1−s)/2}].

In particular we see that [A] = 0, as desired. □

Suppose that π is an irreducible admissible representation of GL_{g}(K). Then we can find

• a parabolic subgroup P ⊂ GL_{n} with a Levi component isomorphic to GL_{g_{1}} × ... × GL_{g_{t}},

• and an irreducible supercuspidal representation π_{i} of GL_{g_{i}}(K); such that π is a subquotient of

n-Ind _{P (K)}^{GL_{g}(K)}(π_{1} × ... × π_{t}).

Moreover the multiset {π_{i}} is independent of all choices. Thus we may define

[r_{l}(π)] = ∑_{i=1}^{t}[r_{l}(π_{i}) ⊗ |Art _{K}^{−1}|^{(g_{i}−g)/2}].

The following lemma follows at once from lemma 11.6.

Lemma 11.8 Suppose that K and K′ are p-adic fields and l ≠ p is a prime.

1. If σ : K →^{∼} K′ is an isomorphism of ℚ_{p}-algebras then for any irreducible admissible representation π′ of GL_{g}(K′) we have

r_{l}(π′ ∘ σ) = r_{l}(π′)^{σ},

where, if we fix an extension σ̃ of σ to an isomorphism σ̃ : K^{ac} →^{∼} (K′)^{ac}, then r_{l}(π′)^{σ}(x) = r_{l}(π′)(σ̃xσ̃^{−1}).

2. If σ ∈ Aut (ℚ_{l}^{ac}) and if π is an irreducible admissible representation of

GL_{g}(K) then

r_{l}(σ(π)) = σ(r_{l}(π)).

3. If π is an irreducible admissible representation of GL_{g}(K) and if ψ is a character of K^{×}/O_{K}^{×} then

r_{l}(π ⊗ (ψ ∘ det)) = r_{l}(π) ⊗ (ψ^{−1} ∘ Art _{K}^{−1}).

We now return to the analysis of [R_{ξ}(π)|_{W_{Fw}}].

Theorem 11.9 Suppose that π is an irreducible admissible representation of G(𝔸^{∞}). Then

n[R_{ξ}(π)|_{W_{Fw}}] = (dim[R_{ξ}(π)])[r_{l}(π_{w}) ⊗ (π_{p,0} ∘ |Art _{F_{w}}^{−1}|)].

Proof: Consider the two homomorphisms

Θ_{1}, Θ_{2} : Groth (GL_{n}(F_{w})) ⟶ Groth_{l}(GL_{n}(F_{w}) × W_{F_{w}})

defined by

Θ_{1}([π]) = [π ⊗ r_{l}(π)]

for any irreducible π and

Θ_{2}([π]) = ∑_{h=0}^{n−1} ∑_{ρ} n-Ind _{P_{h}(F_{w})}^{GL_{n}(F_{w})}n-red_{ρ}^{(h)}[π][Ψ_{F_{w},l,n−h}(ρ) ⊗ |Art _{K}^{−1}|^{−h/2}]

for any admissible π (where ρ runs over irreducible admissible representations of D_{F_{w},n−h}^{×}). By theorem 11.3 we only need to show that Θ_{1} = Θ_{2}. Moreover it follows from lemma A.4.f of [DKV] that we only need check that Θ_{1}([π]) = Θ_{2}([π]) when π is a full induced from square integrable.

Thus suppose that we have positive integers s_{1}, ..., s_{t} and n_{1}, ..., n_{t} such that n = s_{1}n_{1} + ... + s_{t}n_{t}. Suppose also that for i = 1, ..., t we have an irreducible supercuspidal representation π_{i} of GL_{n_{i}}(F_{w}). Let P ⊂ GL_{n} be a parabolic subgroup with Levi component GL_{s_{1}n_{1}} × ... × GL_{s_{t}n_{t}} and write π for

n-Ind _{P (F_{w})}^{GL_{n}(F_{w})}(Sp_{s_{1}}(π_{1}) × ... × Sp_{s_{t}}(π_{t})).

We must check that Θ_{1}([π]) = Θ_{2}([π]).

If h_{1}, ..., h_{t} are positive integers such that h_{i} ≤ s_{i} we will let

• P′_{h_{i}} ⊂ GL_{s_{i}n_{i}} denote a parabolic subgroup with Levi component GL_{h_{i}n_{i}} × GL_{(s_{i}−h_{i})n_{i}},

• h = h_{1}n_{1} + ... + h_{t}n_{t},

• P_{{h_{i}}} ⊂ GL_{h} denote a parabolic subgroup with Levi component GL_{h_{1}n_{1}} × ... × GL_{h_{t}n_{t}},

• P′_{{h_{i}}} ⊂ GL_{n−h} denote a parabolic subgroup with Levi component GL_{(s_{1}−h_{1})n_{1}} × ... × GL_{(s_{t}−h_{t})n_{t}},

By lemma IV.8 we see that

Θ_{2}([π]) =
∑_{h_{i}} ∑_{ρ} vol (D_{F_{w},n−h}^{×}/F_{w}^{×})^{−1}
tr (n-Ind _{P′_{{h_{i}}}(F_{w})}^{GL_{n−h}(F_{w})}(Sp_{s_{1}−h_{1}}(π_{1}) × ... × Sp_{s_{t}−h_{t}}(π_{t})))(φ_{JL (ρ^{∨})})
n-Ind _{P_{h}(F_{w})}^{GL_{n}(F_{w})}[(n-Ind _{P_{{h_{i}}}(F_{w})}^{GL_{h}(F_{w})}Sp_{h_{1}}(π_{1} ⊗ | det |^{s_{1}−h_{1}}) × ...
×Sp_{h_{t}}(π_{t} ⊗ | det |^{s_{t}−h_{t}})) × (Ψ_{F_{w},l,n−h}(ρ) ⊗ |Art _{F_{w}}^{−1}|^{−h/2})],

where h_{1}, ..., h_{t} run over positive integers with h_{i} ≤ s_{i}, where ρ runs over irreducible admissible representations of D_{F_{w},n−h}^{×}. Using the second part of lemma IV.4 we see that this becomes Θ_{2}([π]) = ∑_{i=1}^{t} ∑_{h_{i}=0}^{s_{i}−1} n-Ind _{P (F_{w})}^{GL_{n}(F_{w})}[Sp_{s_{1}}(π_{1}) × ...× (n-Ind _{P′_{h_{i}}(F_{w})}^{GL_{s_{i}n_{i}}(F_{w})}Sp_{h_{i}}(π_{i} ⊗ | det |^{s_{i}−h_{i}}) × (Ψ_{F_{w},l,(s_{i}−h_{i})n_{i}}(JL^{−1}(Sp_{s_{i}−h_{i}}(π_{i})^{∨})) ⊗|Art _{F_{w}}^{−1}|^{(n_{i}(s_{i}−h_{i})−n)/2})) × ...Sp_{s_{t}}(π_{t})]. Now by theorem 11.7 this becomes

Θ_{2}([π]) = ∑_{i=1}^{t} ∑_{h_{i}=0}^{s_{i}−1} ∑_{j=1}^{s_{i}−h_{i}}(−1)^{s_{i}−h_{i}−j}n-Ind _{P (F_{w})}^{GL_{n}(F_{w})}[Sp_{s_{1}}(π_{1}) × ...× (n-Ind _{P′_{h_{i}}(F_{w})}^{GL_{s_{i}n_{i}}(F_{w})}(Sp_{j}(π_{i}) ⊞ (π_{i} ⊗ | det |^{j}) ⊞ ... ⊞ (π_{i} ⊗ | det |^{s_{i}−1−h_{i}})

×Sp_{h_{i}}(π_{i} ⊗ | det |^{s_{i}−h_{i}}))

×... × Sp_{s_{t}}(π_{t})][r_{l}(π_{i} ⊗ | det |^{j−1}) ⊗ |Art _{K}^{−1}|^{(n_{i}−n)/2}]. Next by lemma IV.2 we rewrite this

Θ_{2}([π]) = ∑_{i=1}^{t} ∑_{h_{i}=1}^{s_{i}−1} ∑_{j=1}^{s_{i}−h_{i}}(−1)^{s_{i}−h_{i}−j} n-Ind _{P (F_{w})}^{GL_{n}(F_{w})}[Sp_{s_{1}}(π_{1}) × ... × ω(Γ⃗_{j,s_{i}−h_{i}−j,h_{i}}) × ... × Sp_{s_{t}}(π_{t})] [r_{l}(π_{i} ⊗ | det |^{j−1}) ⊗ |Art _{K}^{−1}|^{(n_{i}−n)/2}]+ ∑_{i=1}^{t} ∑_{h_{i}=0}^{s_{i}−1} ∑_{j=1}^{s_{i}−h_{i}}(−1)^{s_{i}−h_{i}−j} n-Ind _{P (F_{w})}^{GL_{n}(F_{w})}[Sp_{s_{1}}(π_{1}) × ... × ω(Γ⃗_{j,s_{i}−1−h_{i}−j,h_{i}+1}) × ... × Sp_{s_{t}}(π_{t})] [r_{l}(π_{i} ⊗ | det |^{j−1}) ⊗ |Art _{K}^{−1}|^{(n_{i}−n)/2}] = ∑_{i=1}^{t} ∑_{j=1}^{s_{i}} n-Ind _{P (F_{w})}^{GL_{n}(F_{w})}[Sp_{s_{1}}(π_{1}) × ... × ω(Γ⃗_{j,−1,s_{i}+1−j})× ... × Sp_{s_{t}}(π_{t})][r_{l}(π_{i} ⊗ | det |^{j−1}) ⊗ |Art _{K}^{−1}|^{(n_{i}−n)/2}] = [π] ∑_{i=1}^{t} ∑_{j=0}^{s_{i}−1}[r_{l}(π_{i} ⊗ | det |^{j}) ⊗ |Art _{K}^{−1}|^{(n_{i}−n)/2}] = Θ_{1}([π]).

The theorem follows. □

Corollary 11.10 Suppose that π is an irreducible admissible representation of G(𝔸^{∞}). Then either ±[R_{ξ}(π)] is a true representation.

Proof: Suppose not, then by the Weil conjectures for all but finitely many finite places x of F we have [R_{ξ}(π)|_{W_{Fx}}] is neither a true representation, nor the negative of a true representation. This contradicts theorem 11.9. □

We now use our results to extend a theorem of Clozel [Cl1]. Theorem 11.11 Suppose that L is a CM field and that Π is a cuspidal automorphic representation of GL_{g}(𝔸_{L}) satisfying the following conditions:

• Π^{∨} ≅ Π^{c},

• Π_{∞} has the same infinitesimal character as some algebraic representation over ℂ of the restriction of scalars from L to ℚ of GL_{g},

• and for some finite place x of L the representation Π_{x} is square integrable. Then there is a strictly positive integer a(Π) and a continuous representation R_{l}(Π) of Gal (L^{ac}/L) over ℚ_{l}^{ac} such that for any finite place y of L not dividing l we have

[R_{l}(Π)|_{W_{Fy}}] = a(Π)[r_{l}(Π_{y})].

Proof: Suppose first that L = EF^{+} where E is an imaginary quadratic field such that x|_{ℚ} splits in E and where F^{+} is a totally real field with [F^{+} : ℚ] even. In this case we will write F for L.

Choose (B, ∗, τ_{0}, β, Λ_{i}) as in section 1 such that B is a division algebra with centre F such that

• [B : F] = g^{2},

• B splits at all places of F other than x and x^{c}

• and B_{x} and B_{x^{c}} are division algebras. (Here we are using the assumption that [F^{+} : ℚ] is even.) By theorem VI.4, corollary VI.5 and lemma VI.6 we can find

• an algebraic representation ξ of G over ℚ_{l}^{ac},

• and an automorphic representation π of G(𝔸) such that if we set BC (π) = (ψ, Π̃), then

• JL (Π̃) = Π

• and dim[R_{ξ}(π)] ≠ 0.

If e(π) is the sign of dim[R_{ξ}(π)], set

• [R_{l}(Π)] = e(π)g[R_{ξ}(π^{∞}) ⊗ rec(ψ^{c})|_{Gal (F^{ac}/F)}]

• and a(Π) = e(π) dim[R_{ξ}(π^{∞})].

From theorem 11.9 it follows that for all finite places y of F such that

• y |̸xx^{c},

• y|_{ℚ} splits in E

• and y |̸l,

we have

[R_{l}(Π)|_{W_{Fy}}] = a(Π)[r_{l}(Π_{y})].

Next suppose that y|_{ℚ} is inert in E or that y|xx^{c}. Let p denote the rational prime under y. We can find a real quadratic field A such that

• A_{p} ≅ E_{p}

• and x|_{ℚ} splits in A.

Let E′ denote the third quadratic subfield of AE, let (F^{+})′ = AF^{+} and let F′ = E′(F^{+})′ = AF. Note that E′ is an imaginary quadratic field in which both p and x|_{ℚ} split. Let x′ denote a prime of F′ above x and let y′ denote a prime of F′ above y which does not divide x′(x′)^{c}. Note also that F_{x} →^{∼} F′_{x′} and F_{y} →^{∼} F′_{y′}. The x′ component of the automorphic restriction Res_{F′}^{F}(Π) is square integrable and hence Res_{F′}^{F}(Π) is cuspidal. Moreover by strong multiplicity one Res_{F′}^{F}(Π)^{c} ≅ Res_{F′}^{F}(Π)^{∨}, and Res_{F′}^{F}(Π) has the same infinitesimal character as an algebraic representation of GL_{g}(F′ ⊗_{ℚ} ℂ) over ℂ. Thus we can associate to Res_{F′}^{F}(Π) a continuous representation R_{l}(Res_{F′}^{F}(Π)) of Gal (F^{ac}/F′) and a positive integer a(Res_{F′}^{F}(Π)) such that for all places z of F′ for which

• z |̸x′(x′)^{c},

• z|_{ℚ} splits in E′

• and z |̸l,

we have

[R_{l}(Res_{F′}^{F}(Π))|_{W_{F′_{z}}}] = a(Res_{F′}^{F}(Π))[r_{l}(Res_{F′}^{F}(Π)_{z})].

It follows from the Cebotarev density theorem that

a(Π)[R_{l}(Res_{F′}^{F}(Π))] = a(Res_{F′}^{F}(Π))[R_{l}(Π)|_{Gal (F^{ac}/F′)}].

Thus

a(Res_{F′}^{F}(Π))[R_{l}(Π)|_{W_{Fy}}] =
a(Π)[R_{l}(Res_{F′}^{F}(Π))|_{W_{F{′ \atop y′}}}] =
a(Π)a(Res_{F′}^{F}(Π))[r_{l}(Res_{F′}^{F}(Π)_{y′})] =
a(Π)a(Res_{F′}^{F}(Π))[r_{l}(Π_{y})],

and we have established the theorem in the special case.

Now we will turn to the proof in the general case.

If A is an imaginary quadratic field such that x|_{ℚ} splits in E then set

• F_{A}^{+} to be the maximal totally real subfield of AL,

• F_{A} = AF_{A}^{+} = AL,

• x_{A} a prime of F_{A} above x,

• σ_{A} to be the non-trivial element of Gal (F_{A}/L)

• and ϵ_{A} to be the non-trivial character of Gal (A/ℚ).

Then [F_{A}^{+} : ℚ] is even and x_{A}|_{ℚ} splits in A. Moreover L_{x} →^{∼} F_{A}|_{x_{A}} and so as before we see that Res_{F_{A}}^{F}(Π) continues to satisfy the conditions of the theorem, but for F_{A}. Thus, from what we have already proved, there is a continuous representation R_{l}(Res_{F_{A}}^{L}(Π)) of Gal (L^{ac}/F_{A}) and a positive integer a(Res_{F_{A}}^{L}(Π)) such that for all places z |̸l of F_{A} we have

[R_{l}(Res_{F_{A}}^{L}(Π))|_{W_{F_{A,z}}}] = a(Res_{F_{A}}^{L}(Π))[r_{l}(Res_{F_{A}}^{L}(Π)_{z})].

It follows from the Cebotarev density theorem that

[R_{l}(Res_{F_{A}}^{L}(Π))^{σ_{A}}] = [R_{l}(Res_{F_{A}}^{L}(Π))]

and that if A′ is a second such field then

a(Res_{F_{A′}}^{L}(Π))[R_{l}(Res_{F_{A}}^{L}(Π))|_{Gal (L^{ac}/F_{A}F_{A′})}] =
a(Res_{F_{A}}^{L}(Π))[R_{l}(Res_{F_{A′}}^{L}(Π))|_{Gal (L^{ac}/F_{A}F_{A′})}].

Fix one such quadratic extension A_{0}. Let {ρ_{i}} be a set of representatives of the equivalence classes of irreducible continuous representations of Gal (L^{ac}/F_{A_{0}}) on finite dimensional ℚ_{l}^{ac}-vector spaces. Let I be the set of indices such that ρ_{i}^{σ_{A_{0}}} ≅ ρ_{i} and ρ_{i} is a constituent of [R_{l}(Res_{F_{A_{0}}}^{L}(Π))]. Let J be the set of indices such that ρ_{i}^{σ_{A_{0}}} ≅ ρ̸_{i} and ρ_{i} is a constituent of [R_{l}(Res_{F_{A_{0}}}^{L}(Π))]. For i ∈ I choose an extension ρ̃_{i} of ρ_{i} to Gal (L^{ac}/L). Also write

[R_{l}(Res_{F_{A_{0}}}^{L}(Π))] = ∑_{i∈I∪J} b_{i}[ρ_{i}].

Let

ρ = ⨁_{i∈I∪J} ρ_{i}.

Let H denote the Zariski closure of the image of ρ and let H^{0} denote the connected component of the identity in H. Also let M/F_{A_{0}} denote the fixed field of ρ^{−1}H^{0}. If N/L is a finite Galois extension disjoint from M then ρGal (F^{ac}/NA_{0}) is Zariski dense in H and so we have the following results.

1. If i ∈ I ∪ J then ρ_{i}|_{Gal (L^{ac}/NA_{0})} is irreducible.

2. If i ∈ I then ρ̃_{i}|_{Gal (L^{ac}/N)} is irreducible.

3. If i, j ∈ I ∪ J and ρ_{i}|_{Gal (L^{ac}/NA_{0})} ≅ ρ_{j}|_{Gal (L^{ac}/NA_{0})} then i = j.

4. If i, j ∈ I, if δ = 0 or 1 and if ρ̃_{i}|_{Gal (L^{ac}/N)} ≅ ρ̃_{j}|_{Gal (L^{ac}/N)} ⊗ϵ_{A_{0}}^{δ} then i = j and δ = 0.

In particular if A/ℚ is a quadratic extension as above such that F_{A} is linearly disjoint from M over L then

• ρ_{i}|_{Gal (L^{ac}/F_{A_{0}}A)} is irreducible for all i ∈ I ∪ J

• and if, for some i, j ∈ I ∪ J, we have ρ_{i}^{σ_{A_{0}}}|_{Gal (L^{ac}/F_{A_{0}}A)} ≅ ρ_{j}|_{Gal (L^{ac}/F_{A_{0}}A)} then ρ_{i}^{σ_{A_{0}}} = ρ_{j}.

We have also seen that

(dim[R_{l}(Res_{F_{A_{0}}}^{L}(Π))])[R_{l}(Res_{F_{A}}^{L}(Π))|_{Gal (L^{ac}/F_{A_{0}}A)}] =
(dim[R_{l}(Res_{F_{A}}^{L}(Π))]) ∑_{i∈I∪J} b_{i}[ρ_{i}|_{Gal (L^{ac}/F_{A_{0}}A)}].

Thus we must have

(dim[R_{l}(Res_{F_{A_{0}}}^{L}(Π))])[R_{l}(Res_{F_{A}}^{L}(Π))] =
(dim[R_{l}(Res_{F_{A}}^{L}(Π))])(∑_{i∈J}(b_{i}/2)[(Ind _{Gal (L^{ac}/F_{A_{0}})}^{Gal (L^{ac}/L)}ρ_{i})|_{Gal (L^{ac}/F_{A})}]+
∑_{i∈I} b_{i}[(ρ̃_{i} ⊗ ϵ_{A_{0}}^{δ_{Ai}})|_{Gal (L^{ac}/F_{A})}]),

where δ_{Ai} = 0 or 1.

Choose such an extension A_{1} so that F_{A_{1}} is linearly disjoint from M over L. Set

[R_{l}(Π)] = ∑_{i∈J}(b_{i}/2)[Ind _{Gal (L^{ac}/F_{A_{0}})}^{Gal (L^{ac}/L)}ρ_{i}] + ∑_{i∈I} b_{i}[(ρ̃_{i} ⊗ ϵ_{A_{0}}^{δ_{A_{1}i}})].

Also set

a(Π) = a(Res_{F_{A_{0}}}^{L}(Π))

so that

dim[R_{l}(π)] = a(Π)g.

Suppose now that A is such a quadratic extension of ℚ such that F_{A} is linearly disjoint from MA_{1} over L. Then

∑_{i∈I} b_{i}[(ρ̃_{i} ⊗ ϵ_{A_{0}}^{δ_{A_{1}i}})|_{Gal (L^{ac}/F_{A_{1}}A)}]) = ∑_{i∈I} b_{i}[(ρ̃_{i} ⊗ ϵ_{A_{0}}^{δ_{Ai}})|_{Gal (L^{ac}/F_{A}A_{1})}])

and so δ_{Ai} = δ_{A_{1}i} for all i ∈ I. Hence

(dim[R_{l}(Res_{F_{A}}^{L}(Π))])[R_{l}(Π)|_{Gal (L^{ac}/F_{A})}] = (dim[R_{l}(Π)])[R_{l}(Res_{F_{A}}^{L}(Π))].

Given any finite place y |̸l of L we can choose an imaginary quadratic extension A/ℚ such that

• LA is disjoint from MA_{1} over L,

• y splits as y′y′′ in LA

• and x|_{ℚ} splits in A.

Then

(dim[R_{l}(Res_{F_{A}}^{L}(Π))])[R_{l}(Π)|_{W_{Fy}}] =
(dim[R_{l}(Π)])[R_{l}(Res_{F_{A}}^{L}(Π))|_{W_{F_{y′}}}] =
(dim[R_{l}(Π)])a(Res_{F_{A}}^{L}(Π))[r_{l}(Π_{y})],

and so

[R_{l}(Π)|_{W_{Fy}}] = a(Π)[r_{l}(Π_{y})]

as desired. □

(Attempts to construct Galois representations by first constructing them over many quadratic extensions are not new (see for instance [BR]).)

From this one can deduce the following extension of theorem 11.9.

Corollary 11.12 Suppose that π is an automorphic representation of G(𝔸) such π_{∞} is cohomological for ξ. Let BC (π) = (ψ, Π) and suppose that JL (Π) is cuspidal. Then JL (Π) satisfies the hypotheses of theorem 11.11 and

na(JL (Π))[R_{ξ}(π^{∞})] = (dim[R_{ξ}(π^{∞})])[R_{l}(JL (Π)) ⊗ rec(ψ)|_{Gal (F^{ac}/F)}^{−1}].

In particular if y |̸l is a place of F then

n[R_{ξ}(π^{∞})] = (dim[R_{ξ}(π^{∞})])[r_{l}(JL (Π)_{y}) ⊗ (ψ_{y|_{E}}^{−1} ∘ Art _{E_{y}}^{−1})|_{Gal (F^{ac}/F)}].

Proof: Using the Cebotarev density theorem, this follows easily from theorems 11.9 and 11.11. □

We remark that the results of this section depend on the main theorem of [Ko4] via theorem VI.1 and corollary VI.5, both of which rely on theorem A.4.2 of [CL] (the former via theorem A.5.2 of [CL]). However we believe that we could have avoided this logical dependence on the main theorem of [Ko4] at the expense of doing a little more work. More precisely one can calculate directly from the definitions Ψ_{K,l,g}(ρ)^{GL_{g}(O_{K})}. If π is an irreducible admissible representation of G(𝔸^{∞}) this suffices to calculate

∑_{π′}(dim(π′_{w})^{ℤ_{p}^{×}×GL_{n}(O_{Fw})})[R_{ξ}(π′)|_{W_{Fw}}]

where π′ runs over irreducible admissible representations of G(𝔸^{∞}) for which (π′)^{w} ≅ π^{w} (by using an argument similar to the proof of the last theorem). This tells us [R_{ξ}(π)|_{W_{Fw}}] for all but finitely many places w of F which lie over a rational prime which splits in E, and this in turn suffices to prove theorem A.4.2 of [CL].

12 The local Langlands conjecture.

We will start this section by checking various basic functoriality properties of our map r_{l}. Throughout this section K will denote a finite extension of ℚ_{p} and l will denote a prime other than p. Recall that we have fixed an isomorphism

ı : ℚ_{l}^{ac} →^{∼} ℂ.

Lemma 12.1 Suppose that π is an irreducible admissible representation of GL_{g}(K) and that χ is a smooth character of K^{×}. Then

[r_{l}(π ⊗ (χ ∘ det))] = [r_{l}(π) ⊗ (χ^{−1} ∘ Art _{K}^{−1})].

Proof: It is easy to reduce to the case that π is supercuspidal, so suppose that π is supercuspidal. Choose an imaginary quadratic field M in which p splits and a totally real field L^{+} with a place x(+) such that L_{x(+)}^{+} ≅ K. Set L = ML^{+} and choose a place x of L above x(+). Thus L_{x} ≅ K. Let χ̃ be a continuous character of 𝔸_{L}^{×}/L^{×}L_{∞}^{×} such that χχ̃_{x}^{−1} is unramified. By corollary VII.3 we may choose a cuspidal automorphic representation Π of GL_{g}(𝔸_{L}) such that

• Π^{c} ≅ Π^{∨},

• Π_{∞} has the same infinitesimal character as some algebraic representation of RS_{ℚ}^{L}(GL_{g}),

• and Π_{x} ≅ π ⊗ (ψ_{x} ∘ det) for some character ψ_{x} of K^{×}/O_{K}^{×}.

From theorem 11.11 (applied at good places of L) and from the Cebotarev density theorem we see that

a(Π)[R_{l}(Π ⊗ (χ̃ ∘ det))] = a(Π ⊗ (χ̃ ∘ det))[R_{l}(Π) ⊗ rec_{l,ı}(χ̃)^{−1}].

Applying theorem 11.11 at x we conclude that

[r_{l}(Π_{x} ⊗ χ̃_{x})] = [r_{l}(Π_{x}) ⊗ (χ̃_{x}^{−1} ∘ Art _{K}^{−1})].

The lemma now follows from lemma 11.6. □

Lemma 12.2 Suppose that π is an irreducible admissible representation of GL_{g}(K) with central character ψ_{π}. Then

det r_{l}(π) = (ψ_{π} ⊗ | |^{g(g−1)/2})^{−1} ∘ Art _{K}^{−1}.

Proof: Again it is easy to reduce to the case that π is supercuspidal, so suppose that π is supercuspidal. Choose an imaginary quadratic field M in which p splits and a totally real field L^{+} with a place x(+) such that L_{x(+)}^{+} ≅ K. Set L = ML^{+} and choose a place x of L above x(+). Thus L_{x} ≅ K. By corollary VII.3 we may choose a cuspidal automorphic representation Π of GL_{g}(𝔸_{L}) such that

• Π^{c} ≅ Π^{∨},

• Π_{∞} has the same infinitesimal character as some algebraic representation of RS_{ℚ}^{L}(GL_{g}),

• and Π_{x} ≅ π ⊗ (ψ_{x} ∘ det) for some character ψ_{x} of K^{×}/O_{K}^{×}.

From theorem 11.11 (applied at good places of L) and from the Cebotarev density theorem we see that for all σ ∈ Gal (ℚ^{ac}/ℚ) there exist α_{1}(σ), ..., α_{n}(σ) ∈ ℚ_{l}^{ac} such that

• R_{l}(Π)(σ) has eigenvalues α_{1}(σ), ..., α_{n}(σ), each with multiplicity a(Π),

• and α_{1}(σ)...α_{n}(σ) = rec_{l,ı}(ψ_{π} ⊗ | |^{g(g−1)/2})^{−1}(σ).

(See for instance the first paragraph of the proof of proposition 1 of [Tay] for more details of this sort of argument.) Applying theorem 11.11 at x we conclude that

det r_{l}(Π_{x}) = (ψ_{Π,x} ⊗ | |^{g(g−1)/2})^{−1} ∘ Art _{K}^{−1}.

The lemma now follows from lemma 11.6. □

Lemma 12.3 Suppose that π is an irreducible admissible representation of GL_{g}(K). Then

[r_{l}(π^{∨})] = [r_{l}(π ⊗ | det |^{1−g})^{∨}].

Proof: Again it is easy to reduce to the case that π is supercuspidal, so suppose that π is supercuspidal. Choose an imaginary quadratic field M in which p splits and a totally real field L^{+} with a place x(+) such that L_{x(+)}^{+} ≅ K. Set L = ML^{+} and choose a place x of L above x(+). Thus L_{x} ≅ K. By corollary VII.3 we may choose a cuspidal automorphic representation Π of GL_{g}(𝔸_{L}) such that

• Π^{c} ≅ Π^{∨},

• Π_{∞} has the same infinitesimal character as some algebraic representation of RS_{ℚ}^{L}(GL_{g}),

• and Π_{x} ≅ π ⊗ (ψ_{x} ∘ det) for some character ψ_{x} of K^{×}/O_{K}^{×}.

From theorem 11.11 (applied at good places of L) and from the Cebotarev density theorem we see that

a(Π ⊗ | det |^{1−g})[R_{l}(Π^{∨})] = a(Π^{∨})[R_{l}(Π ⊗ | det |^{1−g})^{∨}].

Applying theorem 11.11 at x we conclude that

[r_{l}(Π_{x}^{∨})] = [r_{l}(Π_{x} ⊗ | det |^{1−g})^{∨}].

The lemma now follows from lemma 11.6. □

Now suppose that K′/K is a cyclic Galois extension of prime degree q. If π is an irreducible admissible representation of GL_{g}(K) then one can associate to π its base change lifting Res_{K′}^{K}(π) to K′ (see theorem 6.2 of chapter 1 and the discussion on pages 59 and 60 of [AC]). Also if π′ is a Gal (K′/K)-regular (see section 2.4 of [HH] for the definition of this concept) generic irreducible admissible representation of GL_{g}(K′) one can associate to π′ its automorphic induction Ind _{K′}^{K}(π′) to K (see theorem 2.4 of [HH]). Then Res_{K′}^{K}(π) is an irreducible admissible representation of GL_{g}(K′) and Ind _{K′}^{K}(π′) is an irreducible admissible representation of GL_{qg}(K).

We will need to make use of the global analogues of these constructions, which we now recall. Suppose that L′/L is a cyclic Galois extension of number fields of prime degree q. Let τ denote a generator of Gal (L′/L) and let η denote a non-trivial character of

𝔸_{L}^{×}/L^{×}(L_{∞}^{×})^{0}(N_{L′/L}𝔸_{L′}^{×}).

If Π is a cuspidal automorphic representation of GL_{g}(𝔸_{L}) then we can associate to Π an “induced from cuspidal” (see definition 4.1 of chapter 3 of [AC]) representation Res_{L′}^{L}Π of GL_{g}(𝔸_{L′}) with the following properties.

1. Res_{L′}^{L}(Π) is cuspidal if and only if Π = Π ⊗ (η ∘ det).

2. A cuspidal automorphic representation Π′ of GL_{g}(𝔸_{L′}) is of the form Res_{L′}^{L}Π for some cuspidal automorphic Π if and only if Π′ ≅ Π′ ∘ τ.

3. If x is a place of L which splits in L′ and x̃ is a place of L′ above x then

Res_{L′}^{L}(Π)x̃ ≅ Π_{x}.

4. If x is a finite place of L which is inert in L′ then

Res_{L′}^{L}(Π)_{x} ≅ Res_{L′_{x}}^{L_{x}}(Π_{x}).

(See theorems 4.2 and 5.1 of [AC].) Now suppose that Π′ is a cuspidal automorphic representation of GL_{g}(𝔸_{L′}). Then there is an “induced from cuspidal” representation Ind _{L′}^{L}Π′ of GL_{gq}(𝔸_{L}) with the following properties.

1. Ind _{L′}^{L}(Π′) ⊗ (η ∘ det) ≅ Ind _{L′}^{L}(Π′).

2. If Π′ ≅ Π̸′ ∘ σ then Ind _{L′}^{L}(Π′) is cuspidal.

3. If x is a place L which splits as x_{1}...x_{q} in L′ then

Ind _{L′}^{L}(Π′)_{x} ≅ n-Ind _{Q(L_{x})}^{GL_{qg}(L_{x})}(Π′_{x_{1}} × ... × Π′_{x_{q}}),

where Q is a parabolic subgroup of GL_{qg} with Levi component GL_{g}^{q}.

4. For all but finitely many places x of L which are inert in L′ we have

Ind _{L′}^{L}(Π′)_{x} ≅ Ind _{L′_{x}}^{L_{x}}(Π′_{x}).

(See theorem 6.2, lemma 6.4 and corollary 6.5 of [AC].) The following lemma seems to be well known (see section 1.5 of [HH]), but for lack of an explicit reference we give the proof.

Lemma 12.4 Keep the above notation and suppose that x is a finite place of L which is inert in L′.

1. Π′_{x} is Gal (L′_{x}/L_{x})-regular.

2. The only generic, irreducible, admissible representation π of GL_{qg}(L_{x}) such that

• π ⊗ (η_{x} ∘ det) ≅ π

• and

Res_{L′_{x}}^{L_{x}}(π) ≅ n-Ind _{Q(L′_{x})}^{GL_{qg}(L′_{x})}(Π′_{x} × ... × (Π′_{x} ∘ τ^{q−1}))

(where Q ⊂ GL_{qg} is the parabolic subgroup defined above)

is Ind _{L′_{x}}^{L_{x}}(Π′_{x}).

3. Ind _{L′}^{L}(Π′)_{x} ≅ Ind _{L′_{x}}^{L_{x}}(Π′_{x}).

Proof: Note that the first part follows from lemma 2.3 of [HH]. Also note that the third part follows from the second part and the definition of Ind _{L′}^{L} (see section 6 of chapter 3 of [AC]). Thus it remains to prove the second part.

We can write

Π′_{x} ≅ (⊞_{i∈I′}Sp_{s′_{i}}(π′_{i})) ⊞ (⊞_{i∈J′}Sp_{s′_{i}}(π′_{i}))

where π′_{i} ≅ π′_{i} ∘ τ if i ∈ I′, but not if i ∈ J′. For i ∈ I′ choose an irreducible admissible representation π̃_{i} such that Res_{L′x}^{L_{x}}π̃_{i} ≅ π′_{i}. Then

n-Ind _{Q(L′_{x})}^{GL_{qg}(L′_{x})}Π′_{x} ≅ (⊞_{i∈I′}Sp_{s′_{i}}(π′_{i})^{⊞q}) ⊞ (⊞_{i∈J′} ⊞_{j=0}^{q−1} Sp_{s′_{i}}(π′_{i} ∘ τ^{j}))

(use the fact that Π′_{x} is Gal (L′_{x}/L_{x})-regular generic). Moreover

Ind _{L′_{x}}^{L_{x}}(Π′_{x}) ≅ (⊞_{i∈I′} ⊞_{j=0}^{q−1} Sp_{s′_{i}}(π̃_{i} ⊗ (η_{x}^{j} ∘ det))) ⊞ (⊞_{i∈J′}Sp_{s′_{i}}(Ind _{L′_{x}}^{L_{x}}(π′_{i}))) (see [HH] and assertion 2.6 (a) of [BHK]), and hence

Res_{L′_{x}}^{L_{x}}Ind _{L′_{x}}^{L_{x}}(Π′_{x}) ≅ (⊞_{i∈I′}Sp_{s′_{i}}(π′_{i})^{⊞q}) ⊞ (⊞_{i∈J′} ⊞_{j=0}^{q−1} Sp_{s′_{i}}(π′_{i} ∘ τ^{j})) (see [AC] and assertion 2.6 (b) of [BHK]). In particular

Res_{L′_{x}}^{L_{x}}Ind _{L′_{x}}^{L_{x}}(Π′_{x}) ≅ n-Ind _{Q(L′_{x})}^{GL_{qg}(L′_{x})}Π′_{x}.

Now suppose that π is a generic, irreducible, admissible representation of GL_{qg}(L_{x}) such that

• π ⊗ (η_{x} ∘ det) ≅ π

• and

Res_{L′_{x}}^{L_{x}}(π) ≅ n-Ind _{Q(L′_{x})}^{GL_{qg}(L′_{x})}(Π′_{x} × ... × (Π′_{x} ∘ τ^{q−1})).

We may write

π ≅ (⊞_{i∈I} ⊞_{j=0}^{q−1} Sp_{s_{i}}(π_{i} ⊗ (η_{x}^{j} ∘ det))) ⊞ (⊞_{i∈J}Sp_{s_{i}}(π_{i}))

where π_{i} ≅ π_{i} ⊗ (η_{x} ∘ det) if i ∈ J, but not if i ∈ I. If i ∈ J then we can write π_{i} = Ind _{L′_{x}}^{L_{x}} π̃′_{i}, where π̃′_{i} is an irreducible supercuspidal representation. Then

Res_{L′_{x}}^{L_{x}}π ≅ (⊞_{i∈I}Sp_{s_{i}}(Res_{L′_{x}}^{L_{x}}π_{i})^{⊞q}) ⊞ (⊞_{i∈J} ⊞_{j=0}^{q−1} Sp_{s_{i}}(π̃′_{i} ∘ τ^{j})) (see [AC] and assertion 2.6 (b) of [BHK]). Note that for i ∈ I Res_{L′_{x}}^{L_{x}}π_{i} ≅ (Res_{L′_{x}}^{L_{x}}π_{i}) ∘ τ, while for i ∈ J we have π̃′_{i} ≅ ˜π̸′_{i} ∘ τ. Thus we may identify I with I′ and J with J′ so that

• for i ∈ I we have Res_{L′_{x}}^{L_{x}}π_{i} ≅ π′_{i}

• and for i ∈ J we have π̃′_{i} ≅ π′_{i} ∘ τ^{j(i)} for some j(i). Then

• for i ∈ I we have π_{i} ≅ π′_{i} ⊗ (η_{x} ∘ det)^{j(i)}, for some j(i)

• and for i ∈ J we have π_{i} ≅ π̃_{i}.

Thus

π ≅ Ind _{L′_{x}}^{L_{x}}Π′_{x},

as desired. □

Lemma 12.5 Suppose that π is an irreducible admissible representation of GL_{g}(K). Then

[r_{l}(Res_{K′}^{K}(π))] = [r_{l}(π)|_{W_{K′}}].

Proof: Again one may reduce to the case that π is square integrable, so suppose that π is square integrable. (See section 6.2 of chapter 1 and pages 59 and 60 of [AC].) Choose an imaginary quadratic field M in which p splits. Also choose a cyclic Galois extension (L′)^{+}/L^{+} of totally real fields and a place x(+) of L^{+} such that x(+) is inert in (L′)^{+} and the extension (L′)_{x(+)}^{+}/L_{x(+)}^{+} is isomorphic to the extension K′/K. Set L = ML^{+} (resp. L′ = M(L′)^{+}) and choose a place x of L above x(+). Thus the extension L′_{x}/L_{x} is also isomorphic to K′/K. Choose a place y of L which splits completely in L′ and which lies above a rational prime other than p which splits in M. By corollary VII.3 we may choose a cuspidal automorphic representation Π of GL_{g}(𝔸_{L}) such that

• Π^{c} ≅ Π^{∨},

• Π_{∞} has the same infinitesimal character as some algebraic representation of RS_{ℚ}^{L}(GL_{g}),

• Π_{x} ≅ π ⊗ (ψ_{x} ∘ det) for some character ψ_{x} of K^{×}/O_{K}^{×},

• and Π_{y} is supercuspidal.

Then there is an “induced from cuspidal” representation Res_{L′}^{L}(Π) of GL_{g}(𝔸_{L′}) such that for all places w of L we have

• Res_{L′}^{L}(Π)_{w} = Res_{L′_{w}}^{L_{w}}(Π_{w}) if w is inert in L′,

• and Res_{L′}^{L}(Π)_{w} = Π_{w}^{⊗q} if w splits in L′.

From the second of these conditions we see that Res_{L′}^{L}(Π) is supercuspidal at every place above y and hence is cuspidal automorphic. From the second of these conditions we also see that Res_{L′}^{L}(Π)_{∞} has the same infinitesimal character as some algebraic representation of RS_{ℚ}^{L′}(GL_{g}).

If w is a finite place of L at which Π_{w} is unramified and if w′ is a prime of L′ above w then, from the compatibility of base change with parabolic induction and from the explicit description of base change when g = 1 (see part (d) of theorem 6.2 of chapter 1 of [AC]), we see that if w′ is a prime of L′ above w then

[R_{l}(Res_{L′}^{L}(Π))|_{W_{L{′ \atop w′}}}] = a(Res_{L′}^{L}(Π))[r_{l}(Π_{w})|_{W_{L{′ \atop w′}}}].

Thus, from theorem 11.11 (applied at good places of L′) and from the Cebotarev density theorem we see that

a(Π)[R_{l}(Res_{L′}^{L}(Π)] = a(Res_{L′}^{L}(Π))[R_{l}(Π)|_{Gal ((L′)^{ac}/L′)}].

Applying theorem 11.11 at x we conclude that

[r_{l}(Res_{K′}^{K}(Π_{x}))] = [r_{l}(Π_{x})|_{W_{L{′ \atop x′}}}].

The lemma now follows from lemma 11.6. □

Lemma 12.6 Suppose that π′ is a generic, Gal (K′/K)-regular, irreducible, admissible representation of GL_{g}(K′). Then

[r_{l}(Ind _{K′}^{K}(π′) ⊗ | det |^{g(1−q)/2})] = [(Ind _{W_{K′}}^{W_{K}}r_{l}(π′))].

Proof: One may reduce to the case that π′ = Sp_{2}(π^{0}), where π^{0} is supercuspidal, so suppose that π′ has this form. (See corollary 5.5 and theorem 5.6 of [HH].) Note in particular that in this case g is even. Choose an imaginary quadratic field M in which p splits. Also choose a cyclic Galois extension (L′)^{+}/L^{+} of totally real fields and a place x(+) of L^{+} such that x(+) is inert in (L′)^{+} and the extension (L′)_{x(+)}^{+}/L_{x(+)}^{+} is isomorphic to the extension K′/K. Set L = ML^{+} (resp. L′ = M(L′)^{+}) and choose a place x of L above x(+). Thus the extension L′_{x}/L_{x} is also isomorphic to K′/K. Choose a generator σ of Gal (L′/L). Choose a place y of L which is inert in L′ and which lies above a rational prime other than p which splits in M. Choose a supercuspidal representation π_{y} of GL_{g}(L′_{y}) such that π_{y} ≅ π̸_{y}^{σ} ⊗ ψ for any character ψ of (L′_{y})^{×}/O_{L′,y}^{×}. (To see that this is possible one may argue as follows. First choose any irreducible supercuspidal representation π_{y}^{0} of GL_{g}(L′_{y}). Then take π_{y} = π_{y}^{0} ⊗ χ, where χ is a character of (L′_{y})^{×} such that

(ψ_{π_{y}}χ^{g})|_{O_{L′,y}^{×}} ≠ (ψ_{π_{y}}χ^{g})|_{O_{L′,y}^{×}} ∘ σ.)

By corollary VII.3 we may choose a cuspidal automorphic representation Π of GL_{g}(𝔸_{L}) such that

• Π^{c} ≅ Π^{∨}, • Π_{∞} has the same infinitesimal character as some algebraic representation

of RS_{ℚ}^{L′}(GL_{g}),

• Π_{x} ≅ π ⊗ (ψ_{x} ∘ det) for some character ψ_{x} of (K′)^{×}/O_{K′}^{×},

• and Π_{y} ≅ π_{y} ⊗ (ψ_{y} ∘ det) for some character ψ_{y} of (L′_{y})^{×}/O_{L′,y}^{×}.

Note that in particular Π^{σ} ≅ Π̸.

Consider the cuspidal automorphic representation Ind _{L′}^{L}Π of GL_{qg}(𝔸_{L}). Using the the strong multiplicity one theorem we see that

(Ind _{L′}^{L}Π)^{∨} ≅ (Ind _{L′}^{L}Π)^{c}.

If w is an infinite place of L which splits as w_{1}...w_{q} in L′, then (Ind _{L′}^{L}Π)_{w} is a subquotient of n-Ind _{Q(L_{w})}^{GL_{qg}(L_{w})}(Π_{w_{1}} ×...×Π_{w_{q}}), where Q is a parabolic subgroup of GL_{qg} with Levi component GL_{g}^{q}. In particular this allows one to check that Ind _{L′}^{L}(Π)_{∞} has the same infinitesimal character as some algebraic representation of RS_{ℚ}^{L}(GL_{qg}) (use the fact that g is even). Moreover by proposition 5.5 of [HH] we see that Ind _{L′}^{L}(Π)_{y} is supercuspidal.

From theorem 11.11 (applied at good places of L) and from the Cebotarev density theorem we see that

a(Π)[R_{l}(Ind _{L′}^{L}(Π) ⊗ | det |^{g(1−q)/2})] = a(Ind _{L′}^{L}(Π))[Ind _{Gal ((L′)^{ac}/L′)}^{Gal ((L′)^{ac}/L)}R_{l}(Π)].

Applying theorem 11.11 at x we conclude that

[r_{l}(Ind _{K′}^{K}(Π_{x}) ⊗ | det |^{g(1−q)/2})] = [(Ind _{W_{K′}}^{W_{K}}r_{l}(Π_{x}))].

The lemma now follows from lemma 11.6. □

If π is an irreducible admissible representation of GL_{g}(K) we will set

rec_{l}(π) = r_{l}(π^{∨} ⊗ | det |^{(1−g)/2}).

With this new normalisation we have the following restatement of lemma 11.8 and of the preceding lemmas.

Lemma 12.7 Let K′/K be a cyclic Galois extension of prime degree q and let π be an irreducible admissible representation of GL_{g}(K). Then we have the following results.

1. If τ ∈ Gal (K^{ac}/ℚ_{p}) then rec_{l}(π ∘ τ) = rec_{l}(π)^{τ}.

2. If g = 1 then rec_{l}(π) = π ∘ Art _{K}^{−1}.

3. If ψ_{π} is the central character of π then det rec_{l}(π) = rec_{l}(ψ_{π}).

4. If χ is a character of K^{×} then rec_{l}(π ⊗ (χ ∘ det)) = rec_{l}(π) ⊗ rec_{l}(χ).

5. rec_{l}(Res_{K′}^{K}(π)) = rec_{l}(π)|_{W_{K′}}.

6. If π′ is an irreducible admissible representation of GL_{g}(K′) then rec_{l}(Ind _{K′}^{K}(π′)) = Ind _{W_{K′}}^{W_{K}}rec_{l}(π′).

We next turn to some cases of non-Galois global automorphic induction, which were established by one of us (M.H.) in [Har2]. Indeed in a sense the rest of this section is superfluous, as we could simply refer to section 4 of [Har2]. However we will repeat the arguments here in somewhat greater detail, as we can now be slightly more direct. We will repeat not only arguments of [Har2], but also arguments of Henniart from [BHK] and [He6].

Proposition 12.8 Suppose that L_{3} ⊃ L_{2} ⊃ L_{1} are CM-fields with L_{3}/L_{1} soluble and Galois. Suppose that χ be a character of 𝔸_{L_{2}}^{×}/L_{2}^{×} such that

1. χ^{c} = χ^{−1};

2. for every embedding τ : L_{2} ↪ ℂ giving rise to an infinite place x we

have

χ_{x} : z → → (τz/cτz)^{p_{τ}}

where p_{τ} ∈ ℤ and if τ ≠ τ′ then p_{τ} ≠ p′_{τ};

3. there is a finite place y of L_{1} which is inert in L_{3}, which does not divide l and and for which the stabiliser of the character χ_{y} ∘ N_{L_{3}/L_{2}} of (L_{3})_{y}^{×} in Gal (L_{3}/L_{1}) is Gal (L_{3}/L_{2}).

Let ϕ be a character of 𝔸_{L_{1}}^{×}/L_{1}^{×} such that

1. ϕ^{c} = ϕ^{−1};

2. if [L_{2} : L_{1}] is odd then ϕ_{∞} = 1;

3. if [L_{2} : L_{1}] is even, then for every embedding τ : L_{1} ↪ ℂ giving rise to an infinite place x we have

ϕ_{x} : z → → (τz/|τz|)^{±1};

4. ϕ_{y} is unramified.

Then there is a cuspidal automorphic representation I_{L_{2}}^{L_{1}}(χ) of GL_{[L_{2}:L_{1}]}(𝔸_{L_{1}}) such that

• I_{L_{2}}^{L_{1}}(χ)^{c} ≅ I_{L_{2}}^{L_{1}}(χ)^{∨};

• (I_{L_{2}}^{L_{1}}(χ)⊗(ϕ∘det))_{∞} has the same infinitesimal character as an algebraic representation of RS_{ℚ}^{L_{1}}(GL_{[L_{2}:L_{1}]});

• I_{L_{2}}^{L_{1}}(χ)_{y} is supercuspidal;

• and

[R_{l}(I_{L_{2}}^{L_{1}}(χ) ⊗ (ϕ ∘ det))] = a(I_{L_{2}}^{L_{1}}(χ) ⊗ (ϕ ∘ det))
[Ind _{Gal (L_{3}^{ac}/L_{2})}^{Gal (L_{3}^{ac}/L_{1})}rec_{l,ı}(χ^{−1}(ϕ^{−1} ∘ N_{L_{2}/L_{1}})| |^{(1−[L_{2}:L_{1}])/2})].

Proof: The proof will be by induction on [L_{3} : L_{1}], there being nothing to prove in the case [L_{3} : L_{1}] = 1.

Now consider the inductive step. Because L_{3}/L_{1} is soluble we may choose a subextension L_{3} ⊃ L_{4} ⊃ L_{1} with L_{4}/L_{1} cyclic Galois with prime degree q. Let σ be a generator of Gal (L_{4}/L_{1}) and let σ̃ be a lift of σ to Gal (L_{3}/L_{1}). We will consider separately the cases L_{4} ⊂ L_{2} and L_{4} ∩ L_{2} = L_{1}.

Suppose first that L_{4} ⊂ L_{2}. Set ϕ′ = ϕ ∘ N_{L_{4}/L_{1}} unless [L_{2} : L_{4}] is odd in which case set ϕ′ = 1. Then from the inductive hypothesis we see that there is a cuspidal automorphic representation I_{L_{2}}^{L_{4}}(χ) of GL_{[L_{2}:L_{4}]}(𝔸_{L_{4}}) such that

• I_{L_{2}}^{L_{4}}(χ)^{c} ≅ I_{L_{2}}^{L_{4}}(χ)^{∨};

• (I_{L_{2}}^{L_{4}}(χ)⊗(ϕ′∘det))_{∞} has the same infinitesimal character as an algebraic representation of RS_{ℚ}^{L_{4}}(GL_{[L_{2}:L_{4}]});

• I_{L_{2}}^{L_{4}}(χ)_{y} is supercuspidal;

• and

[R_{l}(I_{L_{2}}^{L_{4}}(χ) ⊗ (ϕ′ ∘ det))] =

a(I_{L_{2}}^{L_{4}}(χ) ⊗ (ϕ′ ∘ det))[Ind _{Gal (L_{3}^{ac}/L_{2})}^{Gal (L_{3}^{ac}/L_{4})}rec_{l,ı}(χ^{−1}(ϕ′)^{−1}| |^{(1−[L_{2}:L_{4}])/2})].

By theorem 11.11 and lemma 12.7 we see that

[rec_{l}(I_{L_{2}}^{L_{4}}(χ)_{y})] = [Ind _{W_{L_{2,y}}}^{W_{L_{4,y}}} rec_{l}(χ_{y})],

and hence that

[rec_{l}(I_{L_{2}}^{L_{4}}(χ)_{y}^{σ})] = [Ind _{W_{σ̃L_{2,y}}}^{W_{L_{4,y}}} rec_{l}(χσ̃ ; y)].

Thus

[rec_{l}(I_{L_{2}}^{L_{4}}(χ)_{y})|_{W_{L_{3,y}}}] = ∑_{τ∈Gal (L_{3}/L_{2})\Gal (L_{3}/L_{4})}[rec_{l}(χ_{y} ∘ N_{L_{3,y}/L_{2,y}} ∘ τ)],

and

[rec_{l}(I_{L_{2}}^{L_{4}}(χ)_{y}^{σ})|_{W_{L_{3,y}}}] = ∑_{τ∈Gal (L_{3}/L_{2})\Gal (L_{3}/L_{4})}[rec_{l}(χ_{y} ∘ N_{L_{3,y}/L_{2,y}} ∘ τσ̃)].

In particular by our assumption on χ_{y} we see that

[rec_{l}(I_{L_{2}}^{L_{4}}(χ)_{y})] ≠ [rec_{l}(I_{L_{2}}^{L_{4}}(χ)_{y}^{σ})]

and conclude that

I_{L_{2}}^{L_{4}}(χ)_{y} ≅ I̸_{L_{2}}^{L_{4}}(χ)_{y}^{σ}.

Now set

I_{L_{2}}^{L_{1}}(χ) = Ind _{L_{4}}^{L_{1}}I_{L_{2}}^{L_{4}}(χ).

By the strong multiplicity one theorem we see that

I_{L_{2}}^{L_{1}}(χ)^{∨} ≅ I_{L_{2}}^{L_{1}}(χ)^{c}.

If w is an infinite place of L_{1} below places x_{1}, ..., x_{q} of L_{4} then I_{L_{2}}^{L_{1}}(χ)_{x} is a

subquotient of Ind _{Q(L_{1,x})}^{GL_{[L_{2}:L_{1}]}(L_{1,x})}(I_{L_{2}}^{L_{4}}(χ)_{x_{1}} × ... × I_{L_{2}}^{L_{4}}(χ)_{x_{q}}), where Q ⊂ GL_{[L_{2}:L_{1}]} is a parabolic subgroup with Levi component GL_{[L_{2}:L_{4}]}^{q}. Using this one can check that (I_{L_{2}}^{L_{1}}(χ) ⊗ (ϕ ∘ det))_{∞} has the same infinitesimal character as an algebraic representation of RS_{ℚ}^{L_{1}}(GL_{[L_{2}:L_{1}]}). Moreover

I_{L_{2}}^{L_{1}}(χ)_{y} = Ind _{L_{4,y}}^{L_{1,y}}(I_{L_{2}}^{L_{4}}(χ)_{y})

is supercuspidal by proposition 5.5 of [HH]. From theorem 11.11 we see that for any finite place x of L_{4} not dividing l and lying below places x_{1}, ..., x_{r} of L_{2} we have

[rec_{l}(I_{L_{2}}^{L_{4}}(χ)_{x} ⊗ (ϕ′_{x} ∘ det))] = ∑_{i=1}^{r}[Ind _{W_{L_{2,x_{i}}}}^{W_{L_{4,x}}} rec_{l}(χ_{x_{i}}(ϕ′_{x} ∘ N_{L_{2,x_{i}}/L_{4,x_{i}}}))].

By lemma 12.7 we conclude that for all but finitely many finite places x of L_{1} lying below places x_{1}, ..., x_{r} of L_{2} we have

[rec_{l}(I_{L_{2}}^{L_{1}}(χ)_{x} ⊗ (ϕ_{x} ∘ det))] = ∑_{i=1}^{r}[Ind _{W_{L_{2,x_{i}}}}^{W_{L_{1,x}}} rec_{l}(χ_{x_{i}}(ϕ_{x} ∘ N_{L_{2,x_{i}}/L_{1,x}}))].

Finally using the Cebotarev density theorem and theorem 11.11 we see that

[R_{l}(I_{L_{2}}^{L_{1}}(χ) ⊗ (ϕ ∘ det)] =

a(I_{L_{2}}^{L_{1}}(χ) ⊗ (ϕ ∘ det))[Ind _{Gal (L_{3}^{ac}/L_{2})}^{Gal (L_{3}^{ac}/L_{1})}rec_{l,ı}(χ^{−1}(ϕ^{−1} ∘ N_{L_{2}/L_{1}})| |^{(1−[L_{2}:L_{1}])/2})].

Now we turn to the case L_{2} ∩ L_{4} = L_{1}. In this case by inductive hypothesis there is a cuspidal automorphic representation I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}}) of GL_{[L_{2}:L_{1}]}(𝔸_{L_{4}}) such that

• I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}})^{c} ≅ I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}})^{∨};

• (I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}}) ⊗ (ϕ ∘ N_{L_{4}/L_{1}} ∘ det))_{∞} has the same infinitesimal character as an algebraic representation of RS_{ℚ}^{L_{4}}(GL_{[L_{2}:L_{1}]});

• I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}})_{y} is supercuspidal;

• and

[R_{l}(I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}}) ⊗ (ϕ ∘ N_{L_{4}/L_{1}} ∘ det))] = a(I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}}) ⊗ (ϕ ∘ N_{L_{4}/L_{1}} ∘ det)) [Ind _{Gal (L_{3}^{ac}/L_{2}L_{4})}^{Gal (L_{3}^{ac}/L_{4})}rec_{l,ı}((χ^{−1} ∘ N_{L_{2}L_{4}/L_{2}})(ϕ^{−1} ∘ N_{L_{2}L_{4}/L_{1}})| |^{(1−[L_{2}:L_{1}])/2})].

By theorem 11.11 and lemma 12.7 we see that for any prime x of L_{4} not dividing l and lying below primes x_{1}, ..., x_{r} of L_{2}L_{4} we have

[rec_{l}(I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}})_{x})] = ∑_{i=1}^{r}[Ind _{W_{(L_{2}L_{4})x_{i}}}^{W_{L_{4,x}}} rec_{l}(χ_{x} ∘ N_{(L_{2}L_{4})_{x_{i}}/L_{2,x_{i}}})].

In particular we see that

[rec_{l}(I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}})_{x})] = [rec_{l}(I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}})_{σx}^{σ})]

and so by the strong multiplicity one theorem

I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}}) ≅ I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}})^{σ}.

Thus by theorem 4.2 of chapter 3 of [AC] there is a cuspidal automorphic representation Π of GL_{[L_{2}:L_{1}]}(𝔸_{L_{1}}) such that Res_{L_{4}}^{L_{1}}(Π) = I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}}).

Again theorem 4.2 of chapter 3 of [AC] tells us that

Π^{∨} ≅ Π^{c} ⊗ (η ∘ det)

for some character η of 𝔸_{L_{1}}^{×}/L_{1}^{×}L_{1,∞}^{×}(N_{L_{4}/L_{1}}𝔸_{L_{4}}^{×}). The norm map gives an isomorphism

N_{L_{1}/L_{1}^{+}} : 𝔸_{L_{1}}^{×}/L_{1}^{×}L_{1,∞}^{×}(N_{L_{4}/L_{1}}𝔸_{L_{4}}^{×}) ⟶^{∼} 𝔸_{L_{1}^{+}}^{×}/(L_{1}^{+})^{×}((L_{1,∞}^{+})^{×})^{0}(N_{L_{4}^{+}/L_{1}^{+}}𝔸_{L_{4}^{+}}^{×}).

On the other hand because L_{4}^{+} is totally real we see that

𝔸_{L_{1}^{+}}^{×}/(L_{1}^{+})^{×}((L_{1,∞}^{+})^{×})^{0}(N_{L_{4}^{+}/L_{1}^{+}}𝔸_{L_{4}^{+}}^{×}) = 𝔸_{L_{1}^{+}}^{×}/(L_{1}^{+})^{×}(L_{1,∞}^{+})^{×}(N_{L_{4}^{+}/L_{1}^{+}}𝔸_{L_{4}^{+}}^{×})

and hence that

N_{L_{1}/L_{1}^{+}} : 𝔸_{L_{1}}^{×}/L_{1}^{×}L_{1,∞}^{×}(N_{L_{4}/L_{1}}𝔸_{L_{4}}^{×}) ↪ 𝔸_{L_{1}}^{×}/L_{1}^{×}L_{1,∞}^{×}(N_{L_{4}^{+}/L_{1}^{+}}𝔸_{L_{4}^{+}}^{×}).

Thus we can find a character ψ of 𝔸_{L_{1}}^{×}/L_{1}^{×}L_{1,∞}^{×} such that ψ ∘ N_{L_{1}/L_{1}^{+}} = η and hence

(Π ⊗ (ψ ∘ det))^{∨} = (Π ⊗ (ψ ∘ det))^{c}.

Note that therefore

(I_{L_{2}L_{4}}^{L_{4}}(χ) ⊗ (ψ ∘ N_{L_{4}/L_{2}} ∘ det))^{∨} = (I_{L_{2}L_{4}}^{L_{4}}(χ) ⊗ (ψ ∘ N_{L_{4}/L_{2}} ∘ det))^{c}.

If x is an infinite place of L_{1} lying under an infinite place x̃ of L_{4} we see that Π_{x} ≅ I_{L_{2}L_{4}}^{L_{4}}(χ ∘ N_{L_{2}L_{4}/L_{2}})x̃. Thus (Π ⊗ (ϕ ∘ det))_{∞} has the same infinitesimal character as an algebraic representation of RS_{ℚ}^{L_{1}}(GL_{[L_{2}:L_{1}]}). From lemma 6.12 of chapter 1 and the discussions on pages 52/53 and 59/60 of [AC] we see that Π_{y} must be supercuspidal.

By lemma 12.7, for all but finitely many pairs (x, x̃) of a place x of L_{1} and a place x̃ of L_{4} above x we have

[rec_{l}((Π ⊗ (ϕψ ∘ det))_{x})|_{W_{L_{4,x̃}}}] = [rec_{l}((I_{L_{2}L_{4}}^{L_{4}}(χ) ⊗ ((ϕψ) ∘ N_{L_{4}/L_{1}} ∘ det))x̃)].

It follows from theorem 11.11 and the Cebotarev density theorem that

a(I_{L_{2}L_{4}}^{L_{4}}(χ) ⊗ (ϕψ ∘ N_{L_{4}/L_{1}} ∘ det))[R_{l}(Π ⊗ (ϕψ ∘ det))|_{Gal (L_{3}^{ac}/L_{4})}] =
a(Π ⊗ (ϕψ ∘ det))[R_{l}(I_{L_{2}L_{4}}^{L_{4}}(χ) ⊗ (ϕψ ∘ N_{L_{4}/L_{1}} ∘ det))]

and hence that

[R_{l}(Π ⊗ (ϕψ ∘ det))|_{Gal (L_{3}^{ac}/L_{4})}] = a(Π ⊗ (ϕψ ∘ det))

[Ind _{Gal (L_{3}^{ac}/L_{2}L_{4})}^{Gal (L_{3}^{ac}/L_{4})}rec_{l,ı}((χ^{−1} ∘ N_{L_{2}L_{4}/L_{2}})(ϕ^{−1}ψ^{−1} ∘ N_{L_{2}L_{4}/L_{1}})| |^{(1−[L_{2}:L_{1}])/2})].

As Ind _{Gal (L_{3}^{ac}/L_{2}L_{4})}^{Gal (L_{3}^{ac}/L_{4})}rec_{l,ı}(χ^{−1} ∘ N_{L_{2}L_{4}/L_{2}})|_{W_{L_{4,y}}} is irreducible we see that

[R_{l}(Π ⊗ (ϕψ ∘ det))] = a(Π ⊗ (ϕψ ∘ det))
[Ind _{Gal (L_{3}^{ac}/L_{2})}^{Gal (L_{3}^{ac}/L_{1})}rec_{l,ı}(χ^{−1}(ϕ^{−1}ψ^{−1}η′ ∘ N_{L_{2}/L_{1}})| |^{(1−[L_{2}:L_{1}])/2})],

for some character η′ of 𝔸_{L_{1}}^{×}/L_{1}^{×}L_{1,∞}^{×}(N_{L_{4}/L_{1}}𝔸_{L_{4}}^{×}). Thus

[(R_{l}(Π ⊗ (ϕψ ∘ det)) ⊗ rec_{l,ı}(ψ ⊗ (η′)^{−1}))^{c}] =
[(R_{l}(Π ⊗ (ϕψ ∘ det)) ⊗ rec_{l,ı}(ψ ⊗ (η′)^{−1}))^{∨}]

and hence by theorem 11.11, lemma 12.7 and the strong multiplicity one theorem we see that

(Π ⊗ (η′ ∘ det))^{c} = (Π ⊗ (η′ ∘ det))^{∨}.

Replacing Π by Π ⊗ (η′ ∘ det), we have that Π^{c} = Π^{∨} and

[R_{l}(Π ⊗ (ϕψ(η′)^{−1} ∘ det))] = a(Π ⊗ (ϕψ(η′)^{−1} ∘ det))
[Ind _{Gal (L_{3}^{ac}/L_{2})}^{Gal (L_{3}^{ac}/L_{1})}rec_{l,ı}(χ^{−1}(ϕ^{−1}ψ^{−1}η′ ∘ N_{L_{2}/L_{1}})| |^{(1−[L_{2}:L_{1}])/2})],

and hence (using theorem 11.11, lemma 12.7 and the Cebotarev density theorem)

[R_{l}(Π ⊗ (ϕ ∘ det))] = a(Π ⊗ (ϕ ∘ det))
[Ind _{Gal (L_{3}^{ac}/L_{2})}^{Gal (L_{3}^{ac}/L_{1})}rec_{l,ı}(χ^{−1}(ϕ^{−1} ∘ N_{L_{2}/L_{1}})| |^{(1−[L_{2}:L_{1}])/2})].

Thus we may set I_{L_{2}}^{L_{1}}(χ) = Π. □

Recall that if r is a continuous representation of W_{K} over ℂ and if Ψ is a continuous additive character of K then we have the following.

• An L-factor

L(r, s) = det((1 − Frob_{p}/(#k(℘_{K}))^{s})|W_{r}^{I_{K}})^{−1},

where W_{r}^{I_{K}} denotes the inertial invariants of r.

• An ϵ-factor

ϵ(r, s, Ψ).

(See for instance [Tat2]. In the notation of [Tat2] we have ϵ(r, s, Ψ) = ϵ(rω_{s}, Ψ, μ_{Ψ}), where μ_{Ψ} is the additive Haar measure on K which is self dual with respect to Ψ.)

• A γ-factor

γ(r, s, Ψ) = L(r^{∨}, 1 − s)ϵ(r, s, Ψ)/L(r, s).

If moreover π_{1} and π_{2} are irreducible admissible representations of GL_{g_{1}}(K) and GL_{g_{2}}(K) then we also have the following.

• An L-factor L(π_{1} × π_{2}, s).

• An ϵ-factor ϵ(π_{1} × π_{2}, s, Ψ).

• A γ-factor

γ(π_{1} × π_{2}, s, Ψ) = L(π_{1}^{∨} × π_{2}^{∨}, 1 − s)ϵ(π_{1} × π_{2}, s, Ψ)/L(π_{1} × π_{2}, s).

(See [JPSS] for the definitions. In the case g_{2} = 1 and π_{2} is trivial we will simply drop it from the notation.) Note that if π is unramified then

L(π, s) = L(rec_{l}(π), s).

Corollary 12.9 Suppose that L_{3} ⊃ L_{2} ⊃ L_{1} are CM-fields with L_{3}/L_{1} soluble and Galois. Suppose that χ be a character of 𝔸_{L_{2}}^{×}/L_{2}^{×} such that

1. χ^{c} = χ^{−1};

2. for every embedding τ : L_{2} ↪ ℂ giving rise to an infinite place x we

have

χ_{x} : z → → (τz/cτz)^{p_{τ}}

where p_{τ} ∈ ℤ and if τ ≠ τ′ then p_{τ} ≠ p′_{τ};

3. and there is a finite place y of L_{1} which is inert in L_{3}, which does not divide l, which is unramified over L_{1}^{+} and for which the stabiliser of the character χ_{y} ∘ N_{L_{3}/L_{2}} of (L_{3})_{y}^{×} in Gal (L_{3}/L_{1}) is Gal (L_{3}/L_{2}).

Then for all but finitely many places x of L_{1} we have

L(I_{L_{2}}^{L_{1}}(χ)_{x}, s) = L((Ind _{Gal (L_{3}^{ac}/L_{2})}^{Gal (L_{3}^{ac}/L_{1})}rec_{l,ı}(χ))|_{W_{L_{1,x}}}, s) = ∏_{x̃|x} L(rec_{l}(χx̃), s).

Proof: We only need show that we can choose a character ϕ as in proposition 12.8. Let ϕ_{∞} be a character of L_{1,∞}^{×} of the form described in proposition 12.8. We have a commutative diagram with exact rows

(0) →  L_{1,∞}^{×} × ∏_{x} O_{L_{1},x}^{×}  →  𝔸_{L_{1}}^{×}  → Cl (L_{1})
↑  ↑  ↑
(0) → O_{L_{1}}^{×}N_{L_{1}/L_{1}^{+}}(L_{1,∞}^{×} × ∏_{x} O_{L_{1},x}^{×}) → L_{1}^{×}N_{L_{1}/L_{1}^{+}}𝔸_{L_{1}}^{×} → Cl (L_{1})^{1+c},

where Cl (L_{1}) denotes the ideal class group of L_{1}. Thus it suffices to define ϕ on ∏_{x} O_{L_{1},x}^{×}/(O_{L_{1},y}^{×}(N_{L_{1}/L_{1}^{+}} ∏_{x} O_{L_{1},x}^{×})) so that it equals ϕ_{∞}^{−1} on O_{L_{1}}^{×}. Let O_{L_{1}^{+}}^{1} denote those elements of O_{L_{1}^{+}}^{×} with norm down to ℚ equal to 1. Then ϕ_{∞} is trivial on O_{L_{1}}^{1}. Thus it suffices to check that

O_{L_{1}}^{×} ∩ (O_{L_{1},y}^{×}(N_{L_{1}/L_{1}^{+}} ∏_{x} O_{L_{1},x}^{×})) ⊂ O_{L_{1}^{+}}^{1}.

So suppose α ∈ O_{L_{1}}^{×} ∩(O_{L_{1},y}^{×}(N_{L_{1}/L_{1}^{+}} ∏_{x} O_{L_{1},x}^{×})). Then, because y is unramified over L_{1}^{+}, we see that α ∈ (L_{1}^{+})^{×} ∩ (N_{L_{1}/L_{1}^{+}}(𝔸_{L_{1}}^{∞})^{×}). But we have a right exact sequence

(L_{1}^{+})^{×} ⟶ ⨁_{x}(L_{1}^{+})_{x}^{×}/N_{L_{1}/L_{1}^{+}}L_{1,x}^{×} ⟶ Gal (L_{1}/L_{1}^{+}) ⟶ (0).

Thus α must fail to be a norm at an even number of infinite places x of L_{1}^{+}, i.e. α is negative at an even number of infinite places x of L_{1}^{+}. Thus the norm down to ℚ of α is positive and hence 1. □

Corollary 12.10 Suppose that L_{3}/L_{1} is a soluble Galois extension of CMfields and suppose that L_{2} and L′_{2} are intermediate fields between L_{3} and L_{1}. Let Ψ = ∏_{x} Ψ_{x} be a non-trivial additive character of 𝔸_{L_{1}}/L_{1}. Suppose that χ (resp. χ′) is a character of 𝔸_{L_{2}}^{×}/L_{2}^{×} (resp. 𝔸_{L′_{2}}^{×}/(L′_{2})^{×}) such that

1. χ^{c} = χ^{−1} (resp. (χ′)^{c} = (χ′)^{−1});

2. for every embedding τ : L_{2} ↪ ℂ (resp. τ′ : L′_{2} ↪ ℂ) giving rise to an infinite place x (resp. x′) we have

χ_{x} : z ⟼ (τz/cτz)^{p_{τ}}

(resp.

χ′_{x′} : z ⟼ (τ′z/cτ′z)^{p′_{τ′}})

where p_{τ} (resp. p′_{τ′}) ∈ ℤ and if τ ≠ τ_{1} (resp. τ′ ≠ τ′_{1}) then p_{τ} ≠ p_{τ_{1}} (resp. p′_{τ′} ≠ p′_{τ′_{1}});

3. there is a finite place y (resp. y′) of L_{1} which is inert in L_{3}, which does not divide l, which is unramified over L_{1}^{+} and for which the stabiliser of the character χ_{y}∘N_{L_{3}/L_{2}} (resp. χ′_{y′}∘N_{L_{3}/L′_{2}} in Gal (L_{3}/L_{1}) is Gal (L_{3}/L_{2}) (resp. Gal (L_{3}/L′_{2})).

Suppose that ψ and ψ′ are algebraic characters of 𝔸_{L_{1}}^{×}. Then for all places x of L_{1} which are inert in L_{3} and which do not divide l we have

[rec_{l}((I_{L_{2}}^{L_{1}}(χ)_{x} ⊗ (ψ_{x} ∘ det))] = [Ind _{W_{L_{2,x}}}^{W_{L_{1,x}}}rec_{l}(χ_{x}(ψ_{x} ∘ N_{L_{2,x}/L_{1,x}}))]

and

[rec_{l}((I_{L′_{2}}^{L_{1}}(χ′)_{x} ⊗ (ψ′_{x} ∘ det))] = [Ind _{W_{L{′ \atop 2,x}}}^{W_{L_{1,x}}}rec_{l}(χ′_{x}(ψ′_{x} ∘ N_{L′_{2,x}/L_{1,x}}))]

and

γ((I_{L_{2}}^{L_{1}}(χ) ⊗ (ψ ∘ det))_{x} × (I_{L′_{2}}^{L_{1}}(χ′) ⊗ (ψ′ ∘ det))_{x}, s, Ψ_{x}) =
γ((Ind _{W_{L_{2,x}}}^{W_{L_{1,x}}}rec_{l}(χ_{x}(ψ_{x} ∘ N_{L_{2,x}/L_{1,x}})))⊗
(Ind _{W_{L{′ \atop 2,x}}}^{W_{L_{1,x}}}rec_{l}(χ′_{x}(ψ′_{x} ∘ N_{L′_{2,x}/L_{1,x}}))), s, Ψ_{x}).

Proof: This follows from the previous corollary and from theorem 4.1 of [He3]. □

Now fix a non-trivial additive character Ψ of K.

Lemma 12.11 Fix a finite Galois extension K_{3}/K. For each pair (K_{2}, χ), where K_{2}/K is a finite subextension of K_{3}/K and where χ is a character of K_{2}^{×} of finite order, we can choose an irreducible admissible representation I_{K_{2}}^{K}(χ) of GL_{[K_{2}:K]}(K) which satisfies the following properties.

1. [rec_{l}(I_{K_{2}}^{K}(χ))] = [Ind _{W_{K_{2}}}^{W_{K}}rec_{l}(χ)].

2. Whenever (K_{2}, χ) and (K′_{2}, χ′) are two such pairs (with both K_{2} and K′_{2} inside the same fixed K_{3}) and ψ is a character of K^{×} of finite order we have

γ((I_{K_{2}}^{K}(χ) ⊗ (ψ ∘ det)) × I_{K′_{2}}^{K}(χ′), s, Ψ) =
γ(Ind _{W_{K_{2}}}^{W_{K}}rec_{l}(χ(ψ ∘ N_{K_{2}/K})) ⊗ Ind _{W_{K′_{2}}}^{W_{K}}rec_{l}(χ′), s, Ψ).

Proof: Choose an extension of totally real fields L_{3}^{0}/L^{0} and a place x_{0} of L^{0}, which is inert in L_{3}^{0} and for which the extension L_{3,x_{0}}^{0}/L_{x_{0}}^{0} is isomorphic to K_{3}/K. (This may be constructed as in lemma 3.6 of [He1]. Using weak approximation one can ensure that all the number fields of that argument can be taken to be totally real). Choose an imaginary quadratic field M and a real quadratic field N such that N is disjoint from L_{3}^{0} over ℚ and such that p splits completely in MN. Let x_{1} and y_{1} denote two places of MN above p which have the same restriction to M. Set L_{1} = MNL^{0} and let x (resp. y) denote the places of L which lie above x_{0} and x_{1} (resp. x_{0} and y_{1}). Also set L_{1} = L^{0}MN and L_{3} = L_{3}^{0}MN. Let x (resp. y) denote the place of L_{1} above x_{0} and x_{1} (resp. y_{1}). Thus x and y are inert in L_{3} and the extension L_{3,x}/L_{1,x} and K_{3}/K and L_{3,y}/L_{1,y} are all isomorphic.

Fix a pair (K_{2}, χ) as in the lemma and let L_{2}/L_{1} be the subfield of L_{3} corresponding to K_{2} ⊂ K_{3} under the isomorphism of L_{3,x}/L_{1,x} and K_{3}/K.

Choose p_{τ} as in corollary 12.10 and all divisible by the inertial degree f_{K_{2}/K}. Let χ_{∞} denote the character of L_{2,∞}^{×} corresponding to this choice of p_{τ}. Also choose a character χ_{y} of K_{2}^{×} for which the stabiliser of χ_{2} ∘ N_{K_{3}/K_{2}} in Gal (K_{3}/K_{1}) is Gal (K_{3}/K_{2}). (For this it suffices to choose a finite order character χ_{y} of O_{K_{2}}^{×} with the same property. Again it suffices to choose a continuous homomorphism O_{K_{2}}^{×} → ℤ_{p} with the same property (then compose it with a character of ℤ_{p} of sufficiently large order). Again it suffices to choose a ℚ_{p}-linear map O_{K_{2}}^{×} ⊗ℤ̂ ℚ_{p} → ℚ_{p} with the same property. But using the p-adic log and the normal basis theorem we find that there is a commutative diagram

O_{K_{3}}^{×} ⊗ℤ̂ ℚ_{p} ⟶^{∼}  K[Gal (K_{3}/K)]
↓  ↓
O_{K_{2}}^{×} ⊗ℤ̂ ℚ_{p} ⟶^{∼} K[Gal (K_{3}/K_{2})\Gal (K_{3}/K)],

where the top horizontal map is Gal (K_{3}/K)-equivariant, where the left hand vertical map is induced by the norm map and where the right hand vertical map is the natural projection. The existence of the desired homomorphism is now immediate.) Now as in the proof of corollary 12.9 we can find a character χ̃ of 𝔸_{L_{2}}^{×} such that

• χ̃^{−1} = χ̃_{c},

• χ̃|_{L_{2,∞}^{×}} = χ_{∞},

• χ̃_{x}χ^{−1} is unramified,

• and χ̃_{y}χ_{y}^{−1} is unramified.

(One must use the fact that x and y are split over the maximal totally real subfield L_{2}^{+} of L_{2}. The argument is easier than in the proof of corollary 12.9 because χ_{∞}|_{O_{L^{+}_{2}}^{×}} = 1.)

Now set ψ_{∞} = χ_{∞}|_{L_{1,∞}^{×}}^{−1/f_{K_{2}/K}} and choose a character ψ of 𝔸_{L_{1}}^{×} which is

unramified at x and which restricts to ψ_{∞} at ∞. One can check that χ̃_{x}(ψ_{x} ∘ N_{L_{2,x}/L_{1,x}}) has finite order and hence is a twist of χ by an unramified character of finite order. Replacing ψ by a twist by a suitable character of finite order one may assume that χ = χ̃_{x}(ψ_{x} ∘ N_{L_{2,x}/L_{1,x}}).

Finally we set

I_{K_{2}}^{K}(χ) = I_{L_{2}}^{L_{1}}(χ̃)_{x} ⊗ (ψ_{x} ∘ det).

The lemma follows from corollary 12.10. □

We will let Cusp (GL_{g}(K)) denote the set of isomorphism classes of irreducible admissible representations of GL_{g}(K) and we will let

Cusp_{K} = ⋃_{g=1}^{∞} Cusp (GL_{g}(K)).

We will let ℤ[Cusp_{K}] denote the free ℤ-module with basis the elements of Cusp_{K}. Then we may extend the definition of L(π_{1} × π_{2}, s), ϵ(π_{1} × π_{2}, s, Φ) and γ(π_{1} × π_{2}, s, Φ) to bilinear maps from ℤ[Cusp_{K}] × ℤ[Cusp_{K}] to the multiplicative abelian group of non-zero meromorphic functions on ℂ. We may also extend

• rec_{l} to a homomorphism ℤ[Cusp_{K}] → Groth_{l}(W_{K}),

• ∨ to a homomorphism ℤ[Cusp_{K}] → ℤ[Cusp_{K}],

• and ⊗(ψ ∘ det) to a ℤ[Cusp_{K}] → ℤ[Cusp_{K}], for any character ψ of K^{×}.

(Note that to any irreducible admissible representation π of GL_{g}(K) we can associate a class [π] ∈ ℤ[Cusp_{K}], i.e. if π is a subquotient of n-Ind (π_{1}×...×π_{r}) with each π_{i} irreducible supercuspidal then [π] = [π_{1}]+...+[π_{r}]. Note however that we do not in general have L(π, s) = L([π], s) etc.)

Lemma 12.12 Fix a finite Galois extension K_{3}/K. We can associate to any irreducible g-dimensional representation r of W_{K}/W_{K_{3}}, an element [π_{K_{3}/K}(r)] in ℤ[Cusp_{K}] with the following properties.

1. For any such r we have rec_{l}[π_{K_{3}/K}(r)] = r.

2. For any irreducible representations r and r′ of W_{K}/W_{K_{3}} and any character ψ of K^{×} of finite order, we have

γ([π_{K_{3}/K}(r)⊗(ψ ∘det)]×[π_{K_{3}/K}(r′)]^{∨}, s, Ψ) = γ(r ⊗rec_{l}(ψ)⊗(r′)^{∨}, s, Ψ).

Proof: This follows from the previous lemma and from Brauer’s theorem that representations induced from characters of subgroups form a ℤ-basis of the Grothendieck group of virtual representations of the finite group W_{K}/W_{K_{3}}. □ Corollary 12.13 Fix a finite Galois extension K_{3}/K. If r is an irreducible representation of W_{K} then [π_{K_{3}/K}(r)] can be represented by a supercuspidal representation π_{K_{3}/K}(r). The map

r ⟼ π_{K_{3}/K}(r)

is an injection from the irreducible representations of W_{K}/W_{K_{3}} to Cusp_{K}.

Proof: Suppose that [π_{K_{3}/K}(r)] = ∑_{i} a_{i}[π_{i}] where the [π_{i}] are distinct elements of Cusp_{K}. Then

γ([π_{K_{3}/K}(r)] × [π_{K_{3}/K}(r)]^{∨}, s, Ψ)

has a zero at s = 0 of order ∑_{i} a_{i}^{2} (see proposition 8.1 of [JPSS]). On the other hand

γ([π_{K_{3}/K}(r)] × [π_{K_{3}/K}(r)]^{∨}, s, Ψ) = γ(r ⊗ r^{∨}, s, Ψ)

and so has a simple zero at s = 1. The corollary follows. □

Corollary 12.14 Fix a finite Galois extension K_{3}/K. If r and r′ are irreducible representations of W_{K}/W_{K_{3}} and if ψ is a continuous character of K^{×} of finite order then

L((π_{K_{3}/K}(r) ⊗ (ψ ∘ det)) × π_{K_{3}/K}(r′)^{∨}, s) = L(r ⊗ rec_{l}(ψ) ⊗ (r′)^{∨}, s)

and

ϵ((π_{K_{3}/K}(r) ⊗ (ψ ∘ det)) × π_{K_{3}/K}(r′)^{∨}, s, Ψ) = ϵ(r ⊗ rec_{l}(ψ) ⊗ (r′)^{∨}, s, Ψ).

In particular π_{K_{3}/K}(r) and r have the same conductor.

Proof: This follows from lemma 12.12 and the previous corollary as in lemma 4.4 and proposition 4.5 of [He3]. □

Lemma 12.15 We can associate to any irreducible continuous g-dimensional representation r of W_{K} with finite image, an irreducible supercuspidal representation π(r) of GL_{g}(K) with the following properties.

1. For any such r we have rec_{l}[π(r)] = r.

2. For any continuous irreducible representations r and r′ of W_{K} with finite images and for any character ψ of K^{×} of finite order, we have

L((π(r) ⊗ (ψ ∘ det)) × π(r′)^{∨}, s) = L(r ⊗ rec_{l}(ψ) ⊗ (r′)^{∨}, s)

and

ϵ((π(r) ⊗ (ψ ∘ det)) × π(r′)^{∨}, s, Ψ) = ϵ(r ⊗ rec_{l}(ψ) ⊗ (r′)^{∨}, s, Ψ).

Proof: As K has finitely many extensions of given degree there are countably many irreducible continuous representations r of W_{K}. List them r_{1}, r_{2}, ... and set g_{i} = dim r_{i}. There are only finitely many irreducible supercuspidal representations of GL_{g_{i}}(K) with the same conductor as r_{i} and with central character det(r_{i}) ∘ Art_{K}. (For instance, as there are only finitely many irreducible representations of D_{K,g_{i}}^{×} which are trivial on a given open compact subgroup and which have given central character, this can be deduced from the Jacquet-Langlands correspondence (see sections 2.5 and 2.6 of [He4]).) For any positive integer I we may find a set K_{I} of finite Galois extensions of K such that

• if K′ and K′′ ∈ K_{0} then either K′ ⊃ K′′ or K′′ ⊃ K′;

• ⋃_{K′∈K_{I}} K′ = K^{ac};

• K_{I} ⊂ K_{I−1};

• and for each K′, K′′ ∈ K_{I} and each i ≤ I we have

π_{K′/K}(r_{i}) = π_{K′′/K}(r_{i}).

(Argue by recursion on I.) Set π(r_{i}) = π_{K′/K}(r_{i}) for any K′ ∈ K_{i}. The lemma now follows easily. □

Corollary 12.16 If r is an irreducible continuous representation of W_{K} with finite image and if ψ is a character of K^{×} of finite order then

π(r ⊗ rec_{l}(ψ)) = π ⊗ (ψ ∘ det).

Proof: Look at the zero at s = 0 of

γ([π(r) ⊗ (ψ ∘ det)] × [π(r ⊗ rec_{l}(ψ))]^{∨}, s, Ψ) = γ(r ⊗ r^{∨}, s, Ψ).

□ Corollary 12.17 The map rec_{l} gives a bijection between isomorphism classes of irreducible supercuspidal representations of GL_{g}(K) with central character of finite order and equivalence classes of g-dimensional irreducible continuous representations of W_{K} with finite image.

Proof: This now follows from the previous lemma and corollary and from theorem 1.2 of [He4]. □

Corollary 12.18 The map rec_{l} gives a bijection between Cusp (GL_{g}(K)) and the set of equivalence classes of g-dimensional irreducible continuous representations of W_{K}. Moreover π_{1} ∈ Cusp (GL_{g_{1}}(K)) and π_{2} ∈ Cusp (GL_{g_{2}}(K)) then

L(π_{1} × π_{2}, s) = L(rec_{l}(π_{1}) ⊗ rec_{l}(π_{2}), s)

and

ϵ(π_{1} × π_{2}, s, Ψ) = ϵ(rec_{l}(π_{1}) ⊗ rec_{l}(π_{2}), s, Ψ).

Proof: Any irreducible supercuspidal representation of GL_{g}(K) is of the form π ⊗ (ψ ∘ det), where ψ_{π} is finite order and ψ is unramified. The corollary follows as in sections 4.2, 4.3 and 4.4 of [He5]. □

Corollary 12.19 The bijection rec_{l} from Cusp (GL_{g}(K)) to the set of equivalence classes of g-dimensional irreducible continuous representations of W_{K} is independent of the choice of l ≠ p and of the choice of isomorphism ı : ℚ_{l}^{ac} →^{∼} ℂ (which we have assumed is chosen so that ı| |_{K}^{1/2} is valued in ℝ_{>0}^{×}).

Proof: This follows from the last corollary and from theorem 4.1 of [He5]. □

As described in section 4.4 of [Rod] one may naturally extend

rec_{l} : Cusp_{K} ⟶^{∼} Irr_{l}(W_{K})

to a series of bijections

rec_{K} : Irr(GL_{g}(K)) ⟶ WDRep_{g}(W_{K})

for all g ∈ ℤ_{>0}. We will let Sp_{g} = (r, N) denote the g-dimensional Weil-Deligne representation of W_{K} on a complex vector space with basis e_{0}, ..., e_{g−1} where

• r(σ)e_{i} = |Art _{K}^{−1}|^{i}e_{i} for all σ ∈ W_{K} and all i = 0, ..., g − 1

• and where Ne_{i} = e_{i+1}.

Then Rodier sets

rec_{K}(Sp_{s_{1}}(π_{1}) ⊞ ... ⊞ Sp_{s_{t}}(π_{t})) = (rec_{l}(π_{1}) ⊗ Sp_{s_{1}}) ⊕ ... ⊕ (rec_{l}(π_{t}) ⊗ Sp_{s_{t}}).

Then we have

rec_{K,l}(π) = (rec_{l}(π), N(π))

for some N(π).

Theorem 12.20 The map rec_{K} is a local Langlands correspondence. (See the introduction for the definition of a local Langlands correspondence.)

Proof: That this follows from what we have already proved seems to be well known, but for lack of an explicit reference we sketch the argument.

It follows from lemma 12.7 and the definition of rec_{K} that

• if π ∈ Irr(GL_{1}(K)) then rec_{K}(π) = π ∘ Art _{K}^{−1};

• if [π] ∈ Irr(GL_{g}(K)) and χ ∈ Irr(GL_{1}(K)) then rec_{K}(π ⊗ (χ ∘ det)) = rec_{K}(π) ⊗ rec_{K}(χ);

• and if [π] ∈ Irr(GL_{g}(K)) then det rec_{K}(π) = rec_{K}(ψ_{π}).

Suppose that π is an irreducible admissible representation of GL_{g}(K). Then we can write

π ≅ Sp_{s_{1}}(π_{1}) ⊞ ... ⊞ Sp_{s_{t}}(π_{t}),

with π_{1}, ..., π_{t} irreducible supercuspidals. Moreover we have

π^{∨} ≅ Sp_{s_{1}}(π_{1}^{∨} ⊗ | det |^{1−s_{1}}) ⊞ ... ⊞ Sp_{s_{t}}(π_{t}^{∨} ⊗ | det |^{1−s_{t}}).

(In the case t = 1 this can be deduced from proposition 1.1 (d) and proposition 2.10 of [Z]. Then the case that π is tempered follows from another application of proposition 1.1 (d) of [Z]. Finally the general case follows from proposition 1.1 (d) of [Z] and corollary 2.7 of chapter XI of [BW].) Hence using lemma 12.7, the definition of rec_{K} and the isomorphism

Sp _{s}^{∨} ≅ Sp_{s} ⊗ |Art _{K}^{−1}|^{1−s}

we see that

• if [π] ∈ Irr(GL_{g}(K)) then rec_{K}(π^{∨}) = rec_{K}(π)^{∨}.

It remains to check that if [π_{1}] ∈ Irr(GL_{g_{1}}(K)) and [π_{2}] ∈ Irr(GL_{g_{2}}(K)) then

L(π_{1} × π_{2}, s) = L(rec_{K}(π_{1}) ⊗ rec_{K}(π_{2}), s)

and

ϵ(π_{1} × π_{2}, s, Ψ) = ϵ(rec_{K}(π_{1}) ⊗ rec_{K}(π_{2}), s, Ψ).

Recall that in [JPSS] the factors L(π_{1} × π_{2}, s) and ϵ(π_{1} × π_{2}, s, Ψ) are only defined directly for π_{1} and π_{2} generic. In this case if

π_{i} ≅ Sp_{s_{1}}(π_{i,1}) ⊞ ... ⊞ Sp_{s_{t_{i}}}(π_{i,t_{i}})

with each π_{i,j} supercuspidal, then

L(π_{1} × π_{2}, s) = ∏_{j_{1}=1}^{t_{1}} ∏_{j_{2}=1}^{t_{2}} L(π_{1,j_{1}} × π_{2,j_{2}}, s)

and

ϵ(π_{1} × π_{2}, s, Ψ) = ∏_{j_{1}=1}^{t_{1}} ∏_{j_{2}=1}^{t_{2}} ϵ(π_{1,j_{1}} × π_{2,j_{2}}, s, Ψ)

(see theorems 3.1 and 9.5 of [JPSS]). In general these formulae are used to define L(π_{1} × π_{2}, s) and ϵ(π_{1} × π_{2}, s, Ψ) for any irreducible admissible π_{1} and π_{2}. As by definition we have

L(r_{1} ⊕ r_{2}, s) = L(r_{1}, s)L(r_{2}, s)

and

ϵ(r_{1} ⊕ r_{2}, s, Ψ) = ϵ(r_{1}, s, Ψ)ϵ(r_{2}, s, Ψ)

for all Weil-Deligne representations r_{1} and r_{2}, we only need to check that

L(Sp_{s_{1}}(π_{1}) × Sp_{s_{2}}(π_{2}), s) = L(rec_{K}(π_{1}) ⊗ rec_{K}(π_{2}) ⊗ Sp_{s_{1}} ⊗ Sp_{s_{2}}, s)

and

ϵ(Sp_{s_{1}}(π_{1}) × Sp_{s_{2}}(π_{2}), s, Ψ) = ϵ(rec_{K}(π_{1}) ⊗ rec_{K}(π_{2}) ⊗ Sp_{s_{1}} ⊗ Sp_{s_{2}}, s, Ψ)

for all irreducible supercuspidal representations π_{1} and π_{2}, and for all positive integers s_{1} ≥ s_{2}.

By theorems 3.1 and 8.2 of [JPSS] (see also equation (14) of section 8.2 of [JPSS]), we see that if s_{1} ≥ s_{2} then

L(Sp_{s_{1}}(π_{1}) × Sp_{s_{2}}(π_{2}), s) = ∏_{i=0}^{s_{2}−1} L(π_{1} × (π_{2} ⊗ | det |^{s_{1}+i−1}), s)

and

γ(Sp_{s_{1}}(π_{1}) × Sp_{s_{2}}(π_{2}), s, Ψ) = ∏_{i=0}^{s_{1}−1} ∏_{j=0}^{s_{2}−1} γ((π_{1} ⊗ | det |^{i}) × (π_{2} ⊗ | det |^{j}), s, Ψ).

Using corollary 12.18, we see that it suffices to check if r_{1} and r_{2} are irreducible Weil-Deligne representations of W_{K} and if s_{1} ≥ s_{2} are positive integers, then

L(r_{1} ⊗ r_{2} ⊗ Sp_{s_{1}} ⊗ Sp_{s_{2}}, s) = ∏_{i=0}^{s_{2}−1} L(r_{1} ⊗ r_{2} ⊗ |Art _{K}^{−1}|^{s_{1}+i−1}), s)

and

γ(r_{1} ⊗ r_{2} ⊗ Sp_{s_{1}} ⊗ Sp_{s_{2}}, s, Ψ) = ∏_{i=0}^{s_{1}−1} ∏_{j=0}^{s_{2}−1} γ(r_{1} ⊗ r_{2} ⊗ |Art _{K}^{−1}|^{i+j}, s, Ψ).

Note that if s_{1} ≥ s_{2} then

Sp_{s_{1}} ⊗ Sp_{s_{2}} ≅ ∑_{i=1}^{s_{2}} Sp_{s_{1}+s_{2}+1−2i} ⊗ |Art _{K}^{−1}|^{i−1}.

The desired equality of L-factors follows at once from the definitions in section 4.1.6 of [Tat2]. The desired equality of γ-factors follows easily if we can show that for any irreducible Weil-Deligne representation r of W_{K} we have

γ(r ⊗ Sp_{t}, s, Ψ) = ∏_{i=0}^{t−1} γ(r ⊗ |Art _{K}^{−1}|^{i}, s, Ψ).

To prove this we consider two cases. If r is ramified then according to section 4.1.6 of [Tat2] we have

γ(r ⊗ Sp_{t}, s, Ψ) = ϵ(⨁_{i=0}^{t−1} r ⊗ |Art _{K}^{−1}|^{i}, s, Ψ) = ∏_{i=0}^{t−1} γ(r ⊗ |Art _{K}^{−1}|^{i}, s, Ψ).

Thus we may suppose that r is unramified and hence that dim r = 1. Again using the formulae of section 4.1.6 of [Tat2] we see that

γ(r ⊗ Sp_{t}, s, Ψ) =
ϵ(⨁_{i=0}^{t−1} r ⊗ |Art _{K}^{−1}|^{i}, s, Ψ)(−1)^{t−1} ∏_{i=0}^{t−2}(r ⊗ |Art _{K}^{−1}|^{i})(Frob_{K})
∏_{i=0}^{t−1} L(r^{∨} ⊗ |Art _{K}^{−1}|^{−i}, 1 − s)(∏_{i=0}^{t−1} L(r ⊗ |Art _{K}^{−1}|^{i}, s))^{−1} =
∏_{i=0}^{t−1} γ(r ⊗ |Art _{K}^{−1}|^{s+i}, s, Ψ) ∏_{i=0}^{t−2}(−(r ⊗ |Art _{K}^{−1}|^{s+i})(Frob_{K})
L(r^{∨} ⊗ |Art _{K}^{−1}|^{−i−1}, 1 − s)/L(r ⊗ |Art _{K}^{−1}|^{i}, s)) =
∏_{i=0}^{t−1} γ(r ⊗ |Art _{K}^{−1}|^{s+i}, s, Ψ).

The theorem follows. □

I Appendix: Generalities on admissible representations.

If G is a group and g ∈ G we will let

• Z(G) denote the centre of G,

• Z_{G}(g) denote the centraliser of g in G,

• and [g] the conjugacy class of g in G.

If π is a representation of G we will let W_{π} denote the vector space on which the image of π acts, and we will let ψ_{π} denote the central character of π (if it has one).

Now suppose that G is a topological group such that every neighbourhood of the identity contains a compact open subgroup. Suppose also that Ω is an algebraically closed field of characteristic 0. then we will let

C_{c}^{∞}(G)

denote the space of locally constant Ω-valued functions on G with compact support. If ψ : Z(G) → Ω^{×} is an admissible character of Z(G) then we will let

C_{c}^{∞}(G, ψ)

denote the space of locally constant Ω-valued functions φ on G such that

• φ(zg) = ψ(z)φ(g) for all z ∈ Z(G) and g ∈ G

• and the image of the support of φ in G/Z(G) is compact.

We may choose a (left or right) Haar measure μ on G such that every compact subgroup of G has measure in ℚ. Then we may speak of an Ωvalued Haar measure meaning a non-zero element of Ωμ. If φ ∈ C_{c}^{∞}(G), if π is an admissible representation G over Ω and if we fix an (Ω-valued) Haar measure on G then we have a well defined endomorphism π(φ) of W_{π}. The endomorphism π(φ) has finite rank and so tr π(φ) makes sense. Similarly if ψ is an admissible character of Z(G), if φ ∈ C_{c}^{∞}(G, ψ^{−1}), if π is an admissible representation G with central character ψ and if we fix Haar measures on G and Z(G) then again we have a well defined endomorphism π(φ) of W_{π}. Again the endomorphism π(φ) has finite rank and so tr π(φ) makes sense. If φ ∈ C_{c}^{∞}(G) or C_{c}^{∞}(G, ψ) and if we fix Haar measures on G and Z_{G}(g) then we will let

O_{g}^{G}(φ) denote the integral

∫_{G/Z_{G}(g)} φ(xgx^{−1})dx,

if this integral converges. All these notations depend on a choice of Haar measure(s) which we are suppressing. We will try to make clear in the accompanying text which measures we have chosen. If the choice of Haar measure on a particular group enters twice into a particular formula we will always suppose that we make the same choice both times, unless there is an explicit statement to the contrary. In such cases it will often be irrelevant which choice we make, only that we make a consistent choice.

Now suppose that H/K is a reductive algebraic group. Let P ⊂ H be a parabolic subgroup with unipotent radical N ⊂ P. If π is an admissible representation of (P/N)(K) then we define an admissible representation Ind _{P}^{H}(π) of H(K) as follows. The underlying space will be the set of function f : H(K) → W_{π} such that

• f(hg) = π(h)(f(g)) for all g ∈ H(K) and h ∈ P (K);

• there exists an open subgroup U ⊂ H(K) such that f(gu) = f(g) for all u ∈ U and g ∈ H(K).

If f is such a function and g ∈ H(K) then we set

(g(f))(h) = f(hg)

for all h ∈ H(K). Conversely if π is an admissible representation of G(K) then the space of N(K)-coinvariants W_{π,N(K)} is naturally an admissible representation π_{N} of (P/N)(K).

Often it is convenient to use instead an alternative normalisation. To describe this, choose a square root | |_{K}^{1/2} : K^{×} → Ω^{×} of | |_{K}. If f_{K} is even we will suppose that | |_{K}^{1/2} takes a uniformiser to p^{−f_{K}/2}. Then we will let n-Ind _{P}^{H}(π) denote the normalised induction as in [BZ]. Thus n-Ind _{P}^{H}(π) = Ind _{P}^{H}(π ⊗ δ_{P}^{1/2}) where δ_{P}^{1/2}(h) = | det(ad (h)|_{Lie N})|_{K}^{1/2}. Similarly if π is an admissible representation of H(K), we will define the Jacquet module J_{N}(π) to be the admissible representation π_{N} ⊗ δ_{P}^{−1/2} of (P/N)(K).

Now return to the general topological group G such that every neighbourhood of the identity in G contains a compact open subgroup. Let Irr(G) denote the set of isomorphism classes of irreducible admissible representations of G over Ω. Let Groth (G) denote the abelian group of formal sums

∑_{Π∈Irr(G)} n_{Π}Π

where n_{Π} ∈ ℤ and where for any open compact subgroup U ⊂ G there are only finitely many Π ∈ Irr(G) with both Π^{U} ≠ (0) and n_{Π} ≠ 0. If (π, V) is an admissible representation of G then we will define

[π] = ∑_{Π∈Irr(G)} n_{Π}(π)Π ∈ Groth (G)

as follows. Given Π ∈ Irr(G) choose an open compact subgroup U ⊂ G such that Π^{U} ≠ (0). Then Π^{U} is an irreducible H(U\G/U)-module. We let n_{Π}(π) denote the number of H(U\G/U)-Jordan-Holder factors of π^{U} isomorphic to Π^{U}. This is independent of the choice of U. (To see this suppose U′ ⊂ U. Let F^{i} be a Jordan-Holder filtration on π^{U}. Let (F^{i})′ = H(U′\G/U)F^{i}. It suffices to show that (F^{i})′/(F^{i+1})′ contains Π^{U′} once or not at all depending on whether F^{i}/F^{i+1} is or is not congruent to Π^{U}. This is easy to verify.) We list some basic properties of this construction.

1. [ ] is additive on short exact sequences.

2. Let K denote a p-adic field. Suppose that G = G_{1} × GL_{n}(K) and that H = G_{1} × P (K) where P ⊂ GL_{n} is a parabolic subgroup. Then there is a unique homomorphism Ind _{P (K)}^{GL_{n}(K)} : Groth (H) → Groth (H) such

that for any admissible representation π of H we have Ind _{P (K)}^{GL_{n}(K)}[π] =

[Ind _{P (K)}^{GL_{n}(K)}π].

3. Suppose that G = G_{1} × G_{2} and that π_{i} is an admissible representation G_{i} for i = 1, 2. Then π_{1} ⊗ π_{2} is an admissible representation of G_{1} × G_{2}. If π_{1} and π_{2} are irreducible so is π_{1} ⊗ π_{2}. We can define a product Groth (G_{1}) ⊗ Groth (G_{2}) → Groth (G_{1} × G_{2}) which sends [Π_{1}] ⊗ [Π_{2}] to [Π_{1} ⊗ Π_{2}] for any irreducible admissibles Π_{1} and Π_{2}. Then for any admissible representations π_{i} of G_{i} for i = 1, 2 we have [π_{1}⊗π_{2}] = [π_{1}][π_{2}].

4. More generally suppose that G = G_{1} ×G_{2} and that we have a continuous homomorphism d : G_{2} → Z(G_{1}) with discrete image. Suppose that π_{1} is an admissible representation of G_{1} and that π_{2} is an admissible representation of G_{2}. Then we define the representation π_{1} ⊗_{d} π_{2} of G_{1} × G_{2} by (π_{1} ⊗_{d} π_{2})(g_{1}, g_{2}) = π_{1}(g_{1}d(g_{2})) ⊗ π_{2}(g_{2}). Then π_{1} ⊗_{d} π_{2} is admissible. If Π_{1} and Π_{2} are irreducible so is Π_{1} ⊗_{d} Π_{2} and so we can define a product Groth (G_{1}) ⊗ Groth (G_{2}) → Groth (G_{1} × G_{2}) which sends [Π_{1}] ⊗ [Π_{2}] to [Π_{1} ⊗_{d} Π_{2}]. We will denote this product ∗_{d}. Then for any admissible representation π_{1} of G_{1} and π_{2} of G_{2} we have [π_{1} ⊗_{d} π_{2}] = [π_{1}] ∗_{d} [π_{2}].

It some special cases it will be convenient to introduce a slight variant of Groth (G). For this suppose that l is a prime number and Ω = ℚ_{l}^{ac}. Suppose that G has an open subgroup H × Γ. By an H × Γ-smooth/continuous representation of G we shall mean a representation π of G such that

• π|_{H} is smooth (i.e. the stabiliser of any element of W_{π} is open)

• we can write W_{π} = lim_{→} W_{i} where W_{i} are finite dimensional Γ invariant subspaces of W_{π} such that the representation

π : Γ ⟶ Aut (W_{i})

is continuous with respect to the l-adic topology on W_{π}^{U}.

By an H × Γ-admissible/continuous representation of G we shall mean a representation π of G such that

• π|_{H} is smooth (i.e. the stabiliser of any element of W_{π} is open)

• and for any open subgroup U ⊂ H, the vector space W_{π}^{U} is finite dimensional and the representation

π : Γ ⟶ Aut (W_{π}^{U})

is continuous with respect to the l-adic topology on W_{π}^{U}.

We will let Irr_{H×Γ,l}(G) denote the set of isomorphism classes of irreducible H × Γ-admissible/continuous representations of G. We will also let Groth_{H×Γ,l}(G) denote the abelian group of formal sums

∑_{Π∈Irr_{H×Γ,l}(G)} n_{Π}Π

where n_{Π} ∈ ℤ and where for any open compact subgroup U ⊂ H there are only finitely many Π ∈ Irr_{H×Γ,l}(G) with both Π^{U} ≠ (0) and n_{Π} ≠ 0. If (π, V) is an H × Γ-admissible/continuous representation of G then we can define [π] ∈ Groth_{H×Γ,l}(G) as before. We will usually suppress the choice of H and Γ. Our examples will all be of one of the following forms. Here K will denote a finite field extension of ℚ_{p}.

1. H is a topological group such that every neighbourhood of the identity in G contains a compact open subgroup, Γ is a Galois group with the Krull topology and G = H × Γ.

2. H is a topological group such that every neighbourhood of the identity in G contains a compact open subgroup, Γ = I_{K} and G = H × W_{K}.

3. G = A_{K,g} (see section 3), H = GL_{g}(O_{K}) × O_{D_{K,g}}^{×} and Γ = I_{K}.

In the latter two cases it is a theorem of Grothendieck that Irr_{l}(G) = Irr(G) and hence that Groth_{l}(G) ⊂ Groth (G).

Finally consider the following special situation. Let K be a p-adic field. For h = 1, ..., n − 1 we will let P_{h} denote the parabolic subgroup of GL_{n}(K) consisting of matrices (g_{ij}) with g_{ij} = 0 if i > n − h and j ≤ n − h. We will let N_{h} denote the unipotent radical of P_{h} and L_{h} the Levi component consisting of matrices (g_{ij}) ∈ P_{h} with g_{ij} = 0 for i ≤ n − h and j > n − h. Also let Z_{h} denote the centre of L_{h}. Abusing notation we will write N_{h}(℘_{K}^{m}) for the elements in P_{h}(F) ∩ ℘_{K}^{m}M_{n}(O_{K}). This is in fact a group.

Lemma I.1 1. If V is an admissible P_{h}(K)-module and if for all u ∈ N_{h}(K) we have (u − 1)^{2} = 0 on V then N_{h}(K) acts trivially on V.

2. If V is a smooth P_{h}(K)-module which is admissible as a L_{h}(K)-module then N_{h}(K) acts trivially on V.

3. If G_{1} is any locally compact totally disconnected group and if V is a smooth G_{1} ×P_{h}(K)-module which ia admissible as a G_{1} ×L_{h}(K)-module then N_{h}(K) acts trivially on V.

Proof: For the first part consider the open compact subgroups U_{m} consisting of elements of P_{h}(O_{K}) which reduce modulo ℘_{K}^{m} to the identity and modulo ℘_{K}^{2m} to an element of N_{h}(O_{K}/℘_{K}^{2m}). Note that U_{m} is normalised by N_{h}(℘_{K}^{−m}). Thus V^{U_{m}} is a finite dimensional smooth N_{h}(℘_{K}^{m})-module, and as N_{h}(℘_{K}^{−m}) is compact, V^{U_{m}} is semi-simple. Thus if u ∈ N(℘_{K}^{−m}) we see that u = 1 on V^{U_{m}}. As N_{h}(K) = ⋃_{m} N_{h}(℘_{K}^{−m}) and V = ⋃_{m} V^{U_{m}}, the first part of the lemma follows.

Now consider the second part of the lemma. If χ is a character of Z_{h}(K) then set

V_{i}^{χ} = ⋂_{z∈Z_{h}(K)} ker(z − χ(z))^{i}

and

V_{∞}^{χ} = ⋃_{i} V_{i}^{χ}.

Because V is an admissible L_{h}(K)-module we have

V = ⨁_{χ} V_{∞}^{χ}.

(This follows because for any open compact subgroup U ⊂ L_{h}(K), V^{U} is a finite dimensional smooth Z_{h}(K)-module, and hence

V^{U} = ⨁_{χ}(V_{χ}^{∞} ∩ V^{U}).)

Thus it suffices to show for each χ and i that V_{i}^{χ} is a P_{h}(K)-module on which N_{h}(K) acts trivially. We will do this by induction on i for fixed χ. For i = 0 there is nothing to prove. Thus assume the result is true for V_{i−1}^{χ} and we will prove it for V_{i}^{χ}. By the first part of this lemma it suffices to show that V_{i}^{χ}/V_{i−1}^{χ} is a P_{h}(K)-submodule of V/V_{i−1}^{χ} on which N_{h}(K) acts trivially. Suppose that v ∈ V/V_{i−1}^{χ}. By smoothness, v is invariant by N_{h}(℘_{K}^{m}) for some m. If u is any element of N_{h}(K) we may choose z ∈ Z_{h}(K) such that zuz^{−1} ∈ N_{h}(℘_{K}^{m}). Then we have

uv = uz^{−1}χ(z)v = χ(z)z^{−1}(zuz^{−1})v = χ(z)z^{−1}v = v.

The second part of the lemma follows.

The third part of the lemma follows easily from the second. □

II Appendix: Vanishing cycles

Here we will collect some facts about vanishing cycles.

Let K/ℚ̂_{p}^{nr} be a finite field extension, let O denote its ring of integers and let l be a prime integer different from p. Suppose that X/O is a proper scheme of finite type.

Lemma II.1 If x is a closed point of X then (R^{i}Ψ_{η}(ℚ_{l}^{ac})_{X})_{x} is a finite dimensional ℚ_{l}^{ac}-vector space with a continuous action of Gal (K^{ac}/K).

Lemma II.2 If L/X is a lisse ℚ_{l}^{ac}-sheaf then

R^{i}Ψ_{η}(L) ≅ (R^{i}Ψ_{η}(ℚ_{l}^{ac})) ⊗ L_{s},

where L_{s} is the restriction of L to the special fibre of X.

Lemma II.3 Suppose that Y/X is a finite cover with an action of a finite group G. Suppose that the generic fibres are a Galois etale cover with group G. Suppose that x is a closed point of X which is totally ramified in Y, and let y be its preimage in Y. Then

(R^{i}Ψ_{η}(ℚ_{l}^{ac})_{X})_{x} →^{∼} (R^{i}Ψ_{η}(ℚ_{l}^{ac})_{Y})_{y}^{G}.

In [Berk3], a vanishing cycles functor is constructed for a certain class of formal schemes over O. The comparison theorem 3.1 of [Berk3] implies that if X is a formal scheme over O, which is isomorphic to the formal completion of a proper scheme of finite type X/O along a subscheme Y of the special fibre of X, then there is a canonical isomorphism of sheaves

R^{i}Ψ_{η}(ℤ/l^{m}ℤ)_{X_{η}} ≅ R^{i}Ψ_{η}(ℤ/l^{m}ℤ)_{X_{η}}|_{Y} .

It follows that the projective system of constructible sheaves R^{i}Ψ_{η}(ℤ/l^{m}ℤ)_{X_{η}} form a ℤ_{l}-sheaf and, therefore, there is a well defined ℚ_{l}^{ac}-sheaf R^{i}Ψ_{η}(ℚ_{l}^{ac})_{X}. It is canonically isomorphic to R^{i}Ψ_{η}(ℚ_{l}^{ac})_{X}|_{Y}, and the automorphism group of the formal scheme X acts on it. More generally the construction of R^{i}Ψ_{η}(ℚ_{l}^{ac})_{X} is functorial in X, i.e. if X and Y are two such formal schemes and if f : X → Y then there is a natural map of ℚ_{l}^{ac}-sheaves on X^{red}

f^{∗} : (f^{red})^{∗}R^{i}Ψ_{η}(ℚ_{l}^{ac})_{Y} ⟶ R^{i}Ψ_{η}(ℚ_{l}^{ac})_{X} .

The continuity theorem from [Berk3] implies that there exists an ideal of definition of X such that any automorphism of X, trivial modulo this ideal, acts trivially on the ℚ_{l}^{ac}-sheaves R^{i}Ψ_{η}(ℚ_{l}^{ac})_{X}. The construction of R^{i}Ψ_{η}(ℚ_{l}^{ac})_{X} extends to any special formal scheme X which is etale locally isomorphic to the formal completion of a proper scheme of finite type 𝕏/O along a subscheme Y of the special fibre of X. Again this construction is functorial in X. The following lemma follows easily by reduction to the algebraic case using Berkovich’s comparison theorem.

Lemma II.4 Suppose that X and Y are special formal schemes each of which is etale locally isomorphic to the formal completion of a proper scheme of finite type over O along a subscheme of its special fibre. Then

R^{i}Ψ_{η}(ℚ_{l}^{ac})_{X ×Y} ≅ ⨁_{j} R^{i}Ψ_{η}(ℚ_{l}^{ac})_{X} ⊗ R^{i}Ψ_{η}(ℚ_{l}^{ac})_{Y}.

We end this section with a lemma of Berkovich’s (see the lemma in [Berk4]).

Lemma II.5 (Berkovich) Let X and Y be special affine formal schemes, say X = Spf A and Y = Spf B. Let J ⊂ B be the maximal ideal of definition for Y, and set Y{N} = Spec (B/J^{N+1}). Assume that X is isomorphic to the formal completion of an affine scheme of finite type X = Spec A′/Spec O along a closed subscheme of its special fibre. Let I′ ⊂ A′ be the maximal ideal of definition of this subscheme. Furthermore assume that we are given projective systems {X_{n}}_{n≥0} and {Y_{n}}_{n≥0} of finite etale coverings of X and Y respectively. Let Y_{n}{N} denote the pull back of Y{N} to Y_{n}. Suppose finally that we are given compatible morphisms

φ_{n} : Y_{n}{n} ⟶ X_{n}.

Then given any positive integer N for m >> 0 (depending on N) we can find a morphism

φ : Y_{m} ⟶ X_{m}

such that

φ|_{Y_{m}{N}} = φ_{m}|_{Y_{m}{N}}.

Proof: As in the proof of the lemma in [Berk4] we see that we have X_{n} = Spf A_{n} and Y_{n} = Spf B_{n}, where A_{n}/A and B_{n}/B are finite etale. We set B_{∞} = lim_{→} B_{n} and J_{∞} = JB_{∞}. Again as in the proof of the lemma of [Berk4] we see that (B_{∞}, J_{∞}) is a Henselian pair. The map φ_{n} induces a homomorphism

φ′_{n} : A′ ⟶ A_{n} ⟶ B_{n}/J^{n+1}B_{n} ⟶ B_{∞}/J_{∞}^{n+1}.

By corollary 1 on page 567 and remark 2 on page 587 of [El] we see that there exists an integer t such that for all n >> 0 there exists a homomorphism

φ̃_{n} : A′ ⟶ B_{∞}

with

φ̃_{n} ≡ φ′_{n} mod J_{∞}^{n+1−t}.

Fix such an n, which we also suppose greater than N + t.

Note that as A′/O is finitely generated φ̃_{n} is in fact valued in some B_{m} for m >> 0. As J_{∞}^{a} ∩ B_{m} = J^{a}B_{m} we see that for m >> 0

φ̃_{n} ≡ φ_{m} mod J^{N+1}B_{m}.

Thus φ̃_{n}(I′) ⊂ JB_{m} and so we may extend φ̃_{n} to a continuous homomorphism A → B_{m} such that

A \underrightarrow{φ̃_{n}}  B_{m}
↓  ↓
A_{m} ⟶^{φ_{m}} B_{m}/J^{N+1}B_{m}

commutes. By lemma 6.2 we see that we get a morphism

φ : A_{m} ⟶ B_{m}

with φ ≡ φ_{m} mod J^{N+1}B_{m}, as desired. □

III Appendix: Abelian varieties over 𝔽_{p}^{ac}.

In this appendix we will explain how the theory of Honda and Tate [Tat1] allows us to classify simple abelian varieties over 𝔽_{p}^{ac}.

By a CM field we will mean a number field M such that for any embedding i : M ↪ ℂ the image iM is stable under complex conjugation and such that the automorphism c of M induced by complex conjugation on iM is independent of the embedding i. Equivalently either M is totally real or a totally imaginary quadratic extension of a totally real field. We will let ℚ[𝔓_{M}] denote the free abelian group on the places of M above p. If i : M ↪ N is a finite extension we get natural maps i_{∗} : ℚ[𝔓_{M}] → ℚ[𝔓_{N}] induced by x ↦ ∑_{y|x} e_{y/x}y and i^{∗} : ℚ[𝔓_{N}] → ℚ[𝔓_{M}] induced by y ↦ f_{y/x}x if y|x. If I is a fractional ideal of M then we set [I] = ∑_{x} x(I)x ∈ ℚ[𝔓_{M}].

By a p-adic type for a CM field M we shall mean an element η ∈ ℚ[𝔓_{M}] such that η + c_{∗}η = [p]. We will call p-adic types η ∈ ℚ[𝔓_{M}] and η′ ∈ ℚ[𝔓_{M′}] equivalent if there is a CM field M′′, a p-adic type η′′ ∈ ℚ[𝔓_{M′′}] and embeddings i : M′′ ↪ M and i′ : M′′ ↪ M′ such that i_{∗}(η′′) = η and i′_{∗}(η′′) = η′. By a p-adic type we shall mean an equivalence class of p-adic types for various CM fields M. Then any p-adic type 𝔟 has a minimal representative (M, η) such that if (M′, η′) ∈ 𝔟 then there exists i : M ↪ M′ such that η′ = i_{∗}η. (To see this choose (M′, η′) ∈ 𝔟 with M′/ℚ Galois, let H denote the subgroup of σ ∈ Gal (M′/ℚ) such that σ_{∗}η′ = η′ and set M = (M′)^{H}.) We call a p-adic type 𝔟 ordinary if for any (M, η) ∈ 𝔟 and any x ∈ 𝔓_{M} we have η_{x}η_{cx} = 0.

Suppose that q = p^{r} is a power of p. By a q-number one means an algebraic number π such that for any embedding i : ℚ[π] ↪ ℂ we have πc(π) = q. To any q-number π we associate a CM-type 𝔟(π) = [(ℚ[π], [π]/r)]. To π Honda and Tate also associate a simple abelian variety A_{π}/𝔽_{q}. The simple factors of A_{π} ×_{𝔽_{q}} 𝔽_{q^{s}} are all isogenous to A_{π^{s}}. Thus all the simple factors of A_{π} ×_{𝔽_{q}} 𝔽_{p}^{ac} are isogenous: we will denote them A′_{π}/𝔽_{p}^{ac}. We see that A′_{π} ∼ A′_{π^{s}}. If 𝔟(π) = 𝔟(π′) then we can find positive integers n and n′ such that (up to Galois conjugation) [π^{n}] = [(π′)^{n′}]. Then (after replacing π by a suitable Galois conjugate) π^{n}/(π′)^{n′} is a unit with all archimedean absolute values 1 and hence a root of unity. Thus we can find a positive integer m such that π^{nm} and (π′)^{n′m} are Galois conjugate. Thus A_{π^{nm}} is isogenous to A_{(π′)^{n′m}} and so A′_{π} ∼ A′_{π′}. This allows us to write A_{𝔟(π)} for A′_{π}/𝔽_{p}^{ac}. If η is is any p-adic type for a CM field M the for some positive integer r we have that rη ∈ ℤ[𝔓_{M}] and hence for a second positive integer h we have that rhη = [α] for some α ∈ O_{M}. Then β = αc(α)/p^{rh} is a unit in the ring of integers of the maximally totally real subfield M^{+} of M. Thus π = α^{2}β^{−1} is a p^{2hr}-number and 𝔟(π) = [(M, η)].

Thus to any p-adic type we can associate a well defined isogeny class of simple abelian varieties A_{𝔟}/𝔽_{p}^{ac}. It follows easily from the theory of Honda and Tate that this gives a bijection between p-types and isogeny classes of simple abelian varieties over 𝔽_{p}^{ac}. The following further results also follow easily from their theory.

1. If 𝔟 is a p-adic type with minimal representative (M, η) then End^{0}(A_{𝔟}) is the division algebra with centre M and invariants

• inv_{x}End^{0}(A_{𝔟}) = 1/2 if x is real;

• inv_{x}End^{0}(A_{𝔟}) = η_{x}f_{x/p} if x|p;

• inv_{x}End^{0}(A_{𝔟}) = 0 otherwise.

2. If 𝔟 is a p-adic type with minimal representative (M, η) then dim A_{𝔟} = [M : ℚ][End^{0}(A_{𝔟}) : M]^{1/2}/2.

3. If 𝔟 is a p-adic type with minimal representative (M, η) and if x is a place of M above p then A_{𝔟}[x^{∞}] has height [M_{x} : ℚ_{p}][End^{0}(A_{𝔟}) : M]^{1/2} and its Newton polygon has pure slope η_{x}/e_{x/p}.

Now fix a CM field F and a central simple F -algebra B. If M/F and M′/F are CM field extensions and if η and η′ are p-adic types for M and M′ respectively the we will call them equivalent over F if there is a CM field extension M′′/F, a p-adic type η′′ ∈ ℚ[𝔓_{M′′}] and embeddings i : M′′ ↪ M and i′ : M′′ ↪ M over F such that i_{∗}(η′′) = η and i′_{∗}(η′′) = η′. By a p-adic type over F we shall mean an F -equivalence class of p-adic types for various CM fields M/F. Then any p-adic type 𝔟 has a minimal representative (M, η) such that if (M′, η′) ∈ 𝔟 then there exists i : M ↪ M′ over F such that η′ = i_{∗}η.

Now let B be an F -division algebra. We will consider the category of pairs (A, i) up to isogeny, where A/𝔽_{p}^{ac} is an abelian variety and i : B ↪ End^{0}(A/𝔽_{p}^{ac}). As in section 3 of [Ko3] we can use the results of the last paragraph to describe the simple objects of this category. They are in bijection with p-adic types over F. If 𝔟 is such a type we will let (A_{𝔟}, i_{𝔟}) denote the corresponding simple object. We have the following additional properties.

1. If 𝔟 is a p-adic type over F with minimal representative (M, η) then End_{B}(A_{𝔟}) is the division algebra with centre M and invariants

• inv_{x}End_{B}(A_{𝔟}) = 1/2 − inv_{x}(B ⊗_{F} M) if x is real;

• inv_{x}End_{B}(A_{𝔟}) = η_{x}f_{x/p} − inv_{x}(B ⊗_{F} M) if x|p;

• inv_{x}End_{B}(A_{𝔟}) = −inv_{x}(B ⊗_{F} M) otherwise.

2. If 𝔟 is a p-adic type over F with minimal representative (M, η) then dim A_{𝔟} = [M : ℚ][B : F]^{1/2}[End_{B}(A_{𝔟}) : M]^{1/2}/2.

3. If 𝔟 is a p-adic type over F with minimal representative (M, η) and if x is a place of M above p then A_{𝔟}[x^{∞}] has height [M_{x} : ℚ_{p}][B : F]^{1/2}[End (A_{𝔟}) : M]^{1/2} and its Newton polygon has pure slope η_{x}/e_{x/p}.

IV Appendix: The local Jacquet-Langlands correspondence, pseudo-coefficients and Zelevinsky’s classification

Let Ω be an algebraically closed field of characteristic 0 and of cardinality equal to that of ℂ. Let K be a finite extension of ℚ_{p}. Suppose that V/Ω is a vector space and that π : GL_{g}(K) → Aut (V) is an irreducible admissible representation with central character ψ_{π}. We will call π supercuspidal if for any v ∈ V and f in the smooth dual of V the function GL_{g}(K) → Ω which sends

x ⟼ f(xv)

is compactly supported modulo the centre K^{×} of GL_{g}(K). Choose an embedding of fields ı : Ω ↪ ℂ. We will call π square integrable if for any v ∈ V and f in the smooth dual of V the function GL_{g}(K)/K^{×} → ℝ which sends

x ⟼ |ı(f(xv))|^{2}|ı(ψ_{π}(det x))|^{−2/g}

is integrable. It follows from Zelevinsky’s classification [Z] that this definition is independent of the choice of ı. We will call π,ı-preunitary if there is a pairing ( , ) from V × V to ℂ such that

• (av_{1} + v_{2}, v_{3}) = ı(a)(v_{1}, v_{3}) + (v_{2}, v_{3}) for all a ∈ Ω and v_{1}, v_{2}, v_{3} ∈ V,

• (v_{1}, v_{2}) = c(v_{2}, v_{1}) for all v_{1}, v_{2} ∈ V (where c denotes complex conjugation),

• (v, v) > 0 for all non-zero v ∈ V,

• (π(x)v_{1}, π(x)v_{2}) = |ı(ψ_{π}(det g))|^{2/g}(v_{1}, v_{2}) for all v_{1}, v_{2} ∈ V and x ∈ GL_{g}(K).

Rogawski ([Rog2]) and Deligne, Kazhdan and Vigneras (see [DKV]) have shown the existence of a unique bijection, which we will denote JL , from irreducible admissible representations of D_{K,g}^{×} to square integrable irreducible admissible representations of GL_{g}(K) such that if ρ is an irreducible admissible representation of D_{K,g}^{×} then the character χ_{JL (ρ)} of JL (ρ) satisfies

• χ_{JL (ρ)}(γ) = 0 if γ ∈ GL_{g}(K) is regular semi-simple but not elliptic,

• χ_{JL (ρ)}(γ) = (−1)^{g−1}tr ρ(δ) if γ ∈ GL_{g}(K) is regular semi-simple and elliptic and if δ is an element of D_{K,g}^{×} with the same characteristic polynomial as γ.

If π is a square integrable irreducible admissible representation of GL_{g}(K) with central character ψ_{π} then it seems that Deligne, Kazhdan and Vigneras also show the existence of a function φ_{π} ∈ C_{c}^{∞}(GL_{g}(K), ψ_{π}^{−1}), which we will call a pseudo-coefficient for π, with the following properties. (We always use associated measures on inner forms of the same group.)

• tr π(φ_{π}) = vol (D_{K,g}^{×}/K^{×}).

• Suppose that

GL_{g_{1}} × ... × GL_{g_{s}} = L ⊂ P ⊂ GL_{g}

is a Levi component of a parabolic subgroup of GL_{g}. Suppose also that for i = 1, ..., s we are given a square integrable irreducible admissible representation π_{i} of GL_{g_{i}}(K) such that π_{i} ≅ π̸ and such that ψ_{π_{1}}...ψ_{π_{s}} = ψ_{π}. Then

tr n-Ind _{P (K)}^{GL_{g}(K)}(π_{1} × ... × π_{s})(φ_{π}) = 0.

• If γ ∈ GL_{g}(K) is a non-elliptic regular semi-simple element then

O_{γ}^{GL_{g}(K)}(φ_{π}) = 0.

• If γ ∈ GL_{g}(K) is an elliptic regular semi-simple element and if δ ∈ D_{K,g}^{×} has the same characteristic polynomial as γ then

O_{γ}^{GL_{g}(K)}(φ_{π}) = (−1)^{g−1}vol (D_{K,g}^{×}/Z_{D_{K,g}^{×}}(δ))tr JL^{−1}(π^{∨})(δ).

(See section A.4 of [DKV], especially the introduction to that section and subsection A.4.l .)

Lemma IV.1 Let π be a square integrable representation of GL_{g}(K) and let φ_{π} be a pseudo-coefficient for π as above.

1. If γ ∈ GL_{g}(K) is a non-elliptic semi-simple element then

O_{γ}^{GL_{g}(K)}(φ_{π}) = 0.

2. If γ ∈ GL_{g}(K) is an elliptic semi-simple element and if δ ∈ D_{K,g}^{×} has the same characteristic polynomial as γ then

O_{γ}^{GL_{g}(K)}(φ_{π}) = (−1)^{g(1−[K(γ):K]^{−1})}vol (D_{K,g}^{×}/Z_{D_{K,g}^{×}}(δ))tr JL^{−1}(π^{∨})(δ).

Proof: Consider the first part. Let T be a maximal torus containing γ. Then

0 = ∑_{u} Γ_{u}(t)O_{γu}^{GL_{g}(K)}(φ_{π})

where u runs over a set of representatives of the unipotent conjugacy classes in Z_{GL_{g}}(γ)(K), where Γ_{u} denotes the Shalika germ associated to u and where and where t is any regular element of T sufficiently close to γ. Then homogeneity ([HC], theorem 14(1)) tells us that

0 = Γ_{1}(t)O_{γu}^{GL_{g}(K)}(φ_{π})

for any regular t ∈ T sufficiently close to γ. By [Rog1], Γ_{1}(t) is not identically zero near γ and the first part of the lemma follows.

Consider now the second part. Let T be an elliptic maximal torus in GL_{g}(K) containing γ. We can and will also think of T ⊂ D_{K,g}^{×}. Then we can take δ to be γ ∈ T ⊂ D_{K,g}^{×}. For t a regular element of T sufficiently close to γ we have

(−1)^{g−1}vol (D_{K,g}^{×}/T)tr JL^{−1}(π^{∨})(t) = ∑_{u} Γ_{u}(t)O_{γu}^{GL_{g}(K)}(φ_{π})

where u runs over a set of representatives of the unipotent conjugacy classes in Z_{GL_{g}}(γ)(K) and where Γ_{u} denotes the Shalika germ associated to u. Again using homogeneity ([HC], theorem 14(1)) we see that

(−1)^{g−1}vol (D_{K,g}^{×}/T)tr JL^{−1}(π^{∨})(δ) = O_{γ}^{GL_{g}(K)}(φ_{π}) lim_{t→γ} Γ_{1}(t).

Thus it suffices to check that

lim_{t→0} Γ_{1}(t) = (−1)^{g/[K(γ):K]−1}vol (Z_{D_{K,g}^{×}}(δ)/T).

This is independent of the choices of measures, as long as we choose associated measures on Z_{GL_{g}}(γ)(K) and Z_{D_{K,g}^{×}}(δ). (Choices of Haar measures on Z_{GL_{g}}(γ)(K) and T are implicit in the definition of Γ_{u}.) Thus we may choose any measure on K^{×} and Euler-Poincaré measure on T and Z_{GL_{g}}(γ)(K) (see section 1 of [Ko5]). Then we must use (−1)^{g/[K(γ):K]−1} times Euler-Poincaré measure on Z_{D_{K,g}^{×}}(δ) (by theorem 1 of [Ko5]). According to [Rog2] with these choices of measures Γ_{1}(t) = 1 for regular t ∈ T sufficiently close to γ. On the other hand according to [Se] with these measures vol (T/K^{×}) = 1 and vol (Z_{D_{K,g}^{×}}(δ)/K^{×}) = 1. The second part of the lemma follows. □

Suppose that s|g is a positive integer and that π is a supercuspidal representation of GL_{g/s}(K). Let Q_{s} denote a parabolic subgroup of GL_{g} with Levi component GL_{g/s}^{s}. Zelevinsky ([Z]) describes the irreducible subquotients of

n-Ind _{Q_{s}(K)}^{GL_{g}(K)}(π × π ⊗ | det | × ... × π ⊗ | det |^{s−1})

as follows. Let Γ(s, π) be the graph with vertices labelled π ⊗ | det |^{j} for j = 0, ..., s − 1 and with one edge between π ⊗ | det |^{j} and π ⊗ | det |^{j+1} for j = 0, ..., s − 2 and no other edges. Zelevinsky shows that there is a bijection between directed graphs Γ⃗ with underlying undirected graph Γ(s, π) and irreducible subquotients of

n-Ind _{Q_{s}(K)}^{GL_{g}(K)}(π × π ⊗ | det | × ... × π ⊗ | det |^{s−1}),

which, following Zelevinsky’s notation, we will denote Γ⃗ ↦ ω(Γ⃗).

Some particular subquotients will be of special importance for us. So we will let Γ⃗_{abc}(π) denote the directed graph with vertices labelled π ⊗ | det |^{j} for j = 0, ...a + b + c − 1 and

• a single edge from π ⊗ | det |^{j} to π ⊗ | det |^{j−1} for j = 1, ..., a − 1 and for j = a + b + 1, ..., a + b + c − 1,

• and a single edge from π ⊗ | det |^{j−1} to π ⊗ | det |^{j} for j = a, ..., a + b.

Similarly we will let Γ⃗′_{abc}(π) denote the directed graph with vertices labelled π ⊗ | det |^{j} for j = 0, ...a + b + c − 1 and

• a single edge from π ⊗ | det |^{j−1} to π ⊗ | det |^{j} for j = 1, ..., a and for j = a + b, ..., a + b + c − 1,

• and a single edge from π ⊗ | det |^{j+1} to π ⊗ | det |^{j} for j = a, ..., a + b − 2.

More over we will denote ω(Γ⃗_{t,s−1−t,0}) = ω(Γ⃗′_{0,t,s−1−t}) by

Sp_{t}(π) ⊞ (π ⊗ | det |^{t}) ⊞ ... ⊞ (π ⊗ | det |^{s−1}),

for any t = 0, ..., s − 1.

Zelevinsky ([Z]) has proved the following results.

• Sp_{s}(π) is square integrable and any square integrable representation is of this form for a unique positive integer s|g and a unique supercuspidal representation π of GL_{g/s}(K).

• The only generic (“non-degenerate” in Zelevinsky’s terminology) subquotient of n-Ind _{Q_{s}(K)}^{GL_{g}(K)}(π × π ⊗ | det | × ... × π ⊗ | det |^{s−1}) is Sp_{s}(π).

Moreover Tadic ([Tad]) has shown that

• for any embedding ı : Ω ↪ ℂ the only ı-preunitary subquotients of n-Ind _{Q_{s}(K)}^{GL_{g}(K)}(π × π ⊗ | det | × ... × π ⊗ | det |^{s−1}) are Sp_{s}(π) and π ⊞ (π ⊗ | det |) ⊞ ... ⊞ (π ⊗ | det |^{s−1}).

Lemma IV.2 Suppose that s_{1} +s_{2} = s|g are positive integers and that π is an irreducible supercuspidal representation of GL_{g/s}(K). Set g_{i} = s_{i}g/s and let P denote a parabolic subgroup of GL_{s} with Levi component GL_{g_{1}} × GL_{g_{2}}. Let Γ⃗_{1} (resp. Γ⃗_{2}) be an oriented graph with unoriented underlying graph Γ(s_{1}, π) (resp. Γ(s_{2}, π ⊗ | det |^{s_{1}})). Let Γ⃗ and Γ⃗′ be the two (distinct) oriented graphs with underlying unoriented graph Γ(s, π) which agree with Γ⃗_{1} on Γ(s_{1}, π) and with Γ⃗_{2} on Γ(s_{2}, π ⊗ | det |^{s_{2}}). Then n-Ind _{P (K)}^{GL_{g}(K)}(ω(Γ⃗_{1}) × ω(Γ⃗_{2})) has a Jordan-Holder series of length two and the two Jordan-Holder factors are ω(Γ⃗) and ω(Γ⃗′).

Proof: We may and will assume that Q_{s} ⊂ P. Let U_{s} denote the unipotent radical of Q_{s}. By section 1.6 and theorem 2.8 of [Z] we can compute the Jordan-Holder factors of J_{U_{s}}(n-Ind _{P (K)}^{GL_{n}(K)}(ω(Γ⃗_{1}) × ω(Γ⃗_{2}))). We find that they are all products (each taken with multiplicity one) of the form

(ρ ⊗ | det |^{j_{1}}) × ... × (ρ ⊗ | det |^{j_{s}}),

where j_{1}, ..., j_{s} runs over all permutations of 0, ..., s − 1 such that j_{i} < j_{i′} if there is an edge of either Γ⃗_{1} or Γ⃗_{2} running from i to i′. This is the same as the union (as sets with multiplicities) of the Jordan-Holder factors of ω(Γ⃗) and ω(Γ⃗′) (see theorem 2.8 of [Z]). The lemma then follows from theorem 2.2 of [Z]. □

Lemma IV.3 Suppose that s|g are positive integers and that π is an irreducible supercuspidal representation of GL_{g/s}(K). For h = 0, ..., g − 1 let N_{h}^{op} be the unipotent subgroup of GL_{g} introduced at the start of section 10.

1. If g |̸sh then J_{N_{h}^{op}}(Sp_{s}(π)) = (0) and J_{N_{h}^{op}}(π ⊞...⊞(π ⊗| det |^{s−1})) = (0).

2. If sh = gh′ for some positive integer h′ then

J_{N_{h}^{op}}(Sp_{s}(π)) = Sp_{h′}(π ⊗ | det |^{s−h′}) × Sp_{s−h′}(π),

and

J_{N_{h}^{op}}(π ⊞ ... ⊞ (π ⊗ | det |^{s−1})) = (π ⊞ ... ⊞ (π ⊗ | det |^{h′−1}))×
×((π ⊗ | det |^{h′}) ⊞ ... ⊞ (π ⊗ | det |^{s−1})).

Proof: These results follow easily from theorem 2.2 of [Z]. □

Lemma IV.4 Suppose that s|g are positive integers and that π is an irreducible supercuspidal representation of GL_{g/s}(K).

1. If π′ is an irreducible admissible representation of GL_{g}(K) which is not a subquotient of n-Ind _{Q_{s}(K)}^{GL_{g}(K)}(π × π ⊗ | det | × ... × π ⊗ | det |^{s−1}) then

tr π′(φ_{Sp_{s}(π)}) = 0.

2. If P is a proper parabolic subgroup of GL_{g} with Levi component GL_{g_{1}} × GL_{g_{2}} and if π_{1} (resp. π_{2}) is an irreducible admissible representation of GL_{g_{1}}(K) (resp. GL_{g_{2}}(K)) then

tr n-Ind _{P (K)}^{GL_{g}(K)}(π_{1} × π_{2})(φ_{Sp_{s}(π)}) = 0.

3. Suppose that Γ⃗ is an oriented graph with underlying unoriented graph Γ(s, π) and that Γ⃗ has s edges oriented from π ⊗ | det |^{j} to π ⊗ | det |^{j+1}. Then

tr ω(Γ)(φ_{Sp_{s}(π)}) = (−1)^{s}vol (D_{K,g}^{×}/K^{×}).

Proof: For the first part note that the proof of lemma A.4.f of [DKV] shows that π′ can be written in Groth (GL_{g}(K)) as an integral linear combination of n-Ind _{P_{i}(K)}^{GL_{g}(K)}(π′_{i}) where P_{i} runs over parabolic subgroups of GL_{g}, where π′_{i} is an irreducible square integrable representation of the Levi component of P_{i}(K) and where no (P_{i}, π′_{i}) is conjugate to (GL_{g}, Sp_{s}(π)). (In fact in the notation of [DKV] we have r(π′_{i}) ≠ r(Sp_{s}(π)).)

For the second part note that it follows from lemma A.4.f of [DKV] that we may write π_{1} ∈ Groth (GL_{g_{1}}(K)) as a finite sum

π_{1} = ∑_{i} a_{1i}n-Ind _{P_{1i}(K)}^{GL_{g_{1}}(K)}π_{1i},

where a_{1i} ∈ ℤ, P_{1i} ⊂ GL_{g_{1}} is a parabolic subgroup and π_{1i} is an irreducible square integrable representation of the Levi component of P_{1i}(K). Similarly we have

π_{2} = ∑_{i} a_{2i}n-Ind _{P_{2i}(K)}^{GL_{g_{2}}(K)}π_{2i}.

For each pair of indices i, j choose a parabolic subgroup P′_{ij} ⊂ P ⊂ GL_{g} such that P′_{ij}(K) ∩ GL_{g_{1}}(K) = P_{1i}(K) and P′_{ij}(K) ∩ GL_{g_{2}}(K) = P_{2i}(K). Note that for all i, j we have P_{ij} ≠ GL_{g}. In Groth (GL_{g}(K)) we have the equality

n-Ind _{P (K)}^{GL_{g}(K)} = ∑_{ij} a_{1i}a_{2j}n-Ind _{P′_{ij}(K)}^{GL_{g}(K)}π_{1i} × π_{2j}.

The second part of the lemma follows.

The third part follows by a simple recursion from the second part and lemma IV.2. □

Corollary IV.5 If s|g are positive integers, if π is an irreducible supercuspidal representation of GL_{g/s}(K) and if π′ is an admissible representation of GL_{g}(K) then

vol (D_{K,g}^{×}/K^{×})^{−1}tr π′(φ_{Sp_{s}(π)}) ∈ ℤ.

Corollary IV.6 If s|g are positive integers, if π is an irreducible supercuspidal representation of GL_{g/s}(K) and if π′ is a generic irreducible admissible representation of GL_{g}(K) such that

tr π′(φ_{Sp_{s}(π)}) ≠ 0

then π′ ≅ Sp_{s}(π).

Corollary IV.7 If s|g are positive integers, if π is an irreducible supercuspidal representation of GL_{g/s}(K), if ı : Ω ↪ ℂ and if π′ is an ı-preunitary irreducible admissible representation of GL_{g}(K) such that

tr π′(φ_{Sp_{s}(π)}) ≠ 0

then either π′ ≅ Sp_{s}(π) or π′ ≅ π ⊞ ... ⊞ (π ⊗ | det |^{s−1}).

Suppose that s_{i} and g_{i} are positive integers for i = 1, ..., t such that g = g_{1}s_{1}+...+g_{t}s_{t}. Suppose moreover that for i = 1, ..., t we are given an irreducible supercuspidal representation π_{i} of GL_{g_{i}}(K). Suppose first that • if i < j then π_{j} ≅ π̸_{i} ⊗ | det |^{a} for any a ∈ ℤ_{≥1} with

1 + s_{i} − s_{j} ≤ a ≤ s_{i}.

Also let P denote the parabolic subgroup of GL_{g} consisting of block diagonal matrices with diagonal blocks of size s_{1}g_{1} × s_{1}g_{1}, ..., s_{t}g_{t} × s_{t}g_{t} from top left to bottom right. Then

n-Ind _{P (K)}^{GL_{g}(K)}(Sp_{s_{1}}(π_{1}) × ... × Sp_{s_{t}}(π_{t}))

has a unique irreducible quotient which we will denote

Sp_{s_{1}}(π_{1}) ⊞ ... ⊞ Sp_{s_{t}}(π_{t}).

If σ is any permutation of {1, 2, ..., t} such that (s_{σ1}, π_{σ1}), ..., (s_{σt}, π_{σt}) still satisfies the above condition then

Sp_{s_{σ1}}(π_{σ1}) ⊞ ... ⊞ Sp_{s_{σt}}(π_{σt}) ≅ Sp_{s_{1}}(π_{1}) ⊞ ... ⊞ Sp_{s_{t}}(π_{t}).

Thus whether or not (s_{1}, π_{1}), ..., (s_{t}, π_{t}) satisfy the above condition we may define

Sp_{s_{1}}(π_{1}) ⊞ ... ⊞ Sp_{s_{t}}(π_{t}) = Sp_{s_{σ1}}(π_{σ1}) ⊞ ... ⊞ Sp_{s_{σt}}(π_{σt}),

for any permutation σ of {1, 2, ..., t} such that (s_{σ1}, π_{σ1}), ..., (s_{σt}, π_{σt}) does satisfy the above condition. It follows from theorem 2.8 of [Z] that this notation is compatible with our previous use of ⊞. Moreover any irreducible admissible representation π of GL_{g}(K) is of this form and, moreover, the multiset {(s_{1}, π_{1}), ..., (s_{t}, π_{t})} is uniquely determined by π. (For these results see section 4.3 of [Rod], and note that a sketch of the unpublished result of I.N.Bernstein (proposition 11 of [Rod]) can be found in [JS].)

We will call the collection {(s_{i}, π_{i})} unlinked if for all i ≠ j the following condition is satisfied.

• If π_{j} ≅ π_{i}⊗| det |^{a} for a positive integer a then either a > s_{i} or a+s_{j} ≤ s_{i}.

Zelevinsky shows ([Z], theorem 9.7) that if {(s_{i}, π_{i})} is unlinked then

Sp_{s_{1}}(π_{1}) ⊞ ... ⊞ Sp_{s_{t}}(π_{t}) = n-Ind _{P (K)}^{GL_{g}(K)}(Sp_{s_{1}}(π_{1}) × ... × Sp_{s_{t}}(π_{t}))

and that this representation is generic. Zelevinsky also shows ([Z], theorem 9.7) that any irreducible generic admissible representation of GL_{g}(K) arises in this way from some unlinked collection {(s_{i}, π_{i})}.

The following result follows from lemma 2.12 of [BZ].

Lemma IV.8 Suppose that s_{i} and g_{i} for i = 1, ..., t are positive integers such that g = g_{1}s_{1} + ... + g_{t}s_{t}. Suppose also that π_{i} is an irreducible supercuspidal representation of GL_{g_{i}}(K) for i = 1, ..., t; and that P ⊂ GL_{g} is a parabolic subgroup with Levi factor GL_{s_{1}g_{1}} × ... × GL_{s_{t}g_{t}}.

For h = 0, ..., g − 1 let N_{h}^{op} < GL_{g} be the unipotent subgroup defined at the start of section 10. Then in Groth (GL_{h}(K) × GL_{g−h}(K)) we have an equality between

[J_{N_{h}^{op}}(n-Ind _{P (K)}^{GL_{g}(K)}(Sp_{s_{1}}(π_{1}) × ... × Sp_{s_{t}}(π_{t}))]

and

∑_{h_{i}}[n-Ind _{P′(K)}^{GL_{h}(K)}(Sp_{h_{1}}(π_{1} ⊗ | det |^{s_{1}−h_{1}}) × ... × Sp_{h_{t}}(π_{t} ⊗ | det |^{s_{t}−h_{t}}))]
[n-Ind _{P′′(K)}^{GL_{n−h}(K)}(Sp_{s_{1}−h_{1}}(π_{1}) × ... × Sp_{s_{t}−h_{t}}(π_{t}))],

where the sum is over all positive integers h_{1}, ..., h_{t} with h_{i} ≤ s_{i} and h = h_{1}g_{1} + ... + h_{t}g_{t} and where

• P′ ⊂ GL_{h} is a parabolic subgroup with Levi component GL_{h_{1}g_{1}} × ... × GL_{h_{t}g_{t}}

• and P′′ ⊂ GL_{n−h} is a parabolic subgroup with Levi component GL_{(s_{1}−h_{1})g_{1}} × ... × GL_{(s_{t}−h_{t})g_{t}}

V Appendix: The global Jacquet-Langlands correspondence.

Let S(B) denote the set of places of F at which B ramifies. Recall that we are assuming that at such a place B_{x} is a division algebra. The following theorem was proved by Vigneras in her unpublished manuscript [V2], which relied on a seminar of Langlands which to the best of our knowledge was never written up. We will explain how it follows easily from an important theorem of Arthur and Clozel [AC].

Theorem V.1 1. If ρ is an irreducible automorphic representation of (B^{op} ⊗𝔸)^{×} then there is a unique irreducible automorphic representation JL (ρ) of GL_{n}(𝔸_{F}), which occurs in the discrete spectrum and for which

JL (ρ)^{S(B)} ≅ ρ^{S(B)}.

2. If x ∈ S(B) and JL (ρ_{x}) = Sp_{s_{x}}(π_{x}) then • either JL (ρ)_{x} ≅ Sp_{s_{x}}(π_{x}),

• or JL (ρ)_{x} ≅ π_{x} ⊞ ... ⊞ (π_{x} ⊗ | det |^{s_{x}−1}).

3. The image of JL is the set of irreducible automorphic representations π of GL_{n}(𝔸_{F}) such that

• π occurs in the discrete spectrum,

• and for every x ∈ S(B) there is a positive integer s_{x}|n and an irreducible supercuspidal representation π′_{x} of GL_{n/s_{x}}(F_{x}) so that either JL (ρ)_{x} ≅ Sp_{s_{x}}(π′_{x}) or JL (ρ)_{x} ≅ π′_{x} ⊞ ... ⊞ (π′_{x} ⊗ | det |^{s_{x}−1}).

4. If ρ_{1} and ρ_{2} are two irreducible automorphic representations of (B^{op} ⊗ 𝔸)^{×} such that for all but finitely many places x of F we have ρ_{1x} ≅ ρ_{2x}, then ρ_{1} = ρ_{2} (i.e. ρ_{1} ≅ ρ_{2} and this representation occurs with multiplicity 1 in the space of automorphic forms).

Proof: Let H denote the algebraic group over ℚ such that H(R) = (B^{op} ⊗_{ℚ} R)^{×} for any ℚ-algebra R. We will confuse irreducible admissible representations occurring discretely in the space of automorphic forms (with fixed central character restricted to ℝ_{>0}^{×}) with their completions in L^{2}. By twisting, we need only consider representations which are trivial on ℝ_{>0}^{×}.

Then by theorem B of [AC] we have, in the notation of [AC],

a_{disc}^{H}(ρ) = ∑_{π} a_{disc}^{GL_{n}×F} (π)δ(π, ρ),

where the sum is over (pre)unitary representations of GL_{n}(𝔸_{F})^{1}. Let us write JL (ρ_{x}) = Sp_{s_{x}(ρ)}(π′_{x}(ρ)) for x ∈ S(B). Moreover for T ⊂ S(B) let π(T, ρ) denote

ρ^{S(B)} × ∏_{x∈T} Sp_{s_{x}(ρ)}(π′_{x}(ρ)) × ∏_{x∈S(B)−T}(π′_{x}(ρ) ⊞ ... ⊞ (π′_{x}(ρ) ⊗ | det |^{s_{x}(ρ)−1})).

Then, using section 8 of [AC], lemma IV.4 and corollary IV.7 we see that δ(π, ρ) = 0 unless π ≅ π(T, ρ) for some T ⊂ S(B), in which case δ(π, ρ) = ±1. Thus the equality of theorem B of [AC] becomes

a_{disc}^{H}(ρ) = ∑_{T ⊂S(B)} ±a_{disc}^{GL_{n}×F} (π(T, ρ)).

The coefficients a_{disc}^{H}(ρ) are just the multiplicity of ρ in the space of automorphic forms trivial on ℝ_{>0}^{×}. The coefficients a_{disc}^{GL_{n}×F} (π(T, ρ)) are defined by the equalities

∑ a_{disc}^{GL_{N} ×F} (π)tr π(f) = ∑_{L∈L} |W_{0}^{L}||W_{0}^{GL_{n}×F} |^{−1}
∑_{s∈W (𝔞_{L})_{reg}} | det(s − 1)_{𝔞_{L}^{GLn×F}}|^{−1}tr (M(s, 0)ρ_{Q,t}(0, f))

in the notation of section 9 of chapter 2 of [AC]. Choose x ∈ S(B). Then for any z in the Bernstein centre for GL_{n}(F_{x}) we see that

∑ a_{disc}^{GL_{N} ×F} (π)tr π(f)π_{x}(z) = ∑_{L∈L} |W_{0}^{L}||W_{0}^{GL_{n}×F} |^{−1}
∑_{s∈W (𝔞_{L})_{reg}} | det(s − 1)_{𝔞_{L}^{GLn×F}}|^{−1}tr (M(s, 0)ρ_{Q,t}(0, f))ρ_{Q,t}(0)_{x}(z).

Let D denote the variety of unramified twists of the representation π′_{x}(ρ)^{s_{x}(ρ)} of GL_{n/s_{x}(ρ)}(F_{x})^{s_{x}(ρ)} and suppose that z corresponds as in [Bern] to a regular function on D/W (GL_{n/s_{x}(ρ)}(F_{x})^{s_{x}(ρ)}, D) (in the notation of [Bern]). If Q ≠ GL_{n} × F then either ρ_{Q,t}(0)_{x}(z) = 0 for all z which correspond to regular functions on D/W (GL_{n/s_{x}(ρ)}(F_{x})^{s_{x}(ρ)}, D), or ρ_{Q,t}(0)_{x} maps to a 0-cycle on D/W (GL_{n/s_{x}(ρ)}(F_{x})^{s_{x}(ρ)}, D) supported away from π′_{x}(ρ) × ... × (π′_{x}(ρ) ⊗ | det |^{s_{x}(ρ)}). Choose z in the Bernstein centre corresponding to a regular function on the space D/W (GL_{n/s_{x}(ρ)}(F_{x})^{s_{x}(ρ)}, D) which is 1 at π′_{x}(ρ)×...×(π′_{x}(ρ)⊗ | det |^{s_{x}(ρ)}) and zero at all other terms occuring in our sum (which is finite for any given f). Then we see that

∑_{π} a_{disc}^{GL_{N} ×F} (π)tr π(f) = ∑_{π} m(π)tr π(f),

where

• m(π) denotes the multiplicity of π in the discrete part of the space of automorphic forms invariant by ℝ_{>0}^{×};

• and where both sums run over irreducible representations π such that π_{w} maps to π′_{x}(ρ) × ... × (π′_{x}(ρ) ⊗ | det |^{s_{x}(ρ)}) in D/W (GL_{n/s_{x}(ρ)}(F_{x})^{s_{x}(ρ)}, D).

We deduce that a_{disc}^{GL_{N} ×F} (π) is just the multiplicity of π in the discrete part of the space of automorphic forms invariant by ℝ_{>0}^{×}.

Using the strong multiplicity one theorem for GL_{n}(𝔸_{F}), the theorem follows. □

Combining this with the facts that any automorphic representation of GL_{n}(𝔸_{F}) which is supercuspidal at one place is cuspidal and that any cuspidal automorphic representation of GL_{n}(𝔸_{F}) is generic we obtain the following corollary.

Corollary V.2 Suppose that ρ is an irreducible automorphic representation of (B^{op}⊗𝔸)^{×} such that for one place x ∉ S(B) the component ρ_{x} is supercuspidal. Then for all places x ∉ S(B) the component ρ_{x} is generic.

VI Appendix: Clozel’s base change.

In this section we will give a description of automorphic representations of G(𝔸) in terms of automorphic representations on (B^{op} ⊗_{ℚ} 𝔸)^{×}. This description is basically due to Clozel (see [Cl1] and [Cl2]), but there were a number of gaps in his argument which were repaired with the help of Labesse (see [Lab] and [CL]).

As we have fixed ı : ℚ_{l}^{ac} →^{∼} ℂ we can think of ξ as an irreducible algebraic representation of G over ℂ. Note that for any ℂ-algebra R we have

RS_{ℚ}^{E}(G × E) × ℂ ≅ (G × ℂ) ×_{ℂ} (G × ℂ),

where the first factor corresponds to τ_{0} : E ↪ ℂ and the second factor to τ_{0} ∘ c. We will let ξ_{E} denote the representation ξ ⊗ ξ of RS_{ℚ}^{E}(G × E) over ℂ. It restricts to the representation ξ ⊗ (ξ ∘ c) of G(E_{∞}) = RS_{ℚ}^{E}(G × E)(ℝ). We will also use ξ_{E} for the restriction of this representation to GL_{n}(F_{∞}) ⊂ E_{∞}^{×} × GL_{n}(F_{∞}) ≅ G(E_{∞}) (see section 1).

We will call an irreducible admissible representation π_{∞} of G(ℝ) (resp. Π_{∞} of GL_{n}(F_{∞})) cohomological for ξ (resp. cohomological for ξ_{E}) if for some i,

H^{i}((Lie G(ℝ)) ⊗_{ℝ} ℂ, U_{∞}, π_{∞} ⊗ ξ) ≠ (0),

(resp.

H^{i}(M_{n}(F_{∞}) ⊗_{ℝ} ℂ, U(n, 0)^{[F^{+}:ℚ]}, Π_{∞} ⊗ ξ_{E}) ≠ (0).)

Suppose that x is a place of ℚ which splits as x = yy^{c} in E. Recall (from section 1) that the choice of a place y|x allows us to consider ℚ_{x} →^{∼} E_{y} as an E-algebra and hence to identify

G(ℚ_{x}) ≅ (B_{y}^{op})^{×} × Q_{x}^{×}.

If π is an irreducible admissible representation of G(ℚ_{x}) we can then decompose

π ≅ π_{y} ⊗ ψ_{π,y^{c}}.

If we vary our choice of y we find that π_{y^{c}} = π_{y}^{#} and that ψ_{π,y} = ψ_{π_{y}}ψ_{π,y^{c}}. (Here we set π_{y}^{#}(g) = π_{y}(g^{−#}).) We define BC (π) to be the representation

π_{y} ⊗ π_{y^{c}} ⊗ (ψ_{π,y^{c}} ∘ c) ⊗ (ψ_{π,y} ∘ c)

of

G(E_{x}) ≅ (B_{x}^{op})^{×} × E_{x}^{×} ≅ (B_{y}^{op})^{×} × (B_{y^{c}}^{op})^{×} × E_{y}^{×} × E_{y^{c}}^{×}.

Now suppose that x is a finite place of ℚ which is inert in E but such that • x is unramified in F;

• (B_{x}^{op}, #) ≅ (M_{n}(F_{x}), †), where g^{†} = w(g^{c})^{t}w^{−1} with w the antidiagonal matrix with ones on the antidiagonal.

These latter two conditions exclude only finitely many places of ℚ. If x is such a place then G(ℚ_{x}) is quasi-split and split over an unramified extension. We will fix a maximal torus T_{x} in a Borel subgroup B_{x} in G × F_{x} so that B_{x}(F_{x}) consists of elements of G(F_{x}) which correspond to upper triangular elements of M_{n}(F_{x}) and T_{x}(F_{x}) consists of elements of G which correspond to diagonal elements of M_{n}(F_{x}). Thus T_{x}(ℚ_{x}) can be identified with the set of elements (d_{0}; d_{1}, ..., d_{n}) ∈ ℚ_{x}^{×} × (F_{x}^{×})^{n} such that d_{0} = d_{i}d_{n+1−i}^{c} for i = 1, ..., n. If ψ is a character of T (ℚ_{x}) we define a character BC (ψ) of E_{x}^{×} × (F_{x}^{×})^{n} by

BC (ψ)(d_{0}; d_{1}, ..., d_{n}) = ψ(d_{0}d_{0}^{c}; d_{0}d_{1}/d_{n}^{c}, ..., d_{0}d_{n}/d_{1}^{c}).

Let B denote the Borel subgroup of upper triangular elements of GL_{n}. If π is an unramified representation of G(ℚ_{x}) which is a subquotient of the induced representation n-Ind _{B_{x}(ℚ_{x})}^{G(ℚ_{x})}(ψ) then we will denote by BC (π) the unique unramified representation of E_{x}^{×} × GL_{n}(F_{x}) which is a subquotient of the normalised induction from E_{x}^{×} × B(F_{x}) of BC (ψ).

If Π is an irreducible automorphic representation of (B^{op}⊗_{ℚ}𝔸)^{×} then define Π^{#} by

Π^{#}(g) = Π(g^{−#}).

Using the strong multiplicity one theorem we see that

JL (Π^{#}) = JL (Π)^{∨} ∘ c.

Theorem VI.1 Suppose that π is an irreducible automorphic representation of G(𝔸) such that π_{∞} is cohomological for ξ. Then there is a unique irreducible automorphic representation BC (π) = (ψ, Π) of 𝔸_{E}^{×} × (B^{op} ⊗_{ℚ} 𝔸)^{×} such that

1. ψ = ψ_{π}|_{𝔸_{E}^{×}}^{c};

2. if x is a place of ℚ which splits in E then BC (π)_{x} = BC (π_{x});

3. for all but finitely many places x of ℚ which are inert in E we have BC (π)_{x} = BC (π_{x});

4. Π_{∞} is cohomological for ξ_{E};

5. ψ_{∞}^{c} = ξ|_{E_{∞}^{×}}^{−1} (where E_{∞}^{×} ⊂ G(ℝ)); 6. ψ_{Π}|_{𝔸_{E}^{×}} = ψ^{c}/ψ;

7. Π^{#} ≅ Π.

Proof: We will deduce this from theorem A.5.2 of [CL].

Let T/ℚ denote the torus RS_{ℚ}^{E}(𝔾_{m}) and let T^{1} ⊂ T denote the kernel of the norm homomorphism T → 𝔾_{m}. We have a natural morphism T × G_{1} → G which is surjective on geometric points. It has kernel T^{1} embedded by t ↦ (t, t^{−1}). If π is an admissible representation of G(𝔸) we obtain an admissible representation π|_{(T ×G_{1})(𝔸)} of (T × G_{1})(𝔸) by composing π with the homomorphism T × G_{1} → G. (Use the fact that (T × G_{1})(𝔸) → G(𝔸) is continuous and open.) If π is automorphic then π|_{(T ×G_{1})(𝔸)} is a direct sum of irreducible subrepresentations. As far as we can see not all these direct summands are automorphic for T × G_{1}. Rather suppose that g_{i} ∈ G(𝔸) form a set of representatives for ν(G(𝔸))/ν(G(ℚ))N(𝔸_{E}^{×}). Then we get a bijection of spaces of automorphic forms

\begin{matrix}A(G(ℚ)\G(𝔸)) & →^{∼} & ⨁_{i} A((T × G_{1})(ℚ)\(T × G_{1})(𝔸))^{g_{i}T^{1}(𝔸)g_{i}^{−1}} \\ f & ⟼ & (g_{i}(f)|_{(T ×G_{1})(𝔸)})_{i}. \end{matrix}

If π′ is an irreducible subquotient of π|_{(T ×G_{1})(𝔸)} then π′ may not be automorphic but π′ composed with conjugation by one of the g_{i} will be. Note that (because G(ℚ) is dense in G(ℝ)) one may assume that g_{i,∞} = 1 for all i.

If x is a place of ℚ which splits in E then we get an exact sequence

(0) ⟶ T^{1}(ℚ_{x}) ⟶ (T × G_{1})(ℚ_{x}) ⟶ G(ℚ_{x}) ⟶ (0).

If π_{x} is an irreducible admissible representation of G(ℚ_{x}) then π_{x}|_{(T ×G_{1})(ℚ_{x})} is also irreducible.

If x is a finite place of ℚ which is inert in E, but such that

• x is unramified in F,

• (B_{x}^{op}, #) ≅ (M_{n}(F_{x}), †), where g^{†} = w(g^{c})^{t}w^{−1} with w the antidiagonal matrix with ones on the antidiagonal;

then we get an exact sequence

(0) ⟶ T^{1}(ℚ_{x}) ⟶ (T × G_{1})(ℚ_{x}) ⟶ G(ℚ_{x}) ⟶ (0),

if n is odd, while if n is even we get an exact sequence

(0) ⟶ T^{1}(ℚ_{x}) ⟶ (T × G_{1})(ℚ_{x}) ⟶ G(ℚ_{x}) ⟶^{x∘ν} (ℤ/2ℤ) ⟶ (0).

In either case if π_{x} is an unramified irreducible representation of G(ℚ_{x}) then π_{x}|_{(T ×G_{1})(ℚ_{x})} contains a unique unramified subquotient which we will denote π_{x}|_{(T ×G_{1})(ℚ_{x})}^{0}.

If [F^{+} : ℚ] > 1 or n > 2 then we have an exact sequence

(0) ⟶ T^{1}(ℝ) ⟶ (T × G_{1})(ℝ) ⟶ G(ℝ) ⟶ (0),

while if F^{+} = ℚ and n = 2 then we get an exact sequence

(0) ⟶ T^{1}(ℝ) ⟶ (T × G_{1})(ℝ) ⟶ G(ℝ) ⟶ {±1} ⟶ (0).

If π_{∞} is a irreducible admissible representation of G(ℝ) then it is cohomological for ξ if and only if

1. π_{∞}|_{E_{∞}^{×}} = ξ|_{E_{∞}^{×}}^{−1},

2. and there is an irreducible constituent π′_{∞} of π_{∞}|_{G_{1}(ℝ)} and an i ∈ ℤ_{≥0}

such that

H^{i}(Lie G_{1}(ℝ) ⊗_{ℝ} ℂ, U_{∞}, π′_{∞} ⊗ ξ|_{G_{1}}) ≠ (0).

Thus if π is an irreducible automorphic representation of G(𝔸) such that π_{∞} is cohomological for ξ we can find an irreducible automorphic representation π′ for (T × G_{1})(𝔸) such that

• π′|_{T^{1}(𝔸)} = 1,

• if x is a place of ℚ which splits in E then π′_{x} = π_{x}|_{(T ×G_{1})(ℚ_{x})};

• for all but finitely many places x of ℚ which are inert in E we have π′_{x} = π_{x}|_{(T ×G_{1})(ℚ_{x})}^{0};

• π′_{∞}|_{E_{∞}^{×}} = ξ|_{E_{∞}^{×}}^{−1};

• and for some i we have

H^{i}(Lie G_{1}(ℝ) ⊗_{ℝ} ℂ, U_{∞}, (π′_{∞} ⊗ ξ)|_{G_{1}(ℝ)}) ≠ (0).

Thus π′ is of the form ψ^{c} ⊗ π′_{1}, where ψ is a character of E^{×}\𝔸_{E}^{×} and π′_{1} is an automorphic representation of G_{1}(𝔸). Moreover ψ|_{T^{1}(𝔸)} = ψ_{π′_{1}}|_{T^{1}(𝔸)}^{−1}.

Now we apply theorem A.5.2 of [CL] to π′_{1} and we conclude that there exists an automorphic representation Π of (B^{op} ⊗ 𝔸)^{×} such that

• if x is a place of ℚ which splits in E then Π_{x} = BC (π_{x})|_{(B_{x}^{op})^{×}}; • for all but finitely many places x of ℚ which are inert in E we have

Π_{x} = BC (π_{x})|_{(B_{x}^{op})^{×}};

• for some i we have

H^{i}(M_{n}(F_{∞}) ⊗_{ℝ} ℂ, U(n, 0)^{[F^{+}:ℚ]}, Π_{∞} ⊗ ξ_{E}) ≠ (0).

(The first of these properties is not explicitly stated in theorem A.5.2 of [CL]. However it follows easily from a slight modification of the proof. In the notation of the proof of theorem A.5.2 in [CL] divide the set S as a disjoint union S = S_{1} ∪S_{2} where S_{1} consists of places which split in E and S_{2} of places which are inert. Now take f_{S} = (∏_{x∈S_{1}} f_{x}) × f_{S_{2}} where for x ∈ S_{1}, f_{x} is any element of C_{c}^{∞}(G_{0}(ℚ_{x})) and f_{S_{2}} is the characteristic function of a sufficiently small open compact subgroup, K_{S_{2}}. Then we can still find a function φ_{S} associated to f_{S}. We conclude as in [CL] that if we fix an unramified representation ρ of G_{0}(𝔸^{S∪{∞}})

∑_{π} ep (π_{∞}) dim π_{S_{2}}^{K_{S_{2}}}tr BC (π_{S_{1}})(φ_{S_{1}}) =
∑_{Π} tr (Π_{∞}(φ_{ep}^{G,θ})Π_{S_{2}}(φ_{S_{2}})I_{θ})tr Π_{S_{1}}(φ_{S_{1}}),

where π runs over automorphic representations of G_{0}(𝔸) with π^{S∪{∞}} ≅ ρ and Π runs over automorphic representations of G_{0}(𝔸_{E}) with Π^{S∪{∞}} ≅ BC (ρ). We deduce that if we fix an irreducible representation ρ of G_{0}(𝔸^{S_{2}∪{∞}}) which is unramified outside S_{1} then

∑_{π} ep (π_{∞}) dim π_{S_{2}}^{K_{S_{2}}} = ∑_{Π} tr (Π_{∞}(φ_{∞})Π_{S_{2}}(φ_{S_{2}})I_{θ}),

where π runs over automorphic representations of G_{0}(𝔸) with π^{S_{2}∪{∞}} ≅ ρ and Π runs over automorphic representations of G_{0}(𝔸_{E}) with Π^{S_{2}∪{∞}} ≅ BC (ρ). The rest of the argument is as in [CL].)

It is now easy to check that (ψ_{π}|_{𝔸_{E}^{×}}, Π) satisfies the first six properties of the theorem. Uniqueness follows from theorem V.1. The final property also follows from V.1, because for all but finitely many places x of ℚ we have Π_{x}^{#} ≅ Π_{x}. □

Corollary VI.2 If π and π′ are irreducible automorphic representations of G(𝔸) such that π_{∞} and π′_{∞} are cohomological for ξ and such that π_{x} ≅ π′_{x} for all but finitely many places x of ℚ, then π_{x} ≅ π′_{x} for all places x of ℚ which split in E.

Proof: This follows from theorems V.1 and VI.1. □

Corollary VI.3 Suppose that π is an irreducible automorphic representation of G(𝔸) such that π_{∞} is cohomological for ξ. Suppose also that there exists a place x of F such that

• x|_{ℚ} splits in E,

• B_{x}^{op} is split,

• and π_{x} is supercuspidal. Then for any place y of F such that

• y|_{ℚ} splits in E

• and B_{y}^{op} is split, π_{y} is generic. Proof: This follows from corollary V.2 and theorem VI.1. □

In the other direction we have the following result. Theorem VI.4 Suppose that Π is an irreducible automorphic representation of (B^{op} ⊗ 𝔸)^{×} and that ψ is a character of 𝔸_{E}^{×}/E^{×} such that

1. Π ≅ Π^{#},

2. ψ_{Π}|_{𝔸_{E}^{×}} = ψ^{c}/ψ,

3. Π_{∞} is cohomological for ξ_{E},

4. and ξ|_{E_{∞}^{×}}^{−1} = ψ|_{E_{∞}^{×}}^{c} (where E_{∞}^{×} ⊂ G(ℝ)). Then there is an irreducible automorphic representation π of G(𝔸) such that

1. BC (π) = (ψ, Π),

2. and π_{∞} is cohomological for ξ.

Proof: This theorem will follow from proposition 2.3 of [Cl2]. (We caution the reader that the proof of proposition 2.3 in [Cl2] seems to us rather sketchy (see in particular the first paragraph of section 2.5 of [Cl2]). However this is remedied in theorem A.3.1 of [CL] where a complete derivation of the key trace identity is given.)

We will use the notation of the proof of theorem VI.1. Let π_{1} be the automorphic representation of G_{1}(𝔸) whose existence is guaranteed by proposition 2.3 of [Cl2]. Then ψ^{c} × π_{1} is an irreducible automorphic representation of (T × G_{1})(𝔸) which vanishes on T^{1}(𝔸) (because if b ∈ 𝔸_{E}^{×} then ψ^{c}(b/b^{c}) = ψ_{Π}(b) = ψ_{π_{1}}(b/b^{c}), the latter equality following from Clozel’s definition of “base change lift”). Thus ψ^{c} × π_{1} is a subrepresentation of the restriction of some automorphic representation π of G(𝔸) to (T ×G_{1})(𝔸). Again from Clozel’s definition of “base change lift” we see that BC (π) = (ψ_{π}|_{𝔸_{E}^{×}}, Π). Finally because π_{1} is cohomological for ξ|_{G_{1}(ℝ)} and because ψ^{c}|_{E_{∞}^{×}}^{−1} = ξ|_{E_{∞}^{×}}^{−1} we see that ψ_{∞}^{c} × π_{1,∞} and hence π_{∞} is cohomological for ξ. □

Corollary VI.5 Keep the notation and assumptions of the theorem. Then

dim[R_{ξ}(π^{∞})] ≠ 0.

Proof: Combine the theorem with a theorem of Kottwitz (see theorem A.4.2 of [CL]). □

It may sometimes be useful to combine theorem VI.4 with the following lemma.

Lemma VI.6 Suppose that Π is an automorphic representation of (B^{op}⊗𝔸)^{×} such that

1. Π ≅ Π^{#},

2. and Π_{∞} has the same infinitesimal character as some algebraic representation of RS_{ℚ}^{F}(GL_{n}) over ℂ.

Then we can find a character ψ of 𝔸_{E}^{×}/E^{×} and an algebraic representation ξ of G over ℂ such that

1. ψ_{Π}|_{𝔸_{E}^{×}} = ψ^{c}/ψ,

2. Π_{∞} is cohomological for ξ_{E},

3. ξ|_{E_{∞}^{×}}^{−1} = ψ_{∞}^{c} (where E_{∞}^{×} ⊂ G(ℝ)),

4. and ψ|_{O_{E,u}^{×}} = 1.

Proof: Because Π_{∞} has the same infinitesimal character as an algebraic representation of RS_{ℚ}^{F}(GL_{n}) over ℂ we see that

ψ_{Π_{∞}} : F_{∞}^{×} ⟶ ℂ^{×}

is of the form

(x_{τ}) ⟼ ∏_{τ} τ(x_{τ})^{a_{τ}} τ(x_{τ}^{c})^{b_{τ}} ,

where a_{τ}, b_{τ} ∈ ℤ and where τ runs over Hom_{ℚ}(F^{+}, ℝ) ≅ Hom_{E,τ_{0}|_{E}}(F, ℂ). As Π_{∞}^{∨} ≅ Π_{∞}^{c} we see that for all τ we have a_{τ} + b_{τ} = 0. Thus if x ∈ E_{∞}^{×} we have

ψ_{Π_{∞}}(x) = (τ_{0}(x)/τ_{0}(x^{c}))^{a},

where a = ∑ a_{τ} ∈ ℤ. Define

ψ_{∞} : E_{∞}^{×} ⟶ ℂ^{×}

by

ψ_{∞}(x) = τ_{0}(x)^{−a}.

then ψ_{∞} × Π_{∞} defines a representation of G(E_{∞}) ≅ E_{∞}^{×} × GL_{n}(F_{∞}) such that

(ψ_{∞} × Π_{∞}) ∘ c = ψ_{∞}^{c}ψ_{Π_{∞}}^{c} × Π_{∞}^{c∨}) ≅ ψ_{∞} × Π_{∞}.

(Note that the first c refers to the G-structure, the second and third to the GL_{1}-structure and the fourth to the GL_{n}-structure.)

We may choose an algebraic representation Ξ of RS_{ℚ}^{E}G over ℂ such that ψ_{∞} × Π_{∞} and Ξ^{∨} have the same infinitesimal character. As (ψ_{∞} × Π_{∞}) ∘ c ≅ ψ_{∞} × Π_{∞} we see that

Ξ ∘ c ≅ Ξ.

(Here c acts on E and so gives a ℂ-morphism from RS_{ℚ}^{E}(G) × ℂ to itself.) Under the isomorphism

(RS_{ℚ}^{E}G) × ℂ ≅ (G × ℂ) ×_{ℂ} (G × ℂ)

we see that Ξ corresponds to ξ⊗ξ for some algebraic representation ξ of G over ℂ, i.e. Ξ = ξ_{E}. Using τ_{0} to view ℂ as an E-algebra we get an identification

G(ℂ) ≅ ℂ^{×} × GL_{n}(F_{∞})

and hence an embedding

i : (E_{∞}^{×})^{2} ↪ G(ℂ)

which sends (x_{1}, x_{2}) to (τ_{0}(x_{1}), x_{2}). Embedding G(ℂ) as the first factor of RS_{ℚ}^{E}(G)(ℂ) ≅ G(ℂ) × G(ℂ) we can extend i to a homomorphism

i : (E_{∞}^{×})^{2} ↪ RS_{ℚ}^{E}(G)(ℂ).

If instead we identify

RS_{ℚ}^{E}(G)(ℂ) ≅ (E ⊗_{ℚ} ℂ)^{×} × GL_{n}(F ⊗_{ℚ} ℂ) ≅ (ℂ ⊕ ℂ)^{×} × GL_{n}(F_{∞} ⊕ F_{∞})

(using the identifications

((τ_{0} ⊗ 1) ⊕ ((τ_{0} ∘ c) ⊗ 1)) : E ⊗_{ℚ} ℂ →^{∼} ℂ^{2}

and

((1 ⊗ τ_{0}^{−1}) ⊕ (1 ⊗ (c ∘ τ_{0}^{−1}))) : F ⊗_{ℚ} ℂ →^{∼} F_{∞}^{2})

then

i(x_{1}, x_{2}) = (τ_{0}(x_{1}), 1; x_{2}, 1).

Thus (ξ ∘ i)(x_{1}, x_{2}) = τ_{0}(x_{1}/x_{2})^{a}. If x ∈ E_{∞}^{×} ⊂ G(ℝ) ⊂ G(ℂ) then we can identify x ∈ G(ℂ) with i(xx^{c}, x). Thus ξ(x) = τ_{0}(x^{c})^{a}, i.e.

ξ|_{E_{∞}^{×}}^{−1} = ψ_{∞}^{c}.

Recall the exact sequence.

(0) ⟶ ℚ^{×}\𝔸^{×} ⟶ E^{×}\𝔸_{E}^{×} ⟶^{1−c} E^{×}\𝔸_{E}^{×} ⟶^{1+c} ℚ^{×}\𝔸^{×} ⟶ Gal (E/ℚ) ⟶ (0).

Note that ψ_{Π}^{−1} = ψ_{Π}^{c} and so ψ_{Π} is trivial on (𝔸_{E}^{×})^{1+c}. From the formulae at the start of this proof we see that ψ_{Π_{∞}}(−1) = 1. But from the exact sequence we see that 𝔸^{×} = ℚ^{×}(𝔸_{E}^{×})^{1+c}ℝ^{×}, and so we deduce that

ψ_{Π}|_{𝔸^{×}} = 1.

Again from the above exact sequence we see that

ψ_{Π} = ψ′ ∘ (c − 1)

for some character ψ′ of (𝔸_{E}^{×})^{c−1}/{±1}. Again from the explicit formulae at the start of this proof we see that

ψ′|_{(E_{∞}^{×})^{c−1}} = ψ_{∞}|_{(E_{∞}^{×})^{c−1}}.

Thus we can choose a character ψ of 𝔸_{E}^{×}/E^{×} such that • ψ|_{(𝔸_{E}^{×})^{c−1}} = ψ′,

• ψ|_{E_{∞}^{×}} = ψ_{∞},

• and ψ|_{O_{E,u}^{×}} = 1.

Then ψ_{Π}|_{𝔸_{E}^{×}} = ψ ∘ (c − 1).

Finally because ξ_{E}^{∨} and Π_{∞} have the same infinitesimal character we see by theorems 6.1 and 7.1 of [En] that Π_{∞} is cohomological for ξ_{E}. □

VII Appendix: The trace formula.

In this appendix we will give an expression for the trace of an element φ ∈ C_{c}^{∞}(G(𝔸^{∞})) on H(X, L_{ξ}). For simplicity we will derive this as a simple specialisation of the main result of [Arth], which relies on arthur’s trace formula. However in our case this expression can be derived in a more elementary fashion by using the Lefschetz trace formula (cf [KS] or [Bew]). We will also give some applications of this expression.

More precisely in this appendix we will prove the following result.

Proposition VII.1 Suppose that φ ∈ C_{c}^{∞}(G(𝔸^{∞}). Then

\begin{matrix}tr (φ|H(X, L_{ξ})) = (−1)^{n}nκ_{B} ∑_{γ}(−1)^{n/[F (γ):F]} \\ [F (γ) : F]^{−1}vol (Z_{G}(γ)(ℝ)_{0}^{1})^{−1}O_{γ}^{G(𝔸^{∞})}(φ)tr ξ(γ), \end{matrix}

unless F^{+} = ℚ and n = 2 in which case we drop the factors [F (γ) : F]^{−1}. Here

• γ runs over a set of representatives of G(𝔸)-conjugacy classes in G(ℚ) which are elliptic in G(ℝ);

• κ_{B} = 2 if [B : ℚ]/2 is odd and κ_{B} = 1 otherwise;

• Z_{G}(γ)(ℝ)_{0} denotes the compact mod centre inner form of Z_{G}(γ)(ℝ) and Z_{G}(γ)(ℝ)_{0}^{1} the kernel of

|ν| : Z_{G}(γ)_{0} ⟶ ℝ_{>0}^{×};

• and we use measures on Z_{G}(γ)(ℝ)_{0}^{1} and Z_{G}(γ)(𝔸^{∞}) compatible with

– Tamagawa measure on Z_{G}(γ)(𝔸), – the decomposition

Z_{G}(γ)(𝔸) = Z_{G}(γ)(𝔸^{∞}) × Z_{G}(γ)(ℝ),

– association of measures on Z_{G}(γ)(ℝ)_{0} and Z_{G}(γ)(ℝ),

– and the measure dt/t on ℝ_{>0}^{×}. Proof: By the discussion prior to lemma 4.1, by theorem 6.1 of [Arth] and

by remark 3 following that theorem we see that tr (φ|H(X, L_{ξ})) = (−1)^{n}δ_{B}# ker^{1}(ℚ, G) ∑_{γ}(−1)^{n/[F (γ):F]}(n/[F (γ) : F]) vol (Z_{G}(γ)(ℚ)ℝ_{>0}^{×}\Z_{G}(γ)(𝔸))vol (ℝ_{>0}^{×}\Z_{G}(γ)(ℝ)_{0})^{−1}O_{γ}^{G(𝔸^{∞})}(φ)tr ξ(γ),

where

• the sum runs over a set of representatives of G(ℚ)-conjugacy classes in G(ℚ);

• δ_{B} = 2 if F^{+} = ℚ and n = 2 and δ_{B} = 1 otherwise;

• and we use associated measures on Z_{G}(γ)(ℝ) and Z_{G}(γ)(ℝ)_{0}. (Note that in the notation of [Arth] we have

• δ_{B} = [K_{ℝ} : K_{ℝ}^{0}],

• q(M_{γ}) = n/[F (γ) : F] − 1,

• |D(M_{γ}, B)| = n/[F (γ) : F], unless n = 2 and F^{+} = ℚ in which case |D(M_{γ}, B)| = 1,

• |i^{G}(γ)| = 1,

• Φ_{G}(γ, ξ) = tr ξ(γ),

• and H_{G}(γ) = O_{γ}^{G(𝔸^{∞})}(φ).) Note that

vol (Z_{G}(γ)(ℚ)ℝ_{>0}^{×}\Z_{G}(γ)(𝔸))vol (ℝ_{>0}^{×}\Z_{G}(γ)(ℝ)_{0})^{−1} =
vol (Z_{G}(γ)(ℚ)\Z_{G}(γ)(𝔸)^{1})vol (Z_{G}(γ)(ℝ)_{0}^{1})^{−1},

where Z_{G}(γ)(𝔸)^{1} denotes the elements g ∈ Z_{G}(γ)(𝔸) with |ν(g)| = 1. By the main theorem of [Ko5] we see that if we use Tamagawa measure then

vol (Z_{G}(γ)(ℚ)\Z_{G}(γ)(𝔸)^{1}) = κ_{B}/# ker^{1}(ℚ, Z_{G}(γ)).

(Using the fact that [B : ℚ]/2 and [Z_{B}(γ) : ℚ]/2 have the same parity, a direct calculation shows that #A(Z_{G}(γ)) = κ_{B}.) Moreover the G(𝔸)-conjugacy class of γ in G(ℚ) contains

# ker(ker^{1}(ℚ, Z_{G}(γ)) ⟶ ker^{1}(ℚ, G))

G(ℚ)-conjugacy classes. As in the discussion (in the third paragraph) following lemma 9.2 we see that

# ker(ker^{1}(ℚ, Z_{G}(γ)) ⟶ ker^{1}(ℚ, G)) = # ker^{1}(ℚ, Z_{G}(γ))/# ker^{1}(ℚ, G).

Thus we may rewrite Arthur’s formula to give the formula in the proposition. □

We will now draw a few corollaries from this lemma which are standard facts.

Corollary VII.2 Suppose that S is a finite set of finite places of F such that

• if x ∈ S then B_{x} is split and x|_{ℚ} splits in E,

• and if x and y are two elements of S with the same restriction to F^{+} then x = y.

For x ∈ S let π_{x}^{0} be a square integrable representation of GL_{n}(F_{x}). Also let ξ be an irreducible representation of G over ℚ_{l}^{ac}. Then we can find an irreducible admissible representation π of G(𝔸^{∞}) such that

• dim[R_{ξ}(π)] ≠ 0,

• for x ∈ S we have π_{x} ≅ π_{x}^{0}⊗(ψ_{x}∘det) for some character ψ_{x} of F_{x}^{×}/O_{F,x}^{×},

• and for x ∈ S we have ψ_{π}|_{O_{E,x|^{c}_{E}}^{×}} = 1.

Proof: We may suppose that for some x ∈ S the representation π_{x}^{0} is supercuspidal. Let S(ℚ) denote the set of places of ℚ below places in S. If y ∈ S(ℚ) then it gives rise to a distinguished place ỹ of E above y, i.e. x|_{E} for any place x ∈ S above y. Decompose

G(𝔸) ≅ G(𝔸^{S(ℚ)∪{∞}}) × ∏_{y∈S(ℚ)}(ℤ_{y}^{×} × ∏_{x|ỹ}(B_{x}^{op})^{×}).

If U ⊂ G(𝔸^{S(ℚ)∪{∞}}) is an open compact subgroup, then set

φ_{U} = char_{U} × ∏_{y∈S(ℚ)}char_{ℤ_{y}^{×}} × ∏_{x|ỹ} φ_{x},

where • if x ∉ S then φ_{x} is the characteristic function of some open compact

subgroup,

• while if x ∈ S then then φ_{x} is the product of a pseudo-coefficient φ_{π_{x}^{0}} for π_{x}^{0} with the characteristic function of det^{−1}(O_{F,x}^{×}).

We may and will choose U sufficiently small so that the only element of finite order in the intersection of G(ℚ) with the support of φ_{U} is 1.

Then proposition VII.1 and lemma IV.1 tell us that

vol (G(ℝ)_{0}^{1})tr (φ_{U}|H(X, L_{ξ})) = nκ_{B}(dim ξ) ∏_{x∈S}(dim JL^{−1}(π_{x}^{0})),

where we use Tamagawa measure on G(𝔸) = G(𝔸^{∞})×G(ℝ) and a measure on G(ℝ)_{0}^{1} compatible with this, the association of measures on G(ℝ) and G(ℝ)_{0}, the exact sequence

(0) ⟶ G(ℝ)_{0}^{1} ⟶ G(ℝ)_{0} ⟶ ℝ_{>0}^{×} ⟶^{|ν|} (0),

and the measure dt/t on ℝ_{>0}^{×}. In particular we see that

∑_{π}(tr π(φ_{U}))(dim[R_{ξ}(π)]) ≠ 0,

where π runs over irreducible admissible representations of G(𝔸^{∞}).

Thus we may choose an irreducible admissible representation π of G(𝔸^{∞}) such that both

• dim[R_{ξ}(π)] ≠ 0

• and tr π(φ_{U}) ≠ 0.

The first of these conditions implies that for some π_{∞} the representation π × π_{∞} is automorphic (apply Matsushima’s formula, lemma 4.1). The second condition implies that

• for y ∈ S(ℚ) we have ψ_{π}|_{O_{E,y^{c}}^{×}} = 1,

• and, for x ∈ S, we have

tr π_{x}(φ_{π_{x}^{0}⊗(ψ_{x}∘det)}) ≠ 0

for some character ψ_{x} of F_{x}^{×}/O_{F,x}^{×} (see the argument for lemma 10.1).

Thus for any x in S with π_{x}^{0} supercuspidal we see that π_{x} ≅ π_{x}^{0} ⊗ (ψ_{x} ∘ det) (use lemma IV.4). As we are assuming that some such place x exists it follows from corollary VI.3 that for all x ∈ S the representation π_{x} is generic and hence it follows from corollary IV.6 that π_{x} ≅ π_{x}^{0} ⊗ (ψ_{x} ∘ det) for all x ∈ S. Thus π is our desired representation of G(𝔸^{∞}). □

Corollary VII.3 Suppose that L is a CM field which is the composite of a totally real field L^{+} and an imaginary quadratic field M. Suppose that S is a finite set of places of L such that

• if x ∈ S then x|_{ℚ} splits in E,

• and if x and y are elements of S with the same restriction to L^{+} then x = y.

For x ∈ S suppose that Π_{x}^{0} is a square integrable representation of GL_{g}(L_{x}). Suppose also that Ξ is an algebraic representation of RS_{ℚ}^{L}(GL_{g}) over ℂ such that Ξ^{c} ≅ Ξ^{∨} (where c acts on RS_{ℚ}^{L}(GL_{g}) via its action on L). Then we can find a cuspidal automorphic representation Π of GL_{g}(𝔸_{L}) such that

1. Π^{c} ≅ Π^{∨},

2. Π_{∞} has the same infinitesimal character as Ξ^{∨}|_{GL_{g}(L_{∞})},

3. and for all x ∈ S there is a character ψ_{x} of L_{x}^{×}/O_{L,x}^{×} such that

Π_{x} ≅ Π_{x}^{0} ⊗ (ψ_{x} ∘ det).

Proof: We may assume that for some x ∈ S the representation Π_{x}^{0} is supercuspidal. Choose (E, F, B, ∗, τ_{0}, β) as in section 1 with E = M, F = L, [B : F] = g^{2} and B_{x} split for all x ∈ S. We can choose an algebraic representation ξ of G over ℂ such that ξ_{E}|_{GL_{n}(L_{∞})} = Ξ|_{GL_{n}(L_{∞})}. The corollary now follows from corollary VII.2 and theorem VI.1. □

Lemma VII.4 Suppose that s|n is a positive integer. Suppose also that π_{w}^{0} is an irreducible supercuspidal representation of GL_{n/s}(F_{w}). Then we can find an irreducible admissible representation π^{∞} of G(𝔸^{∞}) and a character ψ_{w}^{0} of F_{w}^{×}/O_{F,w}^{×} such that

1. dim[R_{ξ}(π^{∞})] ≠ 0,

2. π_{w} ≅ (π_{w}^{0} ⊞ ... ⊞ (π_{w}^{0} ⊗ | det |^{s−1})) ⊗ (ψ_{w}^{0} ∘ det), 3. π_{p,0}|_{ℤ_{p}^{×}} = 1.

Proof: Let S(B) denote the set of places of F at which B is ramified. Note that if x ∈ S(B) then x|_{ℚ} splits in E.

As

RS_{ℚ}^{F}(GL_{n}) × ℂ ≅ (GL_{n} × ℂ)^{Hom (F,ℂ)},

we may choose a maximal torus T ≅ (GL_{1}^{n} × ℂ)^{Hom (F,ℂ)} and a Borel subgroup B ⊃ T consisting of upper triangular matrices in (GL_{n} × ℂ)^{Hom (F,ℂ)}. Then we can identify

X^{∗}(T) ≅ (ℤ^{n})^{Hom (F,ℂ)}

in such a way that the set of B-positive weights, X^{∗}(T)_{+}, consists of vectors (x_{τ,i}) with x_{τ,i} ≥ x_{τ,j} whenever i ≤ j. We will let ρ ∈ X^{∗}(T)_{+} denote half the sum of the positive roots, i.e.

ρ_{τ,i} = (n + 1)/2 − i.

If Ξ is an irreducible algebraic representation of RS_{ℚ}^{F}(GL_{n}) over ℂ we will let x(Ξ) ∈ X^{∗}(T)_{+} denote its heighest weight. Note that Ξ^{c} ≅ Ξ^{∨} (where c acts on RS_{ℚ}^{F}(GL_{n}) via its action on F) if and only if

x(Ξ)_{τ,i} + x(Ξ)_{cτ,n+1−i} = 0

for all τ ∈ Hom (F, ℂ) and all i = 1, ..., n. We will use exactly similar notation for RS_{ℚ}^{F}(GL_{n/s}).

Choose an irreducible algebraic representation Ξ′ of RS_{ℚ}^{F}(GL_{n/s}) over ℂ such that

• (Ξ′)^{∨} ≅ (Ξ′)^{c}

• and x(Ξ′)_{τ,i+1} ≥ x(Ξ′)_{τ,i} + (s − 1) for all τ ∈ Hom (F, ℂ) and all i = 1, ..., n − 1.

By corollary VII.3 we may choose a cuspidal automorphic representation Π′ of GL_{n/s}(𝔸_{F}) such that

• (Π′)^{∨} ≅ (Π′)^{c},

• Π′_{∞} has the same infinitesimal character as (Ξ′)^{∨},

• Π′_{x} is supercuspidal for all x ∈ S(B),

• Π′_{w} ≅ π_{w}^{0} ⊗ (ψ_{w}^{1} ∘ det) for some character ψ_{w}^{1} of F_{w}^{×}/O_{F,w}^{×}.

Also choose a character ϕ of 𝔸_{F}^{×}/F^{×} such that

• ϕ^{−1} = ϕ^{c};

• for every τ : F ↪ ℂ defining a place y of F we have

ϕ_{y} : z ⟼ (τ(z)/|τ(z)|)^{δ_{τ}} ,

where δ_{τ} = 0 if either s or n/s is odd, while δ_{τ} = ±1 if both both s and n/s are even;

• and ϕ_{w} is unramified.

(The existence of such a character ϕ is proved exactly as in the proof of corollary 12.9.) Note that Π′_{∞} ⊗ ϕ_{∞} has infinitesimal character parametrised by x′ ∈ X^{∗}(T), where

x′_{τ,i} = (n/s + 1)/2 − i − x_{τ,n/s+1−i} + δ_{τ}/2.

According to the main theorem of [MW] there is an irreducible automorphic representation Π′ of GL_{n}(𝔸_{F}) which occurs in the discrete spectrum and which is a subquotient of

n-Ind _{Q(𝔸_{F})}^{GL_{n}(𝔸_{F})}(Π′ ⊗ ϕ ⊗ | det |^{(1−s)/2}) × ... × (Π′ ⊗ ϕ ⊗ | det |^{(s−1)/2}),

where Q ⊂ GL_{n} is a parabolic subgroup with Levi component GL_{n/s}^{s}. Moreover we have the following properties.

1. Π^{∨} = Π^{c} (by the strong multiplicity one theorem).

2. Π_{∞} has the same infinitesimal character as the algebraic representation Ξ of RS_{ℚ}^{F}(GL_{n}) over ℂ with

x(Ξ)_{τ,(I−1)s+J} = 1/2(s − 1)(2I − 1 − n/s) + x(Ξ′)_{τ,I} − δ_{τ}/2,

for I = 1, ..., n/s and J = 1, ..., s. (Note that by our assumptions on δ_{τ} and x(Ξ′) we do have x(Ξ) ∈ X^{∗}(T)_{+} and so Ξ^{∨} has infinitesimal character with parameter x ∈ X^{∗}(T) given by

x_{τ,(I−1)s+J} =

(n + 1)/2 − (I − 1)S − J − 1/2(s − 1)(2(n/s + 1 − I) − 1 − n/s)− x(Ξ′)_{τ,n/s+1−I} + δ_{τ}/2 =

(n/s + 1)/2 − I + (s + 1)/2 − J − x(Ξ′)_{τ,n/s+1−I} + δ_{τ}/2, for I = 1, ..., n/s and J = 1, ..., s. This x ∈ X^{∗}(T) also parametrises the infinitesimal character of any subquotient of

n-Ind _{Q(F_{∞})}^{GL_{n}(F_{∞})}(Π′_{∞} ⊗ ϕ_{∞} ⊗ | det |^{(1−s)/2}) × ... × (Π′_{∞} ⊗ ϕ_{∞} ⊗ | det |^{(s−1)/2}).)

3. For each x ∈ S(B), Π_{x} ≅ (π_{x}^{0} ⊞ ... ⊞ (π_{x}^{0} ⊗ | det |^{s−1})), for some supercuspidal representation π_{x}^{0} of GL_{n/s}(F_{x}) (see [MW] and proposition 2.10 of [Z]).

4. Π_{w} ≅ (π_{w}^{0} ⊞ ... ⊞ (π_{w}^{0} ⊗ | det |^{s−1})) ⊗ (ψ_{w}^{0} ∘ det), for some character ψ_{w}^{0} of F_{w}^{×}/O_{F,w}^{×} (see [MW] and proposition 2.10 of [Z]).

Applying theorem V.1, lemma VI.6, theorem VI.4 and corollary VI.5 we see that there is an irreducible automorphic representation π of G(𝔸) such that

• BC (π) = (ψ, JL^{−1}(Π)) (for some character ψ)

• and dim[R_{ξ}(π^{∞})] ≠ 0.

Then π^{∞} is our desired representation of G(𝔸^{∞}). □

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