pacific journal of mathematics Vol. 181, No. 3, 1997

PERIOD RELATIONS AND CRITICAL VALUES OF L-FUNCTIONS

Don Blasius

To the memory of Olga Taussky-Todd

Introduction.

I.1. In this paper we present a general conjecture concerning the arithmetic of critical values of the L-functions of algebraic automorphic forms. While individual critical values seem almost always transcendental, the evidence of Shimura (c.f., esp. [Sh1], [Sh2]) shows that interesting relations between values at different critical integers, and between values of L-functions related by “twisting”, do exist. Furthermore, a general recipe due to Deligne ([D]) enables one to predict many such relations. This recipe further allows one to derive reciprocity laws which certain conjectually algebraic numbers, formed essentially as ratios of critical values, ought to obey.

To give an example, let K be a quadratic imaginary extension of Q, and let χ : 𝔸_{K}^{∗} → ℂ^{∗} be a Hecke character of K with χ(k_{∞}) = k_{∞}^{−w} for k_{∞} ∈ ℂ^{∗} ↪ 𝔸_{K}^{∗}, and w ∈ ℤ. Let ψ be a Hecke character of finite order of K, and let T be the finite extension of K generated by the values of χ and ψ on the finite idèles of K. If w > 0, both L(χ, s) and L(χψ, s) are critical at s = 0. The ratio (defined if L(χ, 0) ≠ 0) is algebraic and satisfies the reciprocity law

σ (\frac{L(χψ, 0)}{L(χ, 0)}) = (ψ |_{T})(σ) (\frac{L(χψ, 0)}{L(χ, 0)})

for σ ∈ Gal(\overline{K}/T) and where we regard ψ as a Galois character. In fact, for any σ,

σ (\frac{L(χψ, 0)}{L(χ, 0)})

can be given exactly if we employ as well the conjugate L-series L(τ(X), s), L(τ(Xψ), s) for the various τ : T → ℂ, and use a non-abelian reciprocity law. Note that, over T, the ratio is a Kummer generator of the class field attached to ψ |_{T}. This type of law was first deduced, for K a CM field, from the results of [B1], themselves in part a refinement of some aspects of [Sh3]. However, the Deligne formalism shows clearly a) that such results in no way require that one restrict to Hecke characters (GL_{1}), and b) that even so, all such reciprocity laws already occur in the setting of Hecke L-series.

On the other hand, if f is a new form of weight k ≥ 2, and ψ is an even Dirichlet character, then Shimura showed

\frac{D(fψ, k)}{D(f, k)} = αG(ψ)

where α ∈ (T_{f} ·T_{ψ})^{∗}, and G(ψ) is the Gaussian sum attached to ψ. Of course, if σ ∈ Gal(\overline{Q}/T_{f}T_{ψ}), then σ(G(ψ)) = ψ(σ)G(ψ), and so we have again a ratio which satisfies a reciprocity law. This paper constructs a common framework for unifying the known results of this sort and for predicting new ones.

The paper has 3 sections. In Section M, we introduce the motivic language, review the construction of Deligne’s periods c^{±}(M), establish some elementary facts about the action of the endomorphism algebra on M_{DR}^{+}, factor c^{±}(M) into more basic periods c_{v}^{±}(M) attached to each place v of K, and establish the reciprocity law which relates a suitable monomial c_{ε_{π}}^{α}(M) in the c_{v}^{±}(M) to c^{±}(M ⊗ π) where π is an Artin motive. At the end of the section, we give some examples. In Section L, we introduce L-functions and, invoking Deligne’s general conjecture, obtain our main conjecture. This says that, for a k which is critical for both M and M ⊗ π,

(1 ⊗ 2πi)^{kd_{α}(M⊗π)} \frac{L(M ⊗ π, T; k)}{c_{ε_{π}}^{α}(M)} = c(δ_{∗}^{α}, det(π))

where c(δ_{∗}^{α}, det(π)) is a quantity characterized mod (T (M)T (π))^{∗} by a reciprocity law depending on δ_{∗}^{α} and det(π). Here δ_{∗}^{α} is a character of T (M)^{∗} and α = (−1)^{k}. Finally, we undertake the formal exercise of transcribing the main conjecture into the setting of algebraic cusp forms, replacing M by an algebraic ∏ on GL_{N}(𝔸_{K}) and supposing now π is on GL_{m}(𝔸_{K}), of Galois type. Since Deligne’s conjecture is known for Hecke L-series of CM fields, this conjecture is a theorem for K CM, and N = m = 1.

A last Section H, gives some special results which arise in the case of GL_{1}. This paper is a revised version of a 1987 MSRI preprint. Since then, Hida

[Hi] has obtained basic results for GL_{2}(𝔸_{K}) where K is any number field; we leave to the reader the task of confirming that Hida’s work is consistent with what we conjecture. Also, both Harris ([Ha]) and Yoshida [Y] have, with independent motivations, pursued the calculation of various period relations.

I would like to thank D. Ramakrishnan for encouraging me to look again at this work and publish it in this revised form.

PERIOD RELATIONS AND CRITICAL VALUES OF L-FUNCTIONS 55

M. Motives and period relations.

M.1. We briefly review our notations and basic objects. See [CV] for more details. Let M be a motive (for absolute Hodge cycles) of pure weight w defined over K and having coefficients in T, both number fields. Then

M = {M_{DR}, M_{B}, M_{f}, I_{∞}, I_{f}}

with T ⊗ K module M_{DR}, a T vector space M_{B}, and a T ⊗ Q_{f} module M_{f}, all free of the same finite rank. Here, for a number field L, L_{f} denotes the finite ideles. These modules are related by T -linear isomorphisms

I_{∞} : M_{B} ⊗_{B} C → M_{DR} ⊗_{K} C
I_{f} : M_{B} ⊗ Q_{f} → M_{f}.

For L as above, G_{L} denotes the Galois group Gal(\overline{L}/L). Then M_{f} is a G_{K} module, and the collection {M_{λ} | λ = finite place of T }, where M_{λ} ⊂ M_{f} is the subspace associated to T_{λ} ⊂ T ⊗ Q_{f}, is a compatible system of rational λ-adic representations of G_{K}.

Also, M_{B} has a Hodge decomposition

M_{B} ⊗ C = ⨁_{\begin{matrix}p,q∈Z \\ p+q=w \end{matrix}} M^{p,q}

with (M^{p,q})^{1⊗ρ} = M^{q,p}, where ρ denotes complex conjugation. For each p ∈ Z, there is a K subspace F^{p}M_{DR} ⊆ M_{DR} such that

I_{∞} (⨁_{p′≥p} M^{p′,q}) = F^{p}M_{DR} ⊗_{K} C.

The Hodge decomposition is stable for the action of T.

M.2. Throughout this paper, we make the strong assumption that for two motives M and N, defined over K, if M_{ℓ} is isomorphic to N_{ℓ}, as a G_{K} module, for a single prime ℓ, then M is isomorphic to N. This hypothesis is a motivic version of Tate’s isogeny conjecture, proved for the H^{1} of abelian varieties by Faltings. See L.3 below for a variant of the conjecture.

M.3. Special motives. For a number field L, let J_{L} and J_{L,R} denote the embeddings of L into C and R, respectively. Let J_{L,C} = ⟨1, ρ⟩\(J_{L} − J_{L,R}), i.e. J_{L,C} is the set of complex places of L. For each σ ∈ J_{K,R}, let F_{σ} : (σM)_{B} → (σM)_{B} denote the action of complex conjugation on the conjugate by σ, σM, of the motive M defined over K. Set (σM)_{B} = (σM)_{B}^{+} ⊕ (σM)_{B}^{−} where (σM)_{B}^{±} = ker(F_{σ} ∓ 1). If w ∈ 2Z, assume M^{w/2,w/2} = {0} unless K is totally real. In this case, assume that each F_{σ} acts on M^{w/2,w/2} by a scalar ε = ±1, independent of σ. We call such motives special, and henceforth any motive denoted by M will be special, defined over K, and with coefficients in T.

Define spaces F^{±}M_{DR} ⊆ M_{DR} by

F^{±}M_{DR} = F^{w/2}M_{DR}  (w ∈ 2Z, ε = ±1)
F^{±}M_{DR} = F^{w/2+1}M_{DR}  (w ∈ 2Z, ε = ∓1)
F^{+}M_{DR} = F^{−}M_{DR} = F^{\frac{w+1}{2}} M_{DR}  (w odd).

Put M_{DR}^{±} = M_{DR}/F^{∓}M_{DR}. Let d = dim_{T} M_{B} and define d^{±} : J_{T} → Z by d^{±}(σ) = dimension of σ-eigenspace for action of T in M_{DR}^{±} ⊗_{K} C. Note that if τ ∈ J_{K}, (τM_{DR})^{±} = τ(M_{DR}^{±}).

M.4. Periods. Let R_{K/Q}M be the motive over Q obtained by applying the restriction of scalars functor to M. Following Deligne ([D]), define c^{±}(M, T) ∈ (T ⊗ C)^{∗} by

c^{±}(M, T) = det_{T} (I^{±})

where I^{±} is the composite

(R_{K/Q}M)_{B}^{±} ⊗ C \underrightarrow{I_{∞}} (R_{K/Q}M_{DR}) ⊗ C → (R_{K/Q}M_{DR})^{±} ⊗ C

and the determinant is computed relative to T bases of each side. When convenient, we will write c^{±}(M) = c^{±}(M, T), leaving T implicit.

M.5. Let End(M) denote the algebra of all endomorphisms of M which are defined over K.

Proposition 1. If M is simple, i.e. has no non-trivial submotives, then End(M) is a division algebra with positive involution.

Proof. The proof follows, exactly as for abelian varieties, from the existence of K-rational polarization of M.

Remark. Hence, by Albert’s classification, End(M) is 1) a totally real number field F, or 2) a quaternion algebra B over F with B ⊗_{Q} R isotypic, or 3) a division algebra over a CM field. (We recall here that a CM field is a totally imaginary quadratic extension of a totally real field.)

Let δ^{±} denote the isomorphism class of the representation of T on M_{DR}^{±}. For a number field L, let I_{L} denote the ring of functions from J_{L} to Z. The elements of I_{L} with non-negative values are identified with the \overline{Q}-rational representations of L as a Q-algebra. If Φ ∈ I_{L} and τ ∈ G_{Q}, define τΦ by (τΦ)(η) = Φ(τ^{−1}η) for η ∈ J_{L}.

Proposition 2.

(a) If either T or K contains no CM subfield, then δ^{±} is a multiple of the regular representation. In any case, δ^{±} + ρδ^{±} is such a multiple.

(b) If δ^{±} is not a multiple of the regular representation, then δ^{+} = δ^{−} and T contains a CM subfield E such that δ^{±}(η_{1}) = δ^{±}(η_{2}) if η_{1} and η_{2} agree on E. Further, for K_{0} = Q(T r δ^{±}(t) | t ∈ T), τδ^{±} = δ^{±} if and only if τ ∈ G_{K_{0}}.

Proof. Assume that M is simple as a motive with coefficients in T. Then δ^{±} is the restriction to T ⊆ End(M) of a representation of End(M). Let T′ be a maximal subfield of End(M) which contains T. Let δ_{T′}^{±} be the representation of T′ on M_{DR}^{±}. Then

δ^{±}(σ) = ∑_{\begin{matrix}η∈J_{T′} \\ η|_{T} =σ \end{matrix}} δ_{T′}^{±}(η).

Regard these functions as defined on G_{Q}. Then δ_{T′}^{±}(ρg) = δ_{T′}^{±}(gρ) for all g ∈ G_{Q}. It follows that the same property holds for δ^{±}, and hence, if E ⊆ T denotes the field attached to H^{±} ⊆ G_{Q}, the right stabilizer of δ^{±}, then E is one of these types of field. Since the period map

(M_{B} ⊕ (ρM)_{B})^{±} ⊗ C \overrightarrow{∼} (M_{DR}^{±} ⊗ C ⊕ (ρM)_{DR}^{±} ⊗ C)

is T ⊗ C linear, δ^{±} + ρδ^{±} is a multiple of the regular representation R. If E is totally real, then δ^{±} = ρδ^{±}, and hence δ^{±} is itself a multiple of R, and E = K_{0} = Q. If E is a CM field, then K_{0} is a CM field, K can contain no real place, and hence M_{DR}^{+} = M_{DR}^{−}, i.e. δ^{+} = δ^{−}. If τ ∈ G_{Q}, then τ fixes K_{0} if and only if (τδ^{±})(t) = δ^{±}(t) for all t ∈ T. This happens if and only if τδ^{±} = δ^{±}.

Remark. The example of an abelian variety of CM type shows that δ^{±} is not always a multiple of R.

M.6. Basic periods. Let M be simple, and let Z be the center of End(M) with [End(M) : Z] = n^{2}. Let T ⊇ Z be a maximal subfield of End(M). Then [T : Z] = n. Let L be a Galois extension of Z which contains T. Let N = R_{K/Q}M. Then

N ⊗_{Z} L = ⊕_{σ∈Hom_{Z}(T,L)} N ⊗_{T,σ} L

where N ⊗_{T,σ} L denotes the extension of coefficients of N via σ : T → L. The N ⊗_{T,σ} L are all L linearly isomorphic since each is isomorphic to the image of M ⊗_{Z} L by a minimal idempotent of End(M) ⊗_{Z} L = M_{n}(L). Hence, the quantities c^{±}(M, T) ⊗_{T,σ} 1 ∈ L ⊗ C all lie in the same coset modulo L^{∗}, i.e. the class of c^{±}(M, T) ⊗ 1 in (L ⊗ C)^{∗}/L^{∗} is fixed under the action of Gal(L/Z) via the first factor. Put z^{±}(σ) = (σ ⊗ 1)(c^{±}(M, T) ⊗ 1)/c^{±}(M, T) ⊗ 1 ∈ L^{∗}. Then z^{±}(σ) is a 1-cocycle for the action of Gal(L/Z) on L^{∗}. Hence there exists b^{±} ∈ L^{∗} such that b^{±} = (σb^{±})z^{±}(σ). Put c_{0}^{±}(M) = b^{±}c^{±}(M, T). Then (σ ⊗ 1)c_{0}^{±}(M) = c_{0}^{±}(M), i.e., c_{0}^{±}(M) belongs to Z ⊗ C. Since the map (Z ⊗ C)^{∗}/Z^{∗} → (L ⊗ C)^{∗}/L^{∗} is injective for every L, we see easily that c_{0}^{±}(M) depends only on M and not the auxiliary choices of T and L.

M.7. Relations. If F is a field contained in End(M), c^{±}(M, F) ∼ N_{T/F} (c_{0}^{±}(M)) mod (F^{∗}) where T ⊇ F is a maximal subfield, N_{T/F} : (T ⊗ C)^{∗} → (F ⊗ C)^{∗} is the norm map, and, for any field L, and α, β ∈ L ⊗ C, with β a unit, α ∼ β mod (L^{∗}) means αβ^{−1} ∈ L = L ⊗ 1 ↪ L ⊗ C.

Proposition. Let M be a simple motive defined over K with coefficients in T. Let Z be the center of End(M), with [End(M) : Z] = n. Then

c^{±}(M, T) ∼ N_{T Z/T} (c_{0}^{±}(M))^{n[T Z:Z]^{−1}} mod (T^{∗}).

Proof. c^{±}(M, T) ∼ N_{L/T} (c_{0}^{±}(M)) where L ⊇ T is a maximal subfield of End(M). Since N_{L/T} = N_{T Z/T} ∘ N_{L/T Z}, c^{±}(M, T) ∼ N_{T Z/Z}(c_{0}^{±}(M))^{[L:T Z]}, because c_{0}^{±}(M) ∈ Z ⊗ C. Since [L : T Z][T Z : Z] = n, the result follows.

Although the periods c_{0}^{±}(M) are more fundamental, we shall work throughout the paper with the quantities c^{±}(M, T).

M.8. Factorization of c^{±}(M, T). Let P_{K} = ⟨1, e⟩\J_{k} denote the set of infinite places of K. Given M and T ⊆ End(M), we will define, for v ∈ P_{K} periods c_{v}^{±}(M, T) ∈ (T ⊗ ℂ)^{∗} such that

c^{±}(M, T) = α^{±}(M) · ∏_{v∈P_{K}} c_{v}^{±}(M, T)

for an elementary computable factor α(M) ∈ (T ⊗ Q)^{∗} depending only certain choices of differentials and characterized mod T^{∗} by a Galois recprocity law.

In the following Φ ⊆ J_{K} denotes a G_{T} orbit of embeddings of K and Ψ denotes a G_{K} orbit of embeddings of T. Identifying, according to context, a set of embeddings Ψ ⊆ J_{T} with the sum Σ_{σ∈Ψ}σ ∈ I_{T}, also denoted Ψ, we have

δ^{±} = ∑_{Ψ} n^{±}(Ψ)Ψ (n(Ψ) ≥ 0).

The algebra T ⊗ K is isomorphic to a direct sum of fields indexed by the orbits of G_{Q} in J_{T} × J_{K}: If Ψ ⊆ J_{T}, we define Ψ^{∗} ⊆ J_{K} by

Ψ^{∗} = π_{2}((G_{Q}(Ψ × 1_{K})) ∩ (1_{T} × J_{T}))

where π_{2} : J_{T} ×J_{K} → J_{K} denotes projection on the second factor. Similarly, we can define Ψ^{∗} ⊆ J_{T} given Ψ ⊆ J_{K}. For a G_{T} orbit Ψ (resp. G_{K} orbit Ψ) let σ^{∼} ∈ Ψ (resp. τ^{∼} ∈ Ψ) be a representative. Then the decomposition becomes

T ⊗ K →^{∼} ⊕_{Φ}(T ⊗ K)^{Φ} →^{∼} ⊕_{σ^{∼}}T σ^{∼}(K)

→^{∼} ⊕_{Ψ}(T ⊗ K)^{Ψ} →^{∼} ⊕_{τ^{∼}}τ^{∼}(T)K,

and the T ⊗ K-module M_{DR}^{±} decomposes as T ⊗ K-module:

M_{DR}^{±} →^{∼} ⊕_{Ψ}(M_{DR}^{±})^{Ψ}

where

(M_{DR}^{±})^{Ψ} →^{∼} ((T ⊗ K)^{Ψ})^{n^{±}(Ψ)}.

Let (δ^{±})^{∗} = Σ_{Ψ}n^{±}(Ψ)Ψ^{∗}, and let d(Ψ^{∗}) denote the number of elements in Ψ^{∗}.

Then (T ⊗ K)^{Ψ} = (T ⊗ K)^{Ψ∗} →^{∼} T τ^{∼}(K) is an extension of T of degree d(Ψ^{∗}), and hence (M_{DR}^{±})^{Ψ} is a (T ⊗ K)^{Ψ} vector space of dimension n^{±}(Ψ).

Let Ω^{±}(Ψ) = {ω_{1}^{±}(Ψ), …, ω_{n^{±}(Ψ)}^{±}(Ψ)} be a (T ⊗ K)^{Ψ}-basis of (M_{DR}^{±})^{Ψ}, and let Ω^{±} = ∪_{Ψ}Ω^{±}(Ψ). Let η ∈ J_{K}.

Case 1. Suppose η ∈ J_{K_{1},ℝ}. The δ^{±} is a multiple n of the regular representation, and η(Ω^{±}) = U_{Ψ}ηΩ^{±}(Ψ) is a T -basis of ηM_{DR}^{±}. Let Γ^{±}(η) = {γ_{1}^{±}(η), …, γ_{n}^{±}(η)} be a T -basis of (ηM)_{B}. Let

I_{∞}^{±}(η) : (ηM)_{B}^{±} ⊗ ℂ →^{∼} (ηM)_{DR}^{±} ⊗_{ηK} ℂ

be the T ⊗ ℂ linear period map, and let

p(η(Ω^{±}(Ψ)), Γ^{±}(η)) = det_{T ⊗ℂ}(P (ηω^{±}(Ψ)_{i}, γ^{±}(η)_{j})) (1 ≤ i, j ≤ n),

where p(ω^{±}(Ψ)_{i}, γ_{j}^{±}(η)) is defined by

I_{∞}^{±}(η)(ηω^{±}(Ψ)_{i}) = ∑_{j=1}^{n} p(ηω^{±}(Ψ)_{i}, γ^{±}(η)_{j}) · γ^{±}(η)_{j}.

Setting v = η, define

c_{v}^{±}(M) = ∏_{Ψ} p(η(Ω^{±}(Ψ)), Γ^{±}(η)).

If we change Ω^{±} and Γ^{±}(η), then c_{v}^{±}(M) undergoes a change c_{v}^{±}(M) → (t ⊗ η(k))c_{v}^{+}(M) with at t ∈ T^{∗}, and a k in K which is independent of η.

Case 2. (η complex). The M_{DR}^{+} = M_{DR}^{−}. Let

Λ(Ψ_{η}) = {λ_{1}, …, λ_{n(Ψ)}, λ′_{1}, …, λ′_{n(Ψ)}}

where

λ_{j} = ηω_{j}(Ψ) + ρηω_{j}(Ψ)
λ′_{j} = (1 ⊗ i)(ηω_{j}(Ψ) − (ρη)ω_{j}(Ψ))

for i = \sqrt{−1}, ω_{j} ∈ Ω(Ψ). Let Γ^{±}(η) = \{γ_{1}(η) + ργ_{1}(η), …, γ_{d}(η) + ργ_{d}(η), (1 ⊗ i)(γ_{1}(η) − ργ_{1}(η)), …, (1 ⊗ i)(γ_{d}(η) − ργ_{d}(η))} where γ_{1}(η), …, γ_{d}(η) is a T -basis of (ηM)_{B}. Let

(I_{∞}^{±}(η) ⊕ I_{∞}^{±}(ρη)) : (((ηM)_{B} ⊗ ℂ)^{ηΨ} ⊕
((ρη)(M)_{B} ⊗ ℂ)^{ρηΨ})^{±} → (ηM_{DR}^{±} ⊗_{η(K)} ℂ ⊕ (ρη)M_{DR}^{±} ⊗_{ρη(K)} ℂ)

be the period isomorphism, and let, as before

p(Λ(Ψ, η), Γ^{±}(η))

be the T ⊗ ℂ-linear determinant of the matrix representing I_{∞}^{±}(η) ⊕ I_{∞}(ρη) relative to be the bases Λ(Ψ, η) and Γ^{±}(η). Let c_{v}^{±}(M) = ∏_{Ψ} p(Λ(Ψ, η), Γ^{±}(η)).

Then c_{v}^{+}(M) undergoes a change of the form c_{v}^{+}(M) → (t ⊗ 1)c_{v}^{+}(M) if we change the basis of (ηM)_{B}^{±}, and a change of the form c_{v}^{+}(M) → (1 ⊗ η(k)(ρη)(k))c_{v}^{+}(M) if we change the Ω(Ψ). Finally, if we replace η by ρη, c_{v}^{±}(M) is unchanged.

For any ε : P_{K} → ℤ and any d ∈ ℤ such that ε(v) = d if v is complex, let

c_{ε}^{±}(M) = ( ∏_{v∈P_{K,ℝ}} c_{v}^{±}(M)^{ε(v)}c_{v}^{∓}(M)^{d−ε(v)})

( ∏_{v∈P_{K}\P_{K,ℝ}} c_{v}^{±}(M))^{d} .

Then, because M is special, c_{ε}^{±}(M, T) is independent of choices up multiplication by an element t ⊗ 1 for t ∈ T.

M.9. We retain the notation of the previous paragraph. Let

{α_{1}(Ψ), …, α_{d(Ψ^{∗})}(Ψ)} = A(Ψ)

be a basis of (T ⊗K)^{Ψ^{∗}} over T. Then AΩ(Ψ) =^{def}{α_{1}(Ψ)Ω(Ψ), α_{2}(Ψ)Ω(Ψ), …, α_{d(Ψ^{∗})}ΩΨ} is a basis of (M_{DR}^{±})^{Ψ} as a T vector space. Then AΩ = ∪_{Ψ}AΩ(Ψ) is an unordered basis of M_{DR}^{±} as a T -vector space.

Since the T -vector space (RM)_{DR}^{±} is canonically identified, ignoring the K action on the latter, with M_{DR}^{±} as T -vector space; AΩ is a T -basis of

RM_{DR}^{±}. Since (RM)_{B}^{±} = ( ⊕_{η∈J_{K}}(ηM)_{B})^{±}, the family of Γ^{±}(η)’s from (M.8)

provides a basis of (((RM)_{B})⊗ℂ)^{±}. Computing c^{±}(M, T) using these bases, and letting ε = 1, the ratio

c_{1}^{±}(M, T)/c^{±}(M, T) = b^{±}(M)

is well-defined modulo T^{∗}, and is computed as the T ⊗ \overline{Q}-linear determinant of the matrix which expresses the basis

(∪_{Ψ} ∪_{η∈J_{K,ℝ}} ηΩ(Ψ)) ∪ (∪_{Ψ} ∪_{η∈R_{K,ℂ}} Λ(Ψ, η))

(where R_{K,ℂ} is a set of representatives in J_{K} for ⟨1, ρ⟩\(J_{K}\J_{K,ℝ})), in terms

of AΩ = ∪_{Ψ}AΩ(Ψ).

To do this, note that if we use the Ω(Ψ) bases to identify M_{DR}^{±} with ⊕_{Ψ}((T ⊗ K)^{Ψ})^{n(Ψ)}, then our task relates to that of calculating, for each Ψ,

the inverse b(Ψ) of the determinant of the matrix in M_{d(Ψ^{∗})}(T ⊗ \overline{Q}) giving the canonical isomorphism

J(Ψ) : (T ⊗ K)^{Ψ} ⊗ \overline{Q} →^{∼} (T ⊗ \overline{Q})^{Ψ^{∗}},

where we regard the right member as the T ⊗\overline{Q}-algebra of maps Ψ^{∗} → T ⊗\overline{Q}, relative to the bases given by A(Ψ) and the set of idempotents 1_{η}, satisfying 1_{η}(η) = 1_{T ⊗\overline{Q}}, 1_{η}(η′) = 0 if η′ ≠ η. Note that

J(Ψ)(α_{j}(Ψ)) = ∑_{η∈Ψ^{∗}} η^{∼}(α_{j}(Ψ)) · 1_{η},

where η^{∼} is the T -linear extension of η to (T ⊗ K)^{Ψ}.

Hence,

b(Ψ) = det(η^{∼}(α_{j}(Ψ)))_{1≤j≤d(Ψ^{∗}), η∈Ψ^{∗}}

and

b^{±} = ∏_{Ψ} b(Ψ)^{n^{±}(Ψ)}.

Then b depends only on T, K and δ^{±} modulo T^{∗}.

This quantity is characterized by a reciprocity law which is easy to compute: Since each Ψ^{∗} is stable under the action of G_{T}, the signature π_{Ψ}(τ) defines a character π_{Ψ}(τ) : G_{T} → {±1}. Define

π_{δ^{±}} = ∏_{Ψ} π_{Ψ}(τ)^{n^{±}(τ)}

and let b(δ^{±}) = (b(δ^{±})_{σ})_{σ∈J_{T}}. Then

τb(δ^{±}) = π_{δ^{±}}(τ)b(δ^{±})_{1}

for τ ∈ G_{T}, and, letting R_{δ^{±}} = Ind_{T}^{Q}(π_{δ^{±}}), we have

(1 ⊗ τ)b(δ^{±}) = R(δ^{±})(τ)b(δ^{±}).

Note that b^{+} ∼ b^{−} unless K is totally real and M_{B}^{p,p} ≠ {0} for some p. Several special cases are worth noting.

1. If T = Q, δ^{±} is a multiple n^{±} of the regular representation of K and π(δ^{±}) = π(J_{K})^{n^{±}}. Hence b^{±} ∼ (\sqrt{D_{K}})^{n} where D_{K} is the discriminant.

2. For more general T, but if δ^{±} is a multiple of the regular representation, then π(δ^{±}) = (sgn(J_{K}) |_{T})^{n}, and again

b^{±} ∼ 1 ⊗ \sqrt{D_{K}}^{n}.

3. If K is a CM field, and δ^{±} is a multiple of a CM type, then G_{T} acts on (δ^{±})^{∗} via the same permutation as it acts on the embeddings J_{K_{0}}, since the restriction δ^{±} → J_{K_{0}} is a bijection, where K_{0} is the maximal real subfield of K. Hence π(δ^{±}) = (sgn_{J_{F}}) |_{T} and so b ∼ 1 ⊗ \sqrt{D_{F}}.

M.10. Artin motives. Let π : G_{K} → Aut(V) be a representation of G_{K} on the rational vector space V. Set π_{B} = V, π_{f} = V ⊗ Q_{f}, and π_{DR} = (V ⊗ \overline{Q})^{G_{K}}. Let I_{∞} : π_{DR} ⊗_{K} C = π_{B} ⊗ C be the identity map, and define I_{f} similarly. The structure π = (π_{DR}, π_{B}, π_{f}, I_{∞}, I_{f}) is called an Artin motive ([D]). We have π_{B} ⊗ C = π^{0,0}, and π admits a field T as coefficients exactly when we can embed T into End(π_{B}) with image in the commutant of the image of G_{K}. Assuming π has coefficients in T, for each σ ∈ J_{K,R}, let ε_{π}(σ) = \frac{1}{2} (d_{π} + T r(F_{σ} | (σπ)_{B})), where d_{π} = dim_{T} π_{B}. For v complex, put ε_{π}(v) = d_{π}. (Here, for σ ∈ G_{Q}, σπ is the Artin motive attached to the representation τ → π(σ^{−1}τσ), τ ∈ G_{σK}.)

M.11. Reciprocity laws. Let H, U, and V be subgroups of a topological group G, with H finite, and U and V of finite index, and U ⊆ V. Put J_{U} = G/U, J_{V} = G/V. Let ψ : J_{V} → Z satisfy, for an H invariant subset S ⊆ J_{V},

∑_{τ∈Hσ} ψ(τ) = \begin{cases}c & σ ∈ S \\ 0 & σ ∉ S \end{cases}

with a constant c depending only upon ψ. Via the natural map J_{U} → J_{V}, regard ψ as a function on J_{U}. Let G_{H} be the subgroup of τ ∈ G for which τHσ = Hτσ for all σ ∈ J_{V}. For each σ ∈ J_{U}, choose a representative w_{σ} ∈ G, and arrange that hw_{σ} = w_{hσ} for all h ∈ H. Let H_{S} = {g ∈ G | gS = S}. Define for τ ∈ G_{H} ∩ H_{S}, t_{ψ}(τ) ∈ U^{ab} by

t_{ψ}(τ) = ∏_{σ∈J_{K}} (w_{τσ}^{−1}τw_{σ})^{ψ(σ)} mod U^{c}

where U^{c} denotes the closure of the commutator subgroup.

Proposition 1. For τ ∈ G_{H} ∩ H_{S}, t_{ψ}(τ) is well-defined.

Proof. If w′_{σ} = w_{σ}u_{σ} for a u_{σ} ∈ U, then u_{hσ} = u_{σ} for all h ∈ H. Hence, it is enough to show

∏_{σ∈J_{U}} u_{τσ}^{ψ(σ)} ≡ ∏_{σ∈J_{U}} u_{σ}^{ψ(σ)} mod U^{c}.

Since τHσ = H_{τσ} for each σ ∈ J_{V}, the result will follow if

∑_{\begin{matrix}η∈J_{U} \\ η→σ \end{matrix}} ∑_{h∈Hτη} ψ(h) = ∑_{\begin{matrix}η∈J_{U} \\ η→σ \end{matrix}} ∑_{h∈Hη} ψ(h).

Since τS = S and τ(J_{V} − S) = J_{V} − S, each side above is either [V : U]c or 0, simultaneously.

Next, let H_{ψ} = {g ∈ G | gψ = ψ}.

Proposition 2. If τ_{1}, τ_{2} ∈ H_{ψ}, t_{ψ}(τ_{1}τ_{2}) = t_{ψ}(τ_{1})t_{ψ}(τ_{2}).

Proof. Let X ⊆ J_{U} be a set of representatives for H_{ψ}\J_{U}. Then

(∗)  t_{ψ}(τ) = ∏_{x∈X} [ ∏_{y∈H_{ψ}x} (w_{τy}^{−1}τw_{y})]^{ψ_{0}(x)}

where ψ_{0} : H_{ψ}\J_{U} → Z is the function defined by ψ. Fixing x, each w_{y} belongs to H_{ψ}w_{x}U = w_{x}(w_{x}^{−1}H_{ψ}w_{x}·U), and hence w_{y} = w_{x}z_{y} with z_{y} ∈ w_{x}^{−1}H_{ψ}w_{x}·U. Hence, if τ ∈ H_{ψ},

∏_{y∈H_{ψ}x} (w_{τy}^{−1}τw_{y}) = ∏_{y∈H_{ψ}x} (z_{τy}^{−1}(w_{x}^{−1}τw_{x})z_{y}).

Now let A and B be subgroups of finite index inside a group C, and let {z_{α} ∈ A·B | α ∈ A·B/B} be a set of representatives in A·B for the quotient A·B/B. If a ∈ A, define t(a) ∈ B^{ab} by

t(a) = ∏_{α∈A·B/B} (z_{aα}^{−1}az_{α}) mod B^{c}.

Then t(a) is well-defined, and

t(a_{1}a_{2}) = ∏_{α∈A·B/B} (z_{a_{1}a_{2}α}^{−1}a_{1}a_{2}z_{α}) mod B^{c}

= ∏_{α∈A·B/B} (z_{a_{1}a_{2}α}^{−1}a_{1}z_{a_{2}α})(z_{a_{2}α}^{−1}a_{2}z_{α}) mod B^{c}

= t(a_{1})t(a_{2}) mod B^{c}.

Applying this remark to a_{1} = w_{x}^{−1}τ_{1}w_{x}, a_{2} = w_{x}^{−1}τ_{2}w_{x}, A = w_{x}^{−1}H_{ψ}w_{x} and B = U, we see that inner term of (*) is a homomorphism and we are done.

Note that if ψ is a constant, taking the value d, then H_{ψ} = G and t_{ψ} : G^{ab} → U^{ab} is the d-th power of the usual transfer map. However, in general, for τ ∉ H_{ψ}, t_{ψ} is not a homomorphism. Rather, we find t_{ψ}(τ_{1}τ_{2}) = t_{τ_{2}ψ}(τ_{1})t_{ψ}(τ_{2}) for general τ_{1}, τ_{2} ∈ G.

Below, we apply the results with H = ⟨1, ρ⟩, U = G_{K} and V = G_{K_{cm}}, where K_{cm} ⊆ K is the maximal CM subfield, or Q, if K contains no CM subfield. Hence, if S = J_{V}, G_{H} = H_{S} = G_{Q}.

M.12. If L and T are number fields with L ⊇ T, we regard any δ ∈ I_{T} as a function in I_{L} via the map J_{L} → J_{T}. Let L be a Galois extension of Q which contains T. Then, for δ^{±} as in M.5, there exists a unique δ_{∗}^{±} ∈ I_{K_{cm}} (c.f. Prop. M.5.2.) such that δ_{∗}^{±}(σ) = δ^{±}(σ^{−1}) for all σ ∈ J_{L}. If τ ∈ G_{Q}, then τδ_{∗}^{±} depends only upon the image of τ in J_{T}.

The ψ ∈ I_{K} which are of interest to us are of the form δ_{∗}^{±}, and hence satisfy

(M.12.1)  ψ(ρσ) + ψ(σ) = w  (σ ∈ J_{K}).
(M.12.2)  ψ(σ_{1}) = ψ(σ_{2})  if σ_{1} = σ_{2} on K_{cm}.
(M.12.3)  τψ = ψ  (τ ∈ J_{T}).

Clearly, for such ψ, the above procedure defines ψ_{∗} ∈ I_{T}, and we have (ψ_{∗})_{∗} = ψ.

M.13. Let ψ be as in M.12, and define r_{ψ} : G_{Q} → (G_{K}^{ab})^{J_{T}} by r_{ψ,η} = t_{ηψ}, for each η ∈ J_{T}. Then, if φ : G_{K}^{ab} → T^{∗} is a character, define, for τ ∈ G_{Q}, φ_{∗}r_{ψ}(τ) ∈ (T ⊗_{Q} \overline{Q})^{∗} by

φ_{∗}r_{ψ}(τ)_{η} = (ηψ) ∘ r_{ψ,η}(τ) (η ∈ J_{T}).

Proposition. There exists an element c(ψ, φ) ∈ (T ⊗ \overline{Q})^{∗}, unique up to a change of the form c(ψ, φ) ↦ (t ⊗ 1)c(ψ, φ) with t ∈ T^{∗}, such that for all τ ∈ G_{Q},

(1 ⊗ τ)c(ψ, φ) = φ_{∗}r_{ψ}(τ)c(ψ, φ).

Proof. Let π_{φ} be an Artin motive, defined over K, with coefficients in Q(φ), the field generated by the values of φ. Set π = π_{φ} ⊗_{Q(φ)} T. Then dim_{T} π_{B} = 1, and hence there exists p(φ) ∈ (T ⊗ K^{ab})^{∗} such that γ = p(φ)·w where 0 ≠ γ ∈ π_{B}, and w is a basis of the free T ⊗ K module π_{DR}. If τ ∈ G_{K}, it follows, from the definition of π_{DR}, that (1 ⊗ τ)p(φ) = (φ(τ) ⊗ 1)p(φ), and p(φ) is independent of choices, up to change of the form p(φ) → αp(φ) with an α ∈ (T ⊗ K)^{∗}.

Now let ψ : J_{T} → Z be as in M.12. with the roles of T and K reversed, and for an a ∈ (T ⊗ \overline{Q})^{∗}, define ψ(a) ∈ T ⊗ \overline{Q} by ψ(a)_{σ} = a_{σ}^{ψ(σ)}. For σ ∈ J_{K}, put p(φ)_{σ} = (1 ⊗ w_{σ})p(φ), and set

P (ψ, φ) = ∏_{σ∈J_{K}}(σψ)(p(φ)_{σ}).

Note that, for τ ∈ G_{Q}, we have (1 ⊗ τ)(ψ(a)) = (τψ)(τa). Then

(1 ⊗ τ)P (ψ, φ) = ∏_{σ∈J_{K}}(τσψ)((1 ⊗ w_{τσ}k(τ, σ))p(φ))

= ∏_{σ∈J_{K}}(τσψ)((φ(k(τ, σ)) ⊗ w_{τσ})p(φ))

= [ ∏_{σ∈J_{K}}(τσψ)(φ(k(τ, σ)) ⊗ 1)] P (ψ, φ),

where we have set τw_{σ} = w_{τσ}k(τ, σ) with k(τ, σ) ∈ G_{K}.

Now, for η ∈ J_{T},

[ ∏_{σ∈J_{K}}(τσψ)(φ(k(τ, σ)) ⊗ 1)]_{η} = (ηφ) ( ∏_{σ∈J_{K}} k(τ, σ)^{(τσψ)(η)})

= (ηφ) ( ∏_{σ∈J_{K}} k(τ, σ)^{(τ^{−1}ηψ_{∗})(σ)})

= ηφ ∘ t_{τ^{−1}ηψ_{∗}}(τ).

Thus (1 ⊗ τ)P (ψ, φ) = φ ∗ r_{ψ_{∗}}(τ)P (ψ, φ).

If we now start with a ψ : J_{K} → Z as in the hypothesis of the Proposition, we prove the above result with the roles of ψ and ψ_{∗} reversed, using (ψ_{∗})_{∗} = ψ. Put c(ψ, φ) = P (ψ_{∗}, φ). Then we are done since the reciprocity law clearly characterizes c(ψ, φ) up to multiplication by t ⊗ 1.

M.14.

Corollary 1. Define c(ψ, φ) ∈ (T ⊗ \overline{Q})^{∗} as in M .13. then

(i) if φ^{n} = 1, then c(ψ, φ)^{n} ∈ T^{∗}, (ii) let E ⊆ T be the field corresponding to H_{ψ}. Then c(ψ, φ)_{1} ∈ E^{ab}, (iii) over E(φ), c(ψ, φ)_{1} generates the Kummer extension attached to the

character φ ∘ t_{ψ} of G_{E}^{ab}.

Proof. Part (i) is obvious, part (ii) follows from Prop. M.13 upon direct calculation of the actions of τ_{1}τ_{2} and τ_{2}τ_{1} for τ_{1}, τ_{2} ∈ G_{E}, and part iii) is elementary.

Corollary 2. The numbers c(ψ, φ)_{η} (η ∈ J_{T}) generate abelian extensions of CM fields.

Proof. By Prop. M.12., E = Q or E is a CM field, and the result follows from (ii) above.

M.15. Let E_{ψ} be the field generated over E by the elements c(ψ, φ)_{1} as φ varies among the characters of finite order of G_{K}^{ab}.

Proposition.

(i) E_{ψ} is the subfield of E^{ab} corresponding to the subgroup of E_{f}^{∗} consisting of the elements e for which ψ_{∗}(e) belongs to the connected component of K_{f}^{∗}/K_{+}^{∗}, where K_{+}^{∗} ⊆ K^{∗} is the subgroup of totally positive elements.

(ii) Suppose that ψ is a CM type, i.e. ψ(σ) + ψ(ρσ) = 1, and ψ(σ) ≥ 0 for all σ ∈ J_{K}, and let F be the maximal totally real subfield of the field E.

Let z ∈ F ⊗ C − F ⊗ R be a CM point of type (E, ψ_{∗}) (c.f. [CV]). Then E_{ψ} is the field generated over E by the values at z of all arithmetic Hilbert modular functions (e.g. elements of A_{0}(Q_{ab}) in the notation of [Sh3]) which are defined at z.

Proof. The argument of Tate in [L] extends immediately to our case to show that r_{K}(ψ_{∗}(e)) = t_{ψ}(σ) if r_{E}(e) = σ ∈ G_{E}^{ab}, and where, for any number field L, r_{L} : L_{f}^{∗} → G_{L}^{ab} denotes the Artin reciprocity law. Thus, (i) is immediate. To see (ii), recall that the field generated by the given values is the compositum of E with the fields of moduli k(P) of the collection of P EL structures P = (A, C, ϑ, v_{1}, …, v_{N}), where (A, C) is a fixed polarized abelian variety with an action ϑ of O_{F}, and v_{1}, …, v_{N} denote a variable set of torsion points. By [SH2], Prop. 5.17, Ek(P) is the field of moduli of (A, C, ϑ_{∗}), where ϑ_{∗} : E_{∗} → End(A) ⊗ Q is an extension of ϑ. By [Sh2], Prop. 5.16, ∪_{P} Ek(P) is the class field attached to

∩_{n≥1} (ψ_{∗}(K^{∗}U_{K}(n)))^{−1} = (ψ_{∗}(∩_{n≥1}(K^{∗}U_{K}(n))))^{−1}

where U_{K}(n) denotes the subgroup of local units in K_{f}^{∗} whose elements are congruent to 1 modulo n. Since ∩_{n≥1} K^{∗}U_{K}(n)/K^{∗} is the connected component of K_{f}^{∗}/K_{+}^{∗}, we are done.

M.16. For M special, irreducible and π an Artin motive, M ⊗ π is special if and only if i) M^{w/2,w/2} = {0} (w ∈ 2Z), or ii), if w ∈ 2Z and M^{w/2,w/2} ≠ {0}, then K is totally real, F_{τ} acts as a scalar independent of τ on each τπ, for all τ ∈ J_{K}, and tr(F_{τ}, (τM)_{B}) is independent of τ. Assume that these conditions are satisfied.

Theorem. For a special motive M and π, as above, both with coefficients in T and defined over K,

c_{1}^{±}(M ⊗ π) ∼ c_{ε_{π}}^{±}(M)c(δ_{∗}^{±}, det π) mod (T^{∗})

where ε_{π} is defined in M .10, det π is the maximal T linear exterior power, and we have omitted notation referring to T.

Proof. For σ ∈ J_{K}, let ξ(σ) = (ξ_{1}(σ), …, ξ_{d_{π}}(σ)) be a basis of (σπ)_{B}. Define Δ_{σ} : (σM)^{d_{π}} → σ(M ⊗ π) by

Δ_{σ,B}(v_{1}, …, v_{d_{π}}) = ∑_{i=1}^{d_{π}} v_{i} ⊗ ξ_{i}(σ)

where v_{1}, …, v_{d_{π}} belong to (σM)_{B}. Passing to quotients, define Δ_{σ}^{±} : ((σM)_{DR}^{±} ⊗_{σK} \overline{Q})^{d_{π}} → (σ(M ⊗ π))_{DR}^{±α} ⊗_{σK} \overline{Q}, via Δ_{σ,DR}, where α = 1 unless M^{w/2,w/2} ≠ {0} and π(ρ) = −1, in which case α = −1.

We consider two commutative diagrams. First, if σ ∈ J_{K,R}, assume that ξ(σ) has been chosen so that ξ(σ)_{i} ∈ (σπ)_{B}^{+} if 1 ≤ i ≤ ε(σ) and ξ(σ)_{i} ∈ (σπ)_{B}^{−} if ε(σ) + 1 ≤ i ≤ d_{π}. Then,

(((σM)_{B}^{±})^{ε(σ)} ⊕ ((σM)_{B}^{∓})^{d_{π}−ε(σ)}) ⊗ C^{\underrightarrow{J_{σ}^{±}}} ((σM)_{DR}^{±} ⊗_{σK} C)^{d_{π}}
↓ Δ_{σ,B}  ↓ Δ_{σ}^{±}
(σ(M ⊗ π))_{B}^{±} ⊗ C  ^{\underrightarrow{I_{σ}^{±}}} (σ(M ⊗ π))_{DR}^{±} ⊗_{σK} C

commutes, where J_{σ}^{±} = (I_{σ}^{±})^{ε(σ)} × (I_{σ}^{∓})^{d_{π}−ε(σ)}. Hence,

c_{σ}^{±}(M ⊗ π) ∼ c_{σ}^{±}(M)^{ε(σ)}c^{∓}(M)^{d_{π}−ε(σ)} × det(Δ_{σ}^{±}),

since det(Δ_{σ,B}) ∈ T^{∗}. If σ ∈ J_{K,C}, assume ξ(ρσ) = ρξ(σ). Then

((σM)_{B}^{d_{π}} ⊕ (ρσM)_{B}^{d_{π}})^{±} ⊗ C  ⟶^{I_{σ}^{dπ} ⊕I_{ρσ}^{dπ}}  \begin{matrix}((σM)_{DR}^{±} ⊗_{σK} C)^{d_{π}}⊕ \\ ((ρσM)_{DR}^{±} ⊗_{ρσK} C)^{d_{π}} \end{matrix}
↓^{Δ_{σ,B}⊕Δ_{ρσ,B}}  ↓^{Δ_{σ}^{±}⊕Δ_{ρσ}^{±}}
((σ(M ⊗ π))_{B} ⊕ (ρσ(M ⊗ π))_{B})^{±} ⊗ C ⟶^{I_{σ}⊕I_{ρσ}} \begin{matrix}((σ(M ⊗ π))_{DR}^{±} ⊗_{σK} C)⊕ \\ ((ρσ(M ⊗ π))_{DR}^{±} ⊗_{ρσK} C) \end{matrix}

commutes. Hence, if v denotes the place of K corresponding to the pair (σ, ρσ), then

c_{v}^{±}(M ⊗ π) ∼ c_{v}^{±}(M)^{d_{π}} det(Δ_{σ}^{±}) det(Δ_{ρσ}^{±}),

where we agree that det(Δ_{σ}^{±}) has component 1 at any η ∈ J_{T} which fails to occur in σM_{DR}^{±}. Thus, we must show

∏_{σ∈J_{K}} det(Δ_{σ}^{±}) ∼ c(δ_{∗}^{±}, det π).

From the definition of Δ_{σ}, we see that if w_{1}, …, w_{d_{π}} belong to M_{DR}^{±},

Δ_{σ}^{±}(σw_{1}, …, w_{d_{π}}) = ∑_{i=1}^{d_{π}} σw_{i} ⊗ ξ_{i}(σ).

Let μ_{1}, …, μ_{d_{π}} be a basis of the free T ⊗ K module π_{DR}, then

ξ_{i}(σ) = ∑_{j=1}^{d_{π}} a_{ij}(σ)σ(μ_{j}),

with a_{ij}(σ) ∈ T ⊗ \overline{Q}. Thus,

Δ_{σ}^{±}(σw_{1}, …, σw_{d_{±}}) = ∑_{i=1}^{d_{π}} ∑_{j=1}^{d_{π}} σ(w_{i} ⊗ μ_{j})a_{ij}(σ).

Decomposing this map into its η eigencomponents for η ∈ J_{T}, we see that

(det(Δ_{σ}^{±}))_{η} = det(A(σ))_{η}^{σδ^{±}(η)}

where A(σ) = (a_{ij}(σ))_{1≤i,j≤d_{π}}.

Without loss of generality, assume that K ⊂ R, if K has a real embedding. Choose ξ(1) as above and let ξ(σ) = w_{σ}ξ(1). Then for each real σ, ξ(σ) satisfies our hypothesis, and ρξ(σ) = ξ(ρσ) if σ is complete. Now, for any Artin motive π, defined over K, and any τ ∈ G_{Q}, the map τ : π_{B} → (τπ)_{B} is the restriction of τ : π_{DR} ⊗_{K} \overline{K} → (τπ)_{DR} ⊗_{τK} \overline{τK}, defined by τ(μ ⊗ k) = τμ ⊗ τk for μ ∈ π_{DR} and k ∈ \overline{K}. Thus,

ξ_{i}(σ) = ∑_{i=1}^{d_{π}} (1 ⊗ w_{σ})a_{ij}(1) · w_{σ}μ_{j}.

Hence det A(σ) = (1 ⊗ w_{σ}) det A(1), and so

∏_{σ∈J_{K}} det(Δ_{σ}^{±}) ∼ ∏_{σ∈J_{K}} (σδ^{±})((1 ⊗ w_{σ}) det(A(1))).

As to det(A(1)), note that (1 ⊗ τ)(ξ_{1}(1) ∧ ⋯ ∧ ξ_{d_{π}}(1)) = det(π) ξ_{1}(1) ∧ ⋯ ∧ ξ_{d_{π}}(1) for τ ∈ G_{K}, and so, from the definition of π_{DR},

(1 ⊗ τ) det(A(1)) = ((det(π))(τ) ⊗ 1) det(A(1)).

Setting det(A(1)) = p(det(π)), as in the proof of Prop. M.13, and recalling that argument, we are done.

M.17. Applications.

Corollary 1. Suppose that K is totally complex. Then

c^{±}(M ⊗ π) ∼ c^{±}(M)^{d_{π}}c(δ_{∗}^{±}, det π) mod (T^{∗}).

Proof. Apply Theorem M.16. and Prop. M.9.

For the next applications, let L be a finite extension of K. For each σ ∈ J_{K}, let r(σ) the number of real embeddings of L which extend σ, and define ε_{L/K} : J_{K} → Z by

ε_{L/K}(σ) = \begin{cases}\frac{[L:K]+r(σ)}{2} & σ real \\ [L : K] & σ not real. \end{cases}

Let M ×_{K} L denote the motive over L obtained from M by extending scalars.

Corollary 2. Suppose that M ×_{K} L is special. Let π_{L/K} be the character of G_{K} which gives the sign of the permutation given by the action of G_{K} on G_{K}/G_{L}. Then, with ε = ε_{L/K},

c^{±}(M ×_{K} L) ∼ c_{ε}^{±}(M)c(δ_{∗}^{±}, π_{L/K}) mod (T^{∗}).

Corollary 3. Suppose that K is totally complex. Then

c^{±}(M ×_{K} L) ∼ c^{±}(M)^{[L:K]}c(δ_{∗}^{±}, π_{L/K}) mod (T^{∗}).

Proof. Apply Corollary 2 and Prop. M.9.

It is easy to give the quantities c(δ_{∗}^{±}, π_{L/K}) an explicit form. Let ℓ_{1}, …, ℓ_{t} be a basis of L as a K vector space. Put

p_{L/K} = det(σ(ℓ_{i}))_{1≤i≤t,σ ∈ G_{K}/G_{L}}.

This determinant is well defined up to an element of K^{∗}, and τp_{L/K} = π_{L/K}(τ)p_{L/K}. As in the proof of Prop. M.13, we can put

c(δ_{∗}^{±}, π_{L/K}) = ∏_{σ∈J_{K}} (σδ^{±})((1 ⊗ w_{σ})(1 ⊗ p_{L/K})).

Suppose next that δ^{±} is a multiple m^{±} of the regular representation of T. Then, for φ : G_{K} → T^{∗}, the reciprocity law is

(1 ⊗ τ)c^{±}(δ_{∗}^{±}, φ) = (φ ∘ tr_{K/Q}(τ)^{m^{±}} ⊗ 1)c^{±}(δ_{∗}^{±}, φ)

where tr_{K/Q} : G_{Q} → G_{K}^{ab} is the transfer homomorphism. Let φ^{∼} : K_{f}^{∗} → T^{∗} be the character of K_{f}^{∗} associated to φ via r_{K}. Let ψ : K_{f} → C^{∗} be a non-trivial additive character. Define

G(φ) = G(φ, ψ) = ∫_{U_{K}} φ^{∼}(u) ⊗ ψ(u)du ∈ (T ⊗ Q_{ab})^{∗}

with the Haar measure du on K_{f} which assigns measure 1 to the integral adeles. Then the identity

(1 ⊗ r_{Q}(z))G(φ) ∼ (φ^{∼}(z) ⊗ 1)G(φ)

is immediate, provided we recall that r_{Q}(z)(e^{2πi/m}) = e^{2πia/m} where a ∈ (Z/(m))^{∗} is the inverse of the image of z. Since the diagram

Q_{A}^{∗} ⟶ G_{Q}^{ab}
↓  ↓^{tr}^{K/Q}
K_{A}^{∗} ⟶ G_{K}^{ab}

commutes, we see that

(1 ⊗ τ)G(φ) = (φ ∘ tr_{K/Q}(τ) ⊗ 1)G(φ)

for all τ ∈ G_{Q}. Thus we have

c(δ_{∗}^{±}, φ) ∼ G(φ)^{m^{±}} mod (T^{∗})

for δ^{±} = m^{±} times the regular representation.

Corollary 4. Let K have a real place. Then

c^{±}(M ⊗ π) ∼ c_{ε_{π}}^{±}(M)(1 ⊗ D_{K}^{1/2})^{d^{±α}·d_{π}}G(det(π))^{m^{±α}} mod (T^{∗})

where α = +1 unless M^{w/2,w/2} ≠ {0}, in which case α is the scalar by which ρ acts on π.

Proof. δ^{±} is m^{±} times the regular representation by Prop. M.5.2, since K contains no CM subfield.

Corollary 5. Let K be totally real. Let L be a totally complex finite extension of K of degree 2a. Suppose that M^{w/2,w/2} = {0} if w ∈ λZ. Then M ×_{K} L is special, and

c^{±}(M ×_{K} L) ∼ (c^{+}(M)c^{−}(M))^{a}G(π_{L/K})^{m^{±}} mod (T^{∗}).

Corollary 6. Let K be totally real. Let L be a totally real extension of K of degree a. Then M ×_{K} L is special, and

c^{±}(M ×_{K} L) ∼ c^{±}(M)^{a}G(π_{L/K})^{m^{±}} mod (T^{∗}).

Example. Let L be a cubic extension of Q. Then either L is cyclic over Q or N, the Galois closure of L, has for its Galois group the symmetric group on 3 letters. In the first case, π_{L/Q} ≡ 1. In the second, there is a unique quadratic extension of Q contained in M such that π_{L/Q} is the non-trivial quadratic character of Gal(N/Q) associated to this extension. L is either totally real, or has one real and one complex place. Hence, Theorem M.16 provides the following possibilities:

c^{±}(M ×_{Q} L) ∼ \begin{cases}c^{±}(M)^{3}(1 ⊗ D_{L/Q}^{1/2})^{d^{±}} & (L totally real, non-cyclic), \\ c^{±}(M)^{3} & (L totally real, cyclic), \\ c^{±}(M)^{2}c^{∓}(1 ⊗ D_{L/Q}^{1/2})^{d^{±}} & (L not totally real). \end{cases}

L. L-functions.

L.1. For a finite place v of K, let D_{v}, I_{v}, and Φ_{v} denote a decomposition group, an inertia subgroup of D_{v}, and a (geometric) Frobenius coset in D_{v}/I_{v}, respectively. In order to define L-functions we henceforth assume that each structure on M as motive with coefficients in T is strictly compatible: For each finite plane λ of T, the polynomial

L_{v}(M, T; X)^{−1} = det_{T} (1 − Φ_{v}X, M_{λ}^{I_{v}}) ∈ T_{λ}[X]

belongs to T [X] and is independent of the choice of λ. Let L_{v}(M, T; s) be the image of L_{v} under the map T [X] → T ⊗ C[N_{v}^{−s}] where N_{v} denotes the norm of v and s ∈ C. Then define

L(M, T; s) = ∏_{v} L_{v}(M, T; s).

If we assume the Riemann hypothesis for M, i.e. that the roots of N_{T/Q}(L_{v}(M, T; X))^{−1} ∈ Q[X] all have absolute value (N_{v})^{w/2}, then L(M, T; s) converges for Re(s) > \frac{w+1}{2} and takes values, for such s, in T ⊗ C. It is standard to conjecture that L(M, T; s) continues to a meromorphic function on C. Further, if M is simple, the continuation should be entire, unless M = Q(−w/2), in which case L(M, Q; s) = ζ_{K}(s − w/2) where ζ_{K} denotes the Dedekind zeta function of K.

L.2. The various L(M, T; s) attached to M for different coefficient structures are related by the following elementary result.

Proposition. Let M be a simple motive defined over K with coefficients in T. Let F be the center of the division algebra End(M). Then there exists a unique Dirichlet series L_{0}(M, s) = ∏_{v} L_{0,v}(M, s), L_{0,v}(M, s) =

L_{0,v}(M, X)|_{X=N_{v}^{−s}} with L_{0,v}(M, X)^{−1} ∈ F [X], such that

L(M, T; s) = N_{T F/T} (L_{0}(M, s))^{n/d}

where n^{2} = [End(M) : F] and d = [T F : F]. In particular, for any T ⊇ F with [T : F] = n, L(M, T; s) = L_{0}(M, s).

Proof. Suppose first that T ⊇ F, [T : F] = n, and let T′ ⊆ T be a Galois extension of F. Let v and λ be finite planes of K and F, respectively, whose restrictions to Q are distinct. Then

T r_{T ⊗F_{λ}}(Φ_{v}, M_{λ}^{I_{v}}) = T r_{T ⊗_{F} T′⊗F_{λ}}(Φ_{v}, M_{λ} ⊗_{F} T′) =
T r_{(T′)^{n}⊗_{F} F_{λ}}(Φ_{v}, (N_{λ}^{I_{v}})^{n}) = T r_{T′⊗F_{λ}}(Φ_{v}, N_{λ}^{I_{v}}) ∈ T′ ⊆ (T′)^{n},

where N_{λ} = e(M_{λ} ⊗_{F} T′) for a minimal idempotent of T ⊗_{F} T′. The first term belongs to T ⊗ 1 while the last term belongs to 1 ⊗ T′. Since T ⊗ 1 ∩ 1 ⊗ T′ = F (inside T ⊗_{F} T′), L(M, T; s) is an Euler product formed of polynomials with coefficients in F. Now End(M ⊗_{F} T′) = End(M) ⊗_{F} T′ = M_{n}(T′) and T ⊗_{F} T′ \underrightarrow{∼} (T′)^{n} ↪ M_{n}(T′) is a maximal semisimple commutative subalgebra. Hence, N_{λ} is independent of the choice of T, and depends only upon the choice of T′ which splits End(M). Thus L(M, T; s) = L(M; T_{1}; s) if T_{1} ⊇ F is any coefficient structure with [T_{1} : F] = n.

Put L_{0}(M, s) = L(M, T; s), with T as above. Next let T_{2} be any coefficient structure for M, and assume, changing T if necessary, that T ⊇ T_{2}. Then L(M, T_{2}; s) = N_{T/T_{2}}(L_{0}(M, s)) = N_{T_{2}F/T_{2}}(N_{T/T_{2}F} (L_{0}(M, s))) = N_{T_{2}F/T_{2}}(L_{0}(M, s))^{n/d} with d = [T_{2}F : F], since [T : T_{2}F] = [T : F][T_{2}F : F]^{−1}. Q.E.D.

L.3. We say that M satisfies the Tate conjecture if, for each prime ℓ, the Q_{ℓ} subalgebra of End(M_{ℓ}) generated by the image of G_{K} is the commutant of End(M) ⊗ Q_{ℓ}.

Proposition. Let M be a simple motive over K with coefficients in T. Let F be the center of End(M). Let F^{T} be the field attached to the subgroup of G_{Q} which stabilizes G_{T} G_{F} /G_{F}. Then i) the polynomials L_{v}(M, T; X)^{−1} have coefficients in F^{T} and, ii) if M satisfies the Tate conjecture, the coefficients of the L_{v}(M, T; X)^{−1} generate F^{T}.

Proof. From the previous proposition, we see that the coefficients lie in the field generated by the elements T r_{T F/T} (α), where α ∈ F varies among the coefficients of the L_{0,v}(M, X)^{−1}. Since the map G_{T} /G_{T F} → G_{Q}/G_{F} is injective, with image G_{T} G_{F} /G_{F}, this field is F^{T}.

To see ii), let A be the commutant of End(M) inside M_{B}. Then the center of A is F. By assumption, and the Cebotarev density theorem, the Q_{ℓ} span of the Frobenius elements Φ_{v} is A ⊗ Q_{ℓ}, for any prime ℓ. Hence the Q_{ℓ} span of their reduced traces T r_{A/F} ⊗ 1 : Q ⊗ Q_{ℓ} → F ⊗ Q_{ℓ} is F ⊗ Q_{ℓ}. Let F_{0} ⊆ F be the field generated over Q by the coefficients of L_{0}(M, s). Then F_{0} coincides with the field generated by the above traces. Thus F_{0} ⊗ Q_{ℓ} = F ⊗ Q_{ℓ}, for all ℓ. Hence F = F_{0}. Now it follows from the analysis above for part i) that the coefficients of the L_{v}(M, T; X) generate F^{T}. The last claim is obvious.

L.4. Remarks. There exists an abelian variety A defined over Q for which End(A) ⊗ Q is a non-commutative division algebra. Set M = H^{1}(A).

Then L(M, s) = N_{F/Q}(L_{0}(M, s))^{n} where n^{2} = [End(A) ⊗ Q : F] and F is the center of End(M). L(M, s) is the usual Hasse-Weil L-function in degree one of A. N_{F/Q}(L_{0}(M, s)) cannot itself occur as L(N, s) for N = H^{1}(B) with an abelian variety B, because then L(N, s)^{n} = L(M, s), and so by Tate’s isogeny conjecture, proved for the motives attached to H^{1} of abelian varieties, A is isogenous to B^{n}, contrary to hypothesis.

L.5. Critical strip. Let M be a critical motive defined over K, and put N = R_{K/Q}M. If N_{B} ⊗ C = N^{w/2,w/2} with w ∈ 2Z, put I_{1} = Z. Otherwise, put I_{1} = {P + 1, …, Q} with

P = max_{p<q}{p|N^{p,q} ≠ 0}.

Recall that if N^{w/2,w/2} ≠ 0, then F_{∞} acts on N^{w/2,w/2} as a scalar (−1)^{ε} for ε = 0 or 1. The critical strip for M is C(M) = C_{ℓ}(M) ∪ C_{r}(M) where

C_{r}(M) = \{λ ∈ I_{1} | λ ≤ \frac{w}{2} and λ ≢ ε(2)\} .
C_{ℓ}(M) = \{λ ∈ I_{1} | λ > \frac{w}{2} and λ ≡ ε(2)\} .

Recall (M.6) that we have attached to M a pair of basic periods c_{0}^{±}(M).

L.6. The following conjectures are due to Deligne [D].

Conjecture 1. Let M be a simple special motive defined over K. Then, for each k ∈ C(M),

L_{0}(M, k) ∼ (1 ⊗ 2πi)^{ke_{α}/n}c_{0}^{α}(M) mod (F^{∗})

where α = (−1)^{k}, F is the center of End(M), [End(M) : F] = n^{2}, and e_{α} = dim_{F} ((R_{K/Q}M)_{B}^{α}).

Conjecture 1 implies the following assertions.

Conjecture 2. Let M be a special motive over K with coefficients in T. Then, for each k ∈ C(M),

L(M, T; k) ∼ (1 ⊗ 2πi)^{kd_{α}}c^{α}(M, T) mod (T^{∗})

where d_{α} = dim_{T} ((R_{K/Q}M)_{B}^{α}).

Conjecture 3. If M is K-simple, and k ∈ C(M),

L(M, T; k) ∼ (1 ⊗ 2πi)^{kd_{α}}c^{α}(M, T) mod ((F^{T})^{∗}).

L.7. Results. Conjecture 2 is known when M = M(χ) is the motive of CM type attached to an algebraic Hecke character χ of a CM field K, by the principal result of [CV]. A method of Harder ([H]) will establish the theorem for general K, by reduction to the former case. See H.12 below. Also, the truth of Conjecture 2, for all T coefficient structures on M implies Conjecture 1 for M, by an easy argument. Thus, all three conjectures are known in this case. Conjecture 2 is also known for the motives attached to classical holomorphic modular forms ([D]) and for the tensor product of two such motives ([Sh1], [BO]). For a partial result in the case of a triple tensor product, see [BO].

L.8. Main conjecture. Let π be an Artin motive with coefficients in T. From Conjecture L.6.2 and Theorem M.16, we obtain:

Conjecture 4. Let k ∈ C(M) ∩ C(M ⊗ π), then

L(M ⊗ π, T; k) ∼ b^{m}(1 ⊗ 2πi)^{kd_{α}(M⊗π)}c_{ε_{π}}^{α}(M)c(δ_{∗}^{α}, det(π)) mod (T^{∗})

where α = (−1)^{k}, where b = b(δ^{±}) unless M_{B}^{w/2,w/2} ≠ {0}, in which case b = b(δ^{αsgn(π)}).

This conjecture is known for Hecke L-series provided dim_{T} π = 1, and results compatible with this conjecture have been obtained by Shimura in [Sh1]. If M is K simple with dim_{T} M > 1, then for all cases in which it is known, δ^{±} is a multiple of the regular representation.

L.9.1. Automorphic L-functions. Fix K and let ∏ be a cuspidal automorphic representation of GL_{N}(𝔸_{K}) whose infinite part π_{∞} = ⊗_{v∈P_{K}}π_{v} is algebraic

in the sense of Clozel ([C]). It is now standard to conjecture (c.f. [C]) that there exists a motive M defined over K, with coefficients in some field T, such that

e_{1}(L_{v}(M, T; s)) = L_{v}(π, s)

for all places v of K; here e_{1} : T ⊗ ℂ → ℂ is the projection determined by 1_{T} ∈ J_{T}. It is thus reasonable to ask for a reformulation of L.8 in purely automorphic terms, invoking additional hypotheses as needed.

L.9.2. Let v ∈ P_{K} and let W_{v} denote the Weil group of K_{v}. To each ∏_{v} is attached a representation

R (π_{v}) : W_{v} → GL_{N}(ℂ)

whose isomorphism class we denote by [R(∏_{v})]; the restriction of R(∏_{v}) to R^{∗} > 0 ⊆ W_{v} is a scalar c ↦ r^{−w} for w ∈ ℤ which is independent of v. For v a complex place of K, let σ ∈ J_{K} be an embedding determining v. Then σ determines an isomorphism

σ^{−1} : ℂ^{∗} →^{∼} K_{v}^{∗}.

We can write

R_{(∏ v)} · σ^{−1} →^{∼} Diag(z^{−a_{1}(σ)}\overline{z}^{−(w−a_{1}(σ))}, …, z^{−a_{N} (σ)}\overline{z}^{−(w−a_{N} (σ))})

with integers a_{i}(σ).

On the other hand, if v = σ is real, the class of R (∏_{v}) ∘ σ^{∼}^{−1} : ℂ^{∗} → GL_{N}(ℂ) is independent of the choice of isomorphism σ^{∼}^{−1} : ℂ →^{∼} \overline{K}_{v}, and R (∏_{v}) ∘ σ^{∼}^{−1} can be diagonalized as above.

L.9.3. Our first task is to define the critical strip C (∏) of ∏. To do this, let

k_{min} = 1 + max_{σ,j} {a_{j}(σ) | a_{j}(σ) < w/2}
d = w − 2k_{min} + 1

and let

C_{1} (∏) = {k_{min}, w − k_{min} + 1}.

If there is a σ and a j for which a_{j}(σ) = \frac{w}{2}, then C (∏) = ∅ unless K is totally real and a signature condition is satisfied: For each 1-dimensional factor χ of R (∏_{v}), the sign χ(j)(= ±1) is independent of v and χ in R (∏_{v}). (Here j ∈ W_{v} satisfies j^{2} = −1.) We denote this sign by sgn(∏) when it is defined.

Now let

C_{ℓ} (∏) = \{k ∈ C_{1} (∏)| k < [\frac{w}{2} ] , sgn (∏) = (−1)^{k+1}\}
C_{r} (∏) = \{k ∈ C_{1} (∏)| k ≥ [\frac{w}{2} ] , sgn (∏) = (−1)^{k}\} .

Then C (∏) = C_{1} (∏) if no a_{j}(σ) = \frac{w}{2}, = C_{1} (∏) ∩ C_{ℓ} (∏) ∩ C_{r} (∏) if some a_{j}(σ) = \frac{w}{2}, K totally real, and sgn(∏) is defined, = ∅ otherwise.

L.9.4. Definition of δ_{∗}^{±} (∏) , d^{±} (∏). Let

d^{±} (∏) = \frac{1}{2} ∑_{v∈P_{K,ℝ}} (N ± tr (R_{(∏ v)} (j))) + ∑_{v∉P_{K,ℝ}} N.

Let δ_{∗}^{±} = d^{±} (∏) ( Σ_{σ∈J_{T}}σ) unless K is totally complex.

If K is totally complex, v ∈ P_{K}, and σ ∈ J_{K} determines v, let f(σ) be the number of indices i (1 ≤ i ≤ N) such that a_{i}(σ) < w − a_{i}(σ), and define a function F : J_{T} × J_{K} → ℤ by F (1, σ) = f(σ) and F (τ|_{T} , τσ) = F (1, σ) for all τ ∈ Gal(\overline{Q}/Q). Finally put

δ^{±}(π) = ∑_{σ∈J_{T}} F (σ, 1) · σ.

L.9.5. Galois forms. Let π be a cuspidal algebraic representation of GL_{m}(𝔸_{K}) such that σ(π_{v}) has finite image for each v in P_{K}. Such a π is called of Galois type since L(π, s) is conjecturally an Artin L-function. Let ω_{π} denote the central character of π.

Define ε_{π} : P_{K} → ℕ by

ε_{π}(v) = m  if v complex
ε_{π}(v) = \frac{1}{2}(m + tr(R(π_{v})(j)))  if v is real.

L.9.6. Tensor product ∏ ⊗π. Let L_{S}(∏ ⊗π, s) denote the usual Rankin product L-function of ∏ and π without Euler factors for v ∈ S, a finite set of places including the infinite ones. Define

d^{±} (∏ ⊗π) = \frac{1}{2} ∑_{v∈P_{K},ℝ} (N_{m} ± trR(π_{v})(j) tr (R \binom{∏}{v} (j))) + ∑_{v∉P_{K},ℝ} N_{m}

and define C (∏ ⊗π) using the tensor product representations R ((∏ ⊗π)_{v}) =^{def} R (∏_{v}) ⊗ R(π_{v}) instead of R (∏_{v}). As before, if some R (∏_{v}) contains an abelian representation, we must have ε(π_{v}) = ±m, with a sign independent of v, if C (∏ ⊗π) ≠ ∅. Let sgn(π) ∈ {±1} be this sign.

L.9.7. Conjugates. Let τ ∈ Aut(ℂ). Then ^{τ} ∏_{f} is defined, where ∏_{f} is the “finite part” of ∏ = ∏_{∞} ⊗ ∏_{f}. If ∏ is algebraic, ∏_{f} should be definable over a finite extension of Q and we let T (∏) be fixed field of H = \{τ ∈ Gal(\overline{Q}/Q) | τ ∏_{j} →^{∼} ∏_{f}\}. Let L_{s}^{∗} (∏, s) be the T ⊗ ℂ valued L-

function such that e_{τ}(L_{s}^{∗}(π, s)) = L_{s} (^{τ} ∏_{f}, s) where e_{τ} : T ⊗ ℂ → ℂ is the projector attached to τ ∈ J_{T}. Similar remarks and definitions apply to the formal product ∏_{f} ⊗π_{f}, and we let now T be any number field containing T (∏) T (π).

L.9.8. Conjecture. There exist quantities c_{v}^{±} (∏) ∈ (T (∏) ⊗ ℂ)^{∗} (v ∈ P_{K}) such that for all cuspidal π of GL_{m}(𝔸_{K}) of Galois type, all k ∈ C (∏) ∩ C (∏ ⊗π), and any finite S containing P_{K},

L_{S}^{∗} (∏ ⊗π, k) ∼ b(δ_{∗}^{αsgn(π)})(1 ⊗ 2πi)^{kd^{α}(∏ ⊗π)}C_{ε_{π}}^{α} (∏) C(δ_{∗}^{α}, w_{π})

mod (T (∏) T (π))^{∗}

where α = (−1)^{k} and

C_{ε_{π}}^{α} (∏) = ( ∏_{v∈P_{K,ℝ}} C_{v}^{α} (∏)^{ε_{π}(v)} C_{v}^{−α} (∏)^{m−ε(v)}) × ( ∏_{v∉P_{K,ℝ}} C_{v}^{α} (∏))^{m} .

H. CM Motives.

H.1. In this chapter, we employ a method of [CV] to derive special period relations which obtain between periods of motives of CM type with respect to T, i.e. dim_{T} M_{B} = 1. Some proofs are only sketched or omitted since they are easy consequences of the methods of [CV] and the earlier results of this paper. Let χ : K_{f}^{∗} → T^{∗} be an (algebraic) Hecke character (cf. [CV], 3.2). Recall that from χ, we construct a motive M = M(χ), over K with coefficients in T, whose T linear isomorphism class depends only upon χ and T. M(χ) is special if and only if, for all α ∈ K^{∗} ⊂ K_{f}^{∗}, putting

(∗)  χ(α) = ∏_{σ∈J_{K}} σ(α)^{n(σ)},

we have n(σ) ≠ n(ρσ) for any σ ∈ J_{K}. As in [CV], (5.1), we say that χ is critical if k = 0 belongs to C(M). Assuming χ is critical, put c^{±}(χ) = c^{±}(M), and L(χ, k) = L(M, T; k). Here L(M, T; s) is simply to T ⊗ C valued series whose components are the Hecke L-series L(ηχ, s) for η ∈ J_{T}. Write L(χ) = L(χ, 0).

H.2. Assume that K is not totally real until H.5. Let K_{CM} ⊆ K be the maximal CM subfield. Define ψ ∈ I_{K} by ψ(σ) = n(σ), with the n(σ) as in H.1. Then ψ ∈ I_{K_{CM}} ⊆ I_{K}. If χ is critical, then either ψ or 1 − ψ (with 1(σ) = 1 for all σ ∈ J_{K}) equals wΦ + μρ − μ where μ, Φ ∈ I_{K_{CM}}, Φ(σ) + Φ(ρσ) = 1, Φ(σ) ≥ 0 and μ(σ) ≥ 0 for all σ in J_{K}, and μ(σ) = 0 unless Φ(σ) = 1. Let E be the field attached to the group H_{Φ} ⊆ G_{Q}, with H_{Φ} as in M.11. Then Φ_{∗} ∈ I_{E} is defined. Let r_{Φ_{∗}} : E_{f}^{∗} → K_{f}^{∗} be defined by r_{Φ_{∗}} = det(Φ_{∗}). For η ∈ J_{E} and τ ∈ G_{Q}, let ε_{ηΦ}(τ) ∈ {1, −1} be the sign of the permutation of < 1, ρ > \J_{K} obtained via the composition

⟨1, ρ⟩\J_{K} \underrightarrow{∼} |ηΦ| \underrightarrow{∼} |τηΦ| \underrightarrow{∼} ⟨1, ρ⟩\J_{K}

where | | denotes support. Note that ε_{Φ} : G_{E} → {1, −1} is a character. For η ∈ J_{T}, let e_{Φ}(η) = (−1)^{t(η)} where t(η) is the number of elements in |ρΦ| ∩ |ηΦ| ⊆ J_{K}. Let χ_{a} : K_{f}^{∗} → T_{f}^{∗} be the map defined by (∗).

H.3. The following result summarizes a basic construction of [CV].

Theorem. Let χ be a critical Hecke character of K whose values on K_{f}^{∗} lie in T. Let N be a motive defined over E with coefficients in T attached to the algebraic Hecke character χ ∘ r_{Φ_{∗}} · ε_{Φ}. Then

a) there exists a collection {γ_{η}^{±} | η ∈ J_{E}, 0 ≠ γ_{η}^{±} ∈ (ηN)_{B}} such that for all η ∈ J_{E}, τ ∈ G_{Q}, and k ∈ K_{f}^{∗} satisfying r_{K}(k) = τ on K_{ab},

τ(I_{f}(γ_{η}^{±})) = χ_{a}(k)^{−1}χ(k)ε_{ηΦ}(τ)e_{Φ}(η)e_{Φ}(τη)I_{f}(γ_{τη}^{±})

this collection is uniquely characterized by this formula up to a change γ_{η}^{±} → tγ_{η}^{±} for a t ∈ T^{∗} which is independent of η ∈ J_{E} and the choice of sign.

b) Put

γ^{±} = ∑_{η∈J_{E}} γ_{η}^{±}

and define F^{∗} = F^{s}(R_{E/Q}N)_{DR} where F^{s} ≠ (R_{E/Q}N)_{DR} but F^{s−1}(R_{E/Q}N)_{DR} = (R_{E/Q}N)_{DR}. Then dim_{T} (R_{E/Q}N)_{DR}/F^{∗} = 1, and if I^{∗} : (R_{E/Q}N)_{B} ⊗ C → ((R_{E/Q}N)_{DR}/F^{∗}) ⊗ C denotes the map constructed from I_{∞}, we have, for 0 ≠ ω ∈ (R_{E/Q}N)_{DR}/F^{∗}

I^{∗}(γ^{±}) ∼ c^{±}(χ) · ω mod (T^{∗}).

Proof. The proof is given, in [CV], 4 and 5, for an analogous result for the dual motive, {γ_{η}^{+} | η ∈ J_{E}}, and where K is a CM field. For the case here, the construction of {γ_{η}^{−} | η ∈ J_{E}} and c^{−}(χ) follows by the same method, if we use elements γ_{σ} − γ_{ρσ} (σ ∈ J_{K}) starting from (5.2.4) of [CV]. For general K, the proof is the same as in [CV], but employs the ε_{ηΦ} where [CV] employed a simpler character ∼ ε of G_{Q}.

H.4.

Proposition. Let M be a motive associated to the χ of Theorem H.3. Then

c^{+}(χ) ∼ e_{Φ} · c^{−}(χ)

where e_{Φ} = \{e_{Φ}(η) | η ∈ J_{E}\} ∈ E ⊗ \overline{Q} ⊆ T ⊗ \overline{Q}.

Proof. From the construction of the γ_{η} out of vectors γ_{σ}, σ ∈ J_{K}, we see at once that γ_{η}^{−} = e_{Φ(η)}γ_{η}^{+}, and the result follows from H.3.

H.5. Let C(χ) = C(M(χ)), the critical strip.

Corollary. For m, n ∈ C(χ),

(1 ⊗ 2πi)^{−mg}L(χ, m) ∼ (1 ⊗ 2πi)^{−ng}L(χ, n)e_{Φ}^{(m−n)},

where g = \frac{1}{2}[K : Q], unless K is totally real, when g = [K : Q].

H.6. If K is totally real, let Φ ∈ I_{K} be the regular representation. Let ε_{Φ} : G_{Q} → {1, −1} be defined by τD_{F}^{1/2} = ε_{Φ}(τ)D_{F}^{1/2}. Let c(K, Φ) ∈ (T ⊗ \overline{Q})^{∗} satisfy τc(K, Φ)_{η} = ε_{ηΦ}(τ)c(K, Φ)_{τη} for all τ ∈ G_{Q} and η ∈ J_{T}.

Proposition. Let 0 < η ∈ Z, and let χ be a critical Hecke character. Then

c^{±}(χ^{n}) ∼ c^{±}(χ)^{n}c(K, Φ)^{n−1} mod (T^{∗}).

Proof. Suppose that K is not totally real. If c^{±}(χ) is defined via {γ_{η}^{±} | η ∈ J_{E}} and ω ∈ (R_{E/Q}N)_{DR}/F^{∗}, as in Theorem H.3, then c^{±}(χ)^{n} is defined using {γ_{η}^{±⊗n} | η ∈ J_{E}} and ω^{⊗n} on R_{E/Q}N^{⊗n}. On the other hand, let N_{n} be the motive attached to χ^{n} as in Theorem H.3. Then N_{n} is attached to the character χ^{n} ∘ r_{Φ_{∗}} · ε_{Φ}, and c^{±}(χ^{n}) is defined via a system {γ_{η}^{±}(n) | η ∈ J_{E}} and w_{n}. The map sending γ_{η}^{±⊗n} to γ_{η}^{±}(n) is a T -linear isomorphism λ^{±} : R_{E/Q}(N^{n}) × \overline{Q} → R_{E/Q}(N_{n}) × \overline{Q}. Since N^{⊗n} is attached to the character χ^{n} ∘ r_{Φ_{∗}} · ε_{Φ}^{n}, λ(ω^{⊗n}) ∼ c(K, Φ)^{n} · ω_{n}, and the claim is proved. If K is totally real, the result is an elementary calculation.

H.7.

Corollary. Let χ be a critical Hecke chracter and let 0 < n ∈ Z. Then

L(χ^{n}) ∼ L(χ)^{n}c(K, Φ)^{n−1} mod (T^{∗}).

H.8.

Proposition. Let χ be a critical Hecke character of K. Suppose that L is a finite extension of K for which χ ∘ N_{L/K} is critical. Then

c^{±}(χ ∘ N_{L/K}) ∼ c^{±}(χ)^{[L:K]}c(Φ, π_{L/K})

with π_{L/K} : G_{K} → {1, −1} as in M.17.

Proof. The proposition just restates M.17, Corollary 3 and Corollary 6, since these are the only possible cases. We use here that Φ = δ_{∗}^{±}.

See M.17., below Corollary 3, for a discussion of c(Φ, π_{L/K}).

H.9.

Corollary. Let χ be a critical Hecke character of K, and assume that Conjecture L.6.2 holds for M(χ) and M(χ ∘ N_{L/K}), with a finite extension L of K for which χ ∘ N_{L/K} is critical. Then

L(χ ∘ N_{L/K}) ∼ L(χ)^{[L:K]}c(Φ, π_{L/K}).

H.10. Now let L be a subfield of K such that χ_{a} ∈ I_{K} lies in the image of I_{L}. Then:

i) If χ is critical, χ|_{L_{f}^{∗}} is also critical.

ii) The following formula holds:

c^{±}(χ) ∼ c^{±}(χ|_{L_{f}^{∗}})c(K, Φ)c(L, ϕ) mod (T^{∗}).

Proof. If K is complex, then, we compute c^{±}(χ) and c^{±}(χ|_{L_{f}^{∗}}) by means of Theorem H.3. as periods of motives R_{E/Q}(M(χ ∘ r_{Φ_{∗}}ε_{Φ})) and R_{E/Q}(M(χ ∘ r_{Φ_{∗}} · ε_{Φ|_{L}})), respectively, where ε_{Φ|_{L}} is the character obtained by regarding Φ as an element of I_{L}. The proof now concludes as in the proof of Proposition H.6. If K is totally real, then the result follows easily from the identity c^{±}(χ) ∼ c^{±}(χ ∘ tr_{K/Q} · π_{K/Q}) with π_{K/Q} as in M.17.

H.11. Remark. Let K_{CM} = L be the maximal CM subfield of K, in the case where K is totally complex. Then c(K_{CM}, Φ) ∼ 1 ⊗ D_{F}^{1/2} mod (T^{∗}), where F is the maximal real subfield of L. It is not hard to check that

c(K, Φ) ∼ (1 ⊗ D_{F}^{1/2})^{[K:L]}β mod (T^{∗})

with β = {β_{η} | η ∈ J_{T} } and

β_{η} = ∏_{σ∈J_{F}} β_{η}(σ)
β_{η}(σ) = det(β_{i}^{τ})_{1≤i≤[K:L],τ∈S_{η,σ}}

for a basis β_{1}, …, β_{[K:L]} of K over L, and where

S_{η,σ} = {τ ∈ |ηΦ| such that τ|_{F} = σ}

and is ordered by first imposing an order upon ⟨1, ρ⟩\J_{K}, and ordering |ηΦ| via its image in this set.

H.12.

Theorem. Let χ be a critical Hecke character of a totally complex field K with maximal CM subfield L. Let T be the field generated by the values of χ on K_{f}^{∗}. Suppose that [K : L] > 1. Then Conjecture L.6.2 is true for M(χ) if and only if

L(χ) ∼ L(χ|_{L_{f}^{∗}})(1 ⊗ D_{F}^{1/2})^{([K:L]−1)}β mod (T^{∗})

with the notations introduced above.

Proof. The hypothesis ensures that L(χ|_{L_{f}^{∗}}) ∈ (T ⊗ C)^{∗}. Since Theorem 9.3.1 of [CV] establishes L.6.2 for χ|_{L_{f}^{∗}}, the theorem follows from H.10 and H.11.

H.13. It appears likely that the method of Harder ([H]) will establish H.12. The paper [H] treats the case where [K : L] = 2.

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University of California Los Angeles, CA 90095-1555 E-mail address: blasius@math.ucla.edu

Note: Corollary 3 in Section M.17 was missing from the paper version. Also, the references there to L.5.2 should be to L.6.2.