On the parity of ranks of Selmer groups

Jan Nekovář and Andrew Plater

0. Introduction

Let f = ∑_{n≥1} a_{n}(f)q^{n} ∈ S_{k_{0}} (Γ_{0}(N)) be a normalized newform of even weight k_{0} ≥ 2. Let F be the number field generated by the coefficients of f and 𝔭 a prime of F lying above a rational prime p. There is a two-dimensional representation V (f) of G_{ℚ} = Gal (\overline{ℚ}/ℚ) over F_{𝔭} associated to f, characterized by the conditions

Tr (Fr(ℓ)_{geom}|V (f)) = a_{ℓ}(f)
det (Fr(ℓ)_{geom}|V (f)) = ℓ^{k_{0}−1}

for all primes ℓ ∤ pN. The Tate twist V_{k_{0}} = V (f)(k_{0}/2) is self dual: there is a skew-symmetric bilinear form

V_{k_{0}} × V_{k_{0}} ⟶ F_{𝔭}(1)

inducing an isomorphism V_{k_{0}} ⟶^{∼} V_{k_{0}}^{∗}(1) = Hom_{F_{𝔭}} (V_{k_{0}}, F_{𝔭}(1)).

The complex L-function L_{∞}(f, s) = ∑_{n≥1} a_{n}(f)n^{−s} satisfies the functional equation

Λ_{∞}(f, s) := (\frac{\sqrt{N}}{2π} )^{s} Γ(s)L_{∞}(f, s) = w_{∞}(f)Λ_{∞}(f, k_{0} − s),

where w_{∞}(f) = ±1 = (−1)^{e_{∞}} for e_{∞} = 0 or 1. Bloch and Kato [Bl-Ka] defined a generalized “Selmer group” H_{f}^{1}(ℚ, V_{k_{0}}) ⊆ H^{1}(ℚ, V_{k_{0}}) and conjectured that

ord_{s=k_{0}/2}L_{∞}(f, s) =^{?} dim_{F_{𝔭}}H_{f}^{1}(ℚ, V_{k_{0}}).

We are interested in a (mod 2) version of this conjecture:

The Parity Conjecture for ranks of Selmer groups

ord_{s=k_{0}/2}L_{∞}(f, s) ≡^{?} dim_{F_{𝔭}}H_{f}^{1}(ℚ, V_{k_{0}}) (mod 2).

Assume that p > 3 and that f is ordinary at p, i.e. that a_{p}(f) ∈ F_{𝔭} is a 𝔭-adic unit. According to Hida’s theory, there is a p-adic family of ordinary modular forms of varying weights containing f (we ignore the phenomenon of “p-stabilization” in this Introduction). In concrete terms, this means that there is an integer c ≥ 0 such that for every integer k ≥ 2 satisfying k ≡ k_{0} (mod (p − 1)p^{c}), there is an ordinary newform f_{k} of weight k on Γ_{0}(N) such that f_{k_{0}} = f and

k ≡ k′ (mod (p − 1)p^{n+c}) implies f_{k} ≡ f_{k′} (mod p^{n}).

Let

ε_{k} = \{ \begin{matrix}1 & if p ∥ N, k = 2, a_{p}(f) = 1, \\ 0 & otherwise. \end{matrix}

The value of ε_{k} indicates the presence of a trivial zero of a suitable p-adic L-function [Ma-Ta-Te]. More precisely, for every weight k ≥ 2, k ≡ k_{0} (mod (p − 1)p^{c}), there is a p-adic L-function L_{p}(f_{k}, s) depending on s ∈ ℤ_{p} such that

L_{p}(f_{k}, k/2) ≐ (Eul_{k})\frac{L_{∞}(f_{k}, k/2)}{Ω_{k}}, L_{p}(f_{k}, k − s) ≐ w_{p}(f_{k})L_{p}(f_{k}, s).

A.P. was supported by an EPSRC grant Here, ≐ means equal up to a non-zero elementary factor, Ω_{k} ∈ ℝ^{×} is a real period of f_{k}, (Eul_{k}) is an Euler factor satisfying

(Eul_{k}) = 0 ⇔ ε_{k} = 1

and

w_{p}(f_{k}) = ±1 = w_{∞}(f_{k})(−1)^{ε_{k}}.

Moreover, there exists a two-variable p-adic L-function L_{p}(k, s) defined for s ∈ ℤ_{p} and k ∈ k_{0} + p^{c}ℤ_{p}, such that L_{p}(k, k − s) = w_{p}L_{p}(k, s), and, if k ≡ k_{0} (mod 2(p − 1)p^{c}), k ≥ 2 is an integer, then

L_{p}(k, s) = C_{k}L_{p}(f_{k}, s) for some C_{k} ≠ 0.

In particular, the value of

w_{p}(f_{k}) = w_{∞}(f_{k})(−1)^{ε_{k}} = w_{p} = (−1)^{e_{p}}

does not depend on k (of course, e_{p} ≡ e_{∞} + ε_{k_{0}} (mod 2)). We shall consider Selmer groups H_{f}^{1}(ℚ, V_{k}) associated to Galois representations V_{k} = V (f_{k})(k/2) and also “extended Selmer groups” sitting in exact sequences

0 ⟶ (F_{𝔭})^{ε_{k}} ⟶ H̃_{f}^{1}(ℚ, V_{k}) ⟶ H_{f}^{1}(ℚ, V_{k}) ⟶ 0

(for technical reasons, it may be necessary to replace F_{𝔭} by a suitable finite extension and V_{k_{0}} by the corresponding base change). Our main result is

Theorem A. Let p > 3 and let f be ordinary at p. If k_{0} ≡ 2 (mod (p − 1)), assume in addition that V (f) has an irreducible residual representation. Then there is an integer n ≥ c such that

dim_{F_{𝔭}} H̃_{f}^{1}(ℚ, V_{k}) ≡ dim_{F_{𝔭}} H̃_{f}^{1}(ℚ, V_{k_{0}}) (mod 2)

whenever k ≥ 2 and k ≡ k_{0} (mod 2(p − 1)p^{n}).

It is known that

(⋆)  ord_{s=k/2}L_{p}(f_{k}, s) = 0 ⇒ H_{f}^{1}(ℚ, V_{k}) = 0  ([Ka])
ord_{s=k/2}L_{p}(f_{k}, s) = 1 ⇒ H_{f}^{1}(ℚ, V_{k}) = 1  for p ∤ N, k > 2  ([Ne 2])

Recall a fundamental non-vanishing conjecture for the two-variable p-adic L-function.

Greenberg’s Conjecture. The generic order of vanishing of L_{p}(k, s) on the line s = k/2 is equal to zero or one (and hence to e_{p}).

In other words, the function of k

\frac{L_{p}(k, s)}{(s − k/2)^{e_{p}}} |_{s=k/2}

should not be identically zero. Theorem A and (⋆) immediately imply

Theorem B. Under the assumptions of Theorem A, Greenberg’s Conjecture implies the parity conjecture

dim_{F_{𝔭}}H_{f}^{1}(ℚ, V_{k_{0}}) ≡ e_{p} − ε_{k_{0}} ≡ e_{∞} (mod 2).

Consider the special case of k_{0} = 2 and F = ℚ. The form f then corresponds to (the isogeny class of) a modular elliptic curve E over ℚ. The Selmer group associated to V_{k_{0}} = T_{p}E ⊗_{ℤ_{p}} ℚ_{p} coincides with the usual Selmer group with ℚ_{p}-coefficients; it sits in an exact sequence

0 ⟶ E(ℚ) ⊗ ℚ_{p} ⟶ H_{f}^{1}(ℚ, V_{k_{0}}) ⟶ T_{p}� (E/ℚ) ⊗_{ℤ_{p}} ℚ_{p} ⟶ 0.

In this case, Theorem B reduces to Theorem C. Let E be a modular elliptic curve over ℚ with ordinary reduction at a prime p > 3. Assume that the p-torsion E_{p}(\overline{ℚ}) is an irreducible 𝔽_{p}[G_{ℚ}]-module and that Greenberg’s conjecture holds for the two-variable p-adic L-function of E. Then

dim_{ℚ} (E(ℚ) ⊗ ℚ) + cork_{ℤ_{p}}� (E/ℚ) ≡ ord_{s=1}L_{∞}(E, s) (mod 2).

See [Gr 3], [Ko 2] (resp. [Bi-St], [Gu], [Ko 1], [Mo]) for other conditional (resp. unconditional) results in this direction. At present, Greenberg’s conjecture is known only if E has complex multiplication and e_{p} = 0 ([Gr 1], [Ro]).

There are several possible approaches to Theorem A, all of which use the existence of a “big Galois representation” T that interpolates suitable Galois invariant lattices T_{k} ⊂ V_{k}. In the most elementary approach, one studies a version of H̃_{f}^{1} for the discrete modules A_{k} = V_{k}/T_{k}. It is relatively easy to show that the p^{n}-torsion subgroup H̃_{f}^{1}(ℚ, A_{k})_{p^{n}} ⊆ H̃_{f}^{1}(ℚ, A_{k}) is locally constant as a function of k. The existence of generalized Cassels-Tate pairings on H̃_{f}^{1}(ℚ, A_{k})/div then gives the desired parity result.

In fact, there is a stronger version of Theorem A. Classical modular forms in (the primitive part of) Hida’s family containing f are parametrized by certain prime ideals P in a suitable factor of the ordinary Hecke algebra. Denote by f_{P} the form corresponding to P; it has weight k(P) ≥ 2 and character χ_{P}. For k(P) even there is a well-defined choice of a square root χ_{P}^{1/2} of χ_{P}. The twisted modular form f̃_{P} = f_{P} ⊗ χ_{P}^{−1/2} has trivial character (i.e. lies in S_{k(P)}(Γ_{0}(Np^{r(P)}) for some r(P)). The Galois representation

V_{[P]} = V (f_{P})(k(P)/2) ⊗ [χ_{P}^{−1/2}] = V (f̃_{P})(k(P)/2)

is two-dimensional over F_{𝔭}(χ_{P}) and self dual in the same way as V_{k_{0}}.

Theorem A′. Under the assumptions of Theorem A, the parity of dim_{F_{𝔭}(χ_{P})}H̃_{f}^{1}(ℚ, V_{[P]}) does not depend of P.

Together with (⋆), this implies

Theorem B′. Let p > 3 be a prime not dividing N and f = ∑_{n≥1} a_{n}q^{n} an ordinary newform of even weight k ≥ 2 and character χ^{−2} on Γ_{1}(Np^{r}), where cond(χ) = p^{s} (necessarily with s = r, if χ ≠ 1). Then f̃ = ∑_{n≥1} a_{n}χ(n)q^{n} is a newform of weight k on Γ_{0}(Np^{2r}) (resp. Γ_{0}(Np^{r})) if χ ≠ 1 (resp. χ = 1). If the tame part of χ^{2} is equal to ω^{k−2}, where ω is the Teichmüller character, assume that V (f) has an irreducible residual representation. If Greenberg’s Conjecture holds for the two-variable p-adic L-function of f, then

dim_{F_{𝔭}}H_{f}^{1}(ℚ, V (f̃)(k/2)) ≡ ord_{s=k/2}L_{∞}(f̃, s) (mod 2).

The proof of Theorem A′ uses “big Selmer groups” associated to T and a suitable big discrete module A. The ultimate explanation of this parity phenomenon relies on the general duality formalism developed in [Ne 3]. What happens is that each Selmer group H̃_{f}^{1}(ℚ, A_{k}) contains a “generic subgroup” H̃_{f}^{1}(ℚ, A_{k})^{gen} that is constant in the whole Hida family. The duality results of [Ne 3] give a symplectic form on the Pontryagin dual of H̃_{f}^{1}(ℚ, A_{k})/H̃_{f}^{1}(ℚ, A_{k})^{gen}, tensored with ℚ (i.e. on the infinite part of the Selmer group!). This can be viewed as a non-classical generalization of the Cassels-Tate pairing to big Selmer groups. Assuming Greenberg’s conjecture, H̃_{f}^{1}(ℚ, A_{k})^{gen} has co-rank equal to e_{p}. In the special case when the local component of the Hecke algebra corresponding to the Hida family in question is equal to the Iwasawa algebra, there is a more elementary argument that uses big Selmer groups introduced in [Pl].

This work was inspired by a lecture of R. Greenberg [Gr 3]. We are grateful to K. Buzzard, F. Diamond, R. Taylor, J. Tilouine and A. Wiles for dispelling some – but certainly not all – of our misconceptions concerning Hecke algebras.

1. Modular forms and Galois representations

(1.1) Modular curves and modular forms

In this section we recall basic notation and normalizations concerning modular curves and modular forms, following the conventions of [KaN] and [Gro].

(1.1.1) For an integer N > 4, X_{1}(N) ⟶ Spec(ℤ[1/N]) is the complete modular curve classifying pairs (E, α : μ_{N} ↪ E_{N}), where E is a generalized elliptic curve ([De-Ra]) and the image of α meets every irreducible component in each geometric fibre of E. We assume that N > 4 from now on.

(1.1.2) For a generalized elliptic curve π : E ⟶ S, let \underline{ω}_{E} be the invertible sheaf \underline{Lie}(E^{reg})^{∨} on S (\underline{ω}_{E} = π_{∗}Ω_{E/S}^{1} if π is smooth). For any ℤ[1/N]-algebra R, the space of holomorphic modular forms of weight k ≥ 1 on Γ_{1}(N) defined over R is, by definition, equal to H^{0} (X_{1}(N)_{/R}, \underline{ω}^{⊗k}). One can interpret a modular form f ∈ H^{0} (X_{1}(N)_{/R}, \underline{ω}^{⊗k}) as a rule assigning an element f(E, α) ∈ \underline{ω}_{E}^{⊗k} to each pair (E, α : μ_{N} ↪ E_{N}) defined over an R-algebra R′, compatible with base change.

(1.1.3) The Tate curve E = 𝔾_{m}/q^{ℤ} is an elliptic curve over ℤ[[q]][q^{−1}] which extends to a generalized elliptic curve over ℤ[[q]]. The exact sequence

0 ⟶ μ_{N} ⟶^{Id_{N}}E_{N} ⟶ ℤ/Nℤ ⟶ 0

gives the tautological embedding Id_{N} : μ_{N} ↪ E_{N}. Evaluation of a modular form f ∈ H^{0} (X_{1}(N)_{/R}, \underline{ω}^{⊗k}) on the Tate curve gives the Fourier expansion F (f) ∈ R[[q]] of f :

f (𝔾_{m}/q^{ℤ}, Id_{N}) = F (f)(q) (\frac{dt}{t})^{⊗k}

where t is the coordinate on 𝔾_{m}. The map F : H^{0} (X_{1}(N)_{/R}, \underline{ω}^{⊗k}) ⟶ R[[q]] is injective ([KaN, 1.6.1]).

(1.1.4) If S ⟶ T is smooth and π : E ⟶ S is an elliptic curve, then the Gauss-Manin connection

∇ : ℝ^{1}π_{∗}ΩE/S^{•} ⟶ (ℝ^{1}π_{∗}ΩE/S^{•}) ⊗ Ω_{S/T}^{1}

and relative Poincaré duality

⟨ , ⟩_{dR} : (ℝ^{1}π_{∗}ΩE/S^{•}) × (ℝ^{1}π_{∗}ΩE/S^{•}) ⟶ ℝ^{2}π_{∗}ΩE/S^{•} ∼^{⟶}^{Tr}O_{S}

define the Kodaira-Spencer map

\begin{matrix}i : \underline{ω}_{E}^{⊗2} & ⟶ Ω_{S/T}^{1} \\ ω ⊗ ν & ↦ ⟨ω, ∇ν⟩_{dR} \end{matrix}

For the Tate curve, i ((dt/t)^{⊗2}) = dq/q. In fact, i extends to an isomorphism of invertible sheaves on X_{1}(N),

\underline{ω}^{⊗2} ⟶^{∼} Ω_{X_{1}(N)}^{1}(cusps).

(1.2) Hecke operators

(1.2.1) For a ∈ (ℤ/Nℤ)^{×}, the diamond operator < a > acts on X_{1}(N) by < a > (E, α) = (E, a · α) and on modular forms by

(< a >_{Alb} f) (E, α) = f(E, a · α),

(the covariant, or “Albanese,” action), and also by

(< a >_{Pic} f) (E, α) = f(E, a^{−1} · α) = (< a^{−1} >_{Alb} f) (E, α)

(the contravariant, or “Picard,” action). If χ : (ℤ/Nℤ)^{×} ⟶ R^{×} satisfies χ(−1) = (−1)^{k}, a modular form f ∈ H^{0} (X_{1}(N)_{/R}, \underline{ω}^{⊗k}) has character χ if < a >_{Alb} f = χ(a)f for all a ∈ (ℤ/Nℤ)^{×}.

(1.2.2) For integers n ≥ 1 with (n, N) = 1, the Hecke operators T (n) act on modular forms by

(T (n)_{Alb}f) (E, α) = \frac{1}{n} ∑_{{λ:E⟶E′ \atop deg(λ)=n}} λ^{∗}(f(E′, λ∘α))

(T (n)_{Pic}f) (E, α) = \frac{1}{n} ∑_{{μ:E′⟶E \atop deg(μ)=n}} μ_{∗}(f(E′, μ^{−1}∘α))

If n = ℓ ∤ N is a prime and

F (f)(q) = ∑_{n≥0} a_{n}q^{n}, F (< ℓ >_{Alb} f) (q) = ∑_{n≥0} b_{n}q^{n},

then a short calculation gives

(T (ℓ)_{Alb}f) (𝔾_{m}/q^{ℤ}, Id_{N}) = \frac{1}{ℓ}f (𝔾_{m}/q^{ℓℤ}, ℓ · Id_{N}) (ℓ\frac{dt}{t})^{⊗k} + \frac{1}{ℓ} ∑_{q′^{ℓ}=q} f (𝔾_{m}/q′^{ℤ}, Id_{N}) (\frac{dt}{t})^{⊗k}

= (ℓ^{k−1} ∑_{n≥0} b_{n}q^{nℓ} + ∑_{n≥0} a_{nℓ}q^{n}) (\frac{dt}{t})^{⊗k}

= (T (ℓ)_{Pic}f) (𝔾_{m}/q^{ℤ}, ℓ · Id_{N}) .

This means that the usual Hecke operator T (ℓ) on Fourier expansions of modular forms corresponds to T (ℓ)_{Alb} =< ℓ >_{Alb} T (ℓ)_{Pic}. Of course, if f has character χ, then b_{n} = χ(n)a_{n} for all n ≥ 1 prime to N.

(1.2.3) For a prime ℓ | N, one defines

(T (ℓ)_{Alb}f) (E, α) = \frac{1}{ℓ} ∑_{λ:E→E′,deg(λ)=ℓ, Ker(λ)∩Im(α)=0} λ^{∗}(f(E′, λ∘α))

and T (ℓ^{m})_{Alb} = T (ℓ)_{Alb}^{m}. The same calculation as in 1.2.2 shows that

(T (ℓ)_{Alb}f) (𝔾_{m}/q^{ℤ}, Id_{N}) = (∑_{n≥0} a_{nℓ}q^{n}) (\frac{dt}{t})^{⊗k} .

One defines T (n)_{Alb}f for any n ≥ 1 by requiring T (mn)_{Alb} = T (m)_{Alb}T (n)_{Alb} whenever (m, n) = 1.

(1.2.4) A more geometric definition of T (ℓ) comes from Hecke correspondences. For a prime ℓ, let X_{1}(N; ℓ) be the curve over Spec(ℤ[1/N]) classifying triples (E, α : μ_{N} ↪ E_{N}, C), where (E, α) is as in 1.1.1 and C ⊂ E_{ℓ} is a locally free subgroup scheme of rank ℓ such that Im(α) ∩ C = 0 and Im(α) × C meets every irreducible component of each geometric fibre of E. There are finite maps s, t : X_{1}(N; ℓ) ⟶ X_{1}(N) given over the affine curve Y_{1}(N) = X_{1}(N)−{cusps} by

(E, α) ⟵^{s} (E, α, C) ⟼^{t} (E′ = E/C, α′ = λ∘α),

where λ : E ⟶ E/C is the degree ℓ isogeny associated to C. The maps s, t define a correspondence

X_{1}(N; ℓ)

[figure: s; t]

X_{1}(N)  \overline{T (ℓ)} X_{1}(N)

which acts on various cohomology groups, both covariantly: T (ℓ)_{Alb} = t_{∗}∘s^{∗} (the “Albanese action”) and contravariantly: T (ℓ)_{Pic} = s_{∗}∘t^{∗} (the “Picard action”).

(1.2.5) Similarly, the diamond operators < a >: X_{1}(N) ⟶ X_{1}(N), a ∈ (ℤ/Nℤ)^{×}, act on cohomology by

< a >_{Alb}=< a >_{∗}, < a >_{Pic}=< a >^{∗}=< a^{−1} >_{∗}=< a^{−1} >_{Alb} .

(1.2.6) For modular forms of weight two, the Kodaira-Spencer isomorphism

i : H^{0} (X_{1}(N), \underline{ω}^{⊗2}) ⟶^{∼} H^{0} (X_{1}(N), Ω_{X_{1}(N)}^{1}(cusps))

is compatible with the actions of T (l)_{Alb} and < a >_{Alb} on both sides (1.2.2 and 1.2.3 on the left hand side, 1.2.4 and 1.2.5 on the right hand side). The factor 1/n in the formulas of 1.2.2 comes from the equality

⟨λ^{∗}(ν), λ^{∗}(∇ν′)⟩_{dR}^{E} = deg(λ)⟨ν, ∇ν′⟩_{dR}^{E′}

valid for any isogeny λ : E ⟶ E′.

(1.2.7) There are two definitions of the Weil pairing

e_{N} = e_{N,E} : E_{N} × E_{N} ⟶ μ_{N}

which differ by a sign. We use the one normalized by e_{N,E}(ζ, q^{1/N}) = ζ for all ζ ∈ μ_{N} and all q^{1/N} for the Tate curve E = 𝔾_{m}/q^{ℤ}.

(1.2.8) For a primitive N-th root of unity ζ ∈ μ_{N}, the Fricke involution W_{ζ} : X_{1}(N) ⟶ X_{1}(N) (defined over ℤ[1/N, μ_{N}]) is given on Y_{1}(N) by W_{ζ}(E, α : μ_{N} ↪ E_{N}) = (E′, β_{ζ}), where E′ = E/Im(α) and β_{ζ} : μ_{N} ↪ E′ is characterized by

e_{N,E}(α(ζ), any lift of β_{ζ}(ζ) to E_{N}) = ζ.

For a ∈ (ℤ/Nℤ)^{×}, one has β_{ζ}(ζ) = β_{ζ^{a}}(ζ^{a}) = aβ_{ζ^{a}}(ζ). Hence,

(1.2.8.1)  W_{ζ} =< a > ∘W_{ζ^{a}},  W_{ζ} ∘ < a >=< a >^{−1} ∘W_{ζ} = W_{ζ^{a}},  as W_{ζ}^{2} = id.

If g ∈ Gal(ℚ(μ_{N})/ℚ) acts on μ_{N} by ζ ↦ ζ^{a}, then

(1.2.8.2)  g∘W_{ζ} = W_{ζ^{a}} ∘g = W_{ζ} ∘ < a > ∘g.

For every prime ℓ, there is an equality of correspondences on X_{1}(N), W_{ζ} ∘T (ℓ) = T (ℓ)^{t}∘W_{ζ}. this implies that the induced action on cohomology satisfies

(1.2.8.3)  T (ℓ)_{Pic} = W_{ζ} ∘T (ℓ)_{Alb}∘W_{ζ}

for all primes ℓ.

(1.2.9) For integers d ≥ 1, M, N > 4 such that dM | N, there are finite degeneration maps π_{d} : X_{1}(N) ⟶ X_{1}(M) given on Y_{1}(M) by

π_{d}(E, α : μ_{N} ↪ E_{N}) = (E′ = E/α(μ_{d}), α′ : μ_{M} ↪ E′_{M})

with

α′ : μ_{M} ↪ μ_{N/d} ∼^{⟵}^{d} μ_{N}/μ_{d} ↪^{α mod μ_{d}} E/α(μ_{d}) = E′.

In particular,

π_{1}(E, α : μ_{N} ↪ E_{N}) = (E, α |_{μ_{M}})

and

π_{d} (𝔾_{m}/q^{ℤ}, Id_{N}) = (𝔾_{m}/(q^{d})^{ℤ}, Id_{M}) .

If ζ_{N} ∈ μ_{N} is a primitive N-th root of unity and ζ_{M} = ζ_{N}^{N/M}, then π_{d}∘W_{ζ_{N}} = W_{ζ_{M}} ∘π_{N/dM}.

(1.2.10) We now consider the induced action on cohomology. π_{1}^{∗} commutes with < a >_{Alb} (and therefore with < a >_{Pic} too) for a ∈ (ℤ/Nℤ)^{×}, and commutes with T (n)_{Alb} (and T (n)_{Pic} too) for (n, N) = 1, and with T (ℓ)_{Alb} for primes ℓ | M. The trace operator π_{1∗} commutes with < a >_{Alb} (and hence with < a >_{Pic}) for a ∈ (ℤ/Nℤ)^{×}, and with T (n)_{Alb} (and hence with T (n)_{Pic}) for (n, N) = 1. Moreover, if ℓ | M is a prime, then both squares in the commutative diagram

X_{1}(N) ⟵^{s} X_{1}(N; ℓ) ⟶^{t} X_{1}(N)
↓^{π_{1}}  ↓^{π_{1}}  ↓^{π_{1}}
X_{1}(M) ⟵^{s} X_{1}(M; ℓ) ⟶^{t} X_{1}(M)

are cartesian, which implies that s^{∗}∘π_{1∗} = π_{1∗}∘s^{∗}, t^{∗}∘π_{1∗} = π_{1∗}∘t^{∗}, and

T (ℓ)_{Pic}∘π_{1∗} = s_{∗}∘t^{∗}∘π_{1∗} = s_{∗}∘π_{1∗}∘t^{∗} = π_{1∗}∘s_{∗}∘t^{∗} = π_{1∗}∘T (ℓ)_{Pic}
T (ℓ)_{Alb}∘π_{1∗} = t_{∗}∘s^{∗}∘π_{1∗} = t_{∗}∘π_{1∗}∘s^{∗} = π_{1∗}∘t_{∗}∘s^{∗} = π_{1∗}∘T (ℓ)_{Alb}.

In particular, if N = ℓM for a prime ℓ | M, then π_{1∗} commutes with both T (ℓ)_{Alb} and T (ℓ)_{Pic}, and π_{ℓ∗} = W_{ζ_{M}} ∘π_{1∗}∘W_{ζ_{N}} commutes with both T (ℓ)_{Pic} = W_{ζ} ∘T (ℓ)_{Alb}∘W_{ζ} and T (ℓ)_{Alb} = W_{ζ} ∘T (ℓ)_{Pic}∘W_{ζ}.

(1.2.11) The Eichler-Shimura relation on X_{1}(N) defined as in 1.1.1 has the following form: for every prime ℓ ∤ N,

T (ℓ) ≡ Fr(ℓ)^{t}+ < ℓ > Fr(ℓ) (mod ℓ)

as a correspondence on X_{1}(N)_{/𝔽_{ℓ}}. This implies that the induced contravariant action on étale cohomology H^{1}(\overline{X_{1}(N)}_{et}, ℤ_{p}) (where p ≠ ℓ and \overline{X_{1}(N)} = X_{1}(N) ⊗_{ℚ} \overline{ℚ}) satisfies

T (ℓ)_{Pic} = ℓ · Fr(ℓ)_{geom}^{−1}+ < ℓ >_{Pic} Fr(ℓ)_{geom}.

Consequently, Fr(ℓ)_{geom} is a root of X^{2} − T (ℓ)_{Alb}X + ℓ < ℓ >_{Alb}= 0.

(1.3) Galois representations

Fix algebraic closures \overline{ℚ} (resp. \overline{ℚ}_{p}) of ℚ (resp. ℚ_{p}), and embeddings i_{∞} : \overline{ℚ} ↪ ℂ, i_{p} : \overline{ℚ} ↪ \overline{ℚ}_{p} (for a given prime p).

(1.3.1) Let f = ∑_{n≥1} a_{n}q^{n} ∈ S_{k} (Γ_{1}(N), χ) be a cusp form on Γ_{1}(N) of weight k ≥ 2 and character χ : (ℤ/Nℤ)^{×} → ℂ^{×}, defined over ℂ. It is a normalized eigenform if a_{1} = 1 and T (ℓ)_{Alb}f = λ_{ℓ}f for all primes ℓ (necessarily with λ_{ℓ} = a_{ℓ}). A normalized newform is a normalized eigenform such that the set of eigenvalues {λ_{ℓ} : ℓ ∤ N} does not occur for any eigenform of weight k on Γ_{1}(M), for any proper divisor M | N.

(1.3.2) Assume that f from 1.3.1 is a normalized newform. Let F be a finite extension of ℚ in \overline{ℚ} containing all i_{∞}^{−1}(a_{n}) and all values of i_{∞}^{−1}∘χ. The embedding i_{p} induces on F a prime 𝔭 above p. Let S = {primes ℓ : ℓ ∤ pN} ∪ {∞}. The p-adic Galois representation associated to f

ρ = ρ_{f,𝔭} : G_{ℚ,S} ⟶ GL_{2}(F_{𝔭})

(where G_{ℚ,S} denotes the Galois group with restricted ramification; cf. 2.1.1 below) is characterized by the conditions

Tr (ρ(Fr(ℓ)_{geom})) = i_{p}(a_{ℓ})
det (ρ(Fr(ℓ)_{geom})) = i_{p}(χ(ℓ)ℓ^{k−1})

for all ℓ ∉ S. It was constructed in [Ei],[Sh] for k = 2 and [De] for k > 2. Ribet [Ri 1, Thm. 2.3] showed that ρ is irreducible (hence absolutely irreducible, by the same argument as in 1.5.3(3) below).

(1.3.3) Scholl [Sc] constructed a (Grothendieck) motive M = M(f) associated to f, pure of weight k − 1 and of rank two over F. Geometrically, M ⊂ h^{k−1}(Z) ⊗ F, where Z is a suitable smooth compactification of the (k − 1)-dimensional Kuga-Sato variety over Y (N) (at least for N ≥ 3). The p-adic étale realization M_{p} ⊂ H^{k−1}(\overline{Z}_{et}, F ⊗ ℚ_{p}) of M is free of rank two over F ⊗ ℚ_{p} = ∏_{v|p} F_{v}; its 𝔭-component M_{𝔭} gives the representation ρ_{f,𝔭}. By [La], [Ca] the L-series of the motive M agrees with

L_{∞}(f, s) = ∑_{n≥1} a_{n}n^{−s} = ∏_{ℓ|N}(1 − a_{ℓ}ℓ^{−s})^{−1} ∏_{ℓ∤N}(1 − a_{ℓ}ℓ^{−s} + χ(ℓ)ℓ^{k−1−2s})^{−1}

even at Euler factors at bad primes ℓ | N.

(1.3.4) A suitably twisted Poincaré duality on Z gives a non-degenerate skew-symmetric pairing

M_{𝔭} × M_{𝔭} ⟶ M_{𝔭} ⊗_{F_{𝔭}} M_{𝔭} ⟶ F_{𝔭}(1 − k) ⊗_{F_{𝔭}} [χ],

where [χ] is the one-dimensional representation of G_{ℚ,S} over F_{𝔭} given by [χ](Fr(ℓ)_{geom}) = χ(ℓ), (ℓ ∤ pN). In the special case when k is even and χ = 1, then V = M_{𝔭}(k/2) is pure of weight −1 and the above pairing defines a non-degenerate skew-symmetric pairing

V × V ⟶ V ⊗_{F_{𝔭}} V ⟶ F_{𝔭}(1),

which induces an isomorphism V ⟶^{∼} V^{∗}(1) = Hom_{F_{𝔭}}(V, F_{𝔭}(1)).

(1.3.5) A normalized eigenform f from 1.3.1 is ordinary (with respect to i_{∞}, i_{p}) if i_{p}(a_{p}) ∈ F_{𝔭} is a 𝔭-adic unit. In particular, a_{p} ≠ 0, which implies ([Mi, Thm. 4.6.17]) that either

(i) ord_{p}(N) = ord_{p}(cond(χ)) or (ii) p ∥ N, k = 2, p ∤ cond(χ), a_{p}^{2} = χ(p).

Furthermore, the Galois representation M_{𝔭} restricted to G_{ℚ_{p}} = Gal(\overline{ℚ}_{p}/ℚ_{p}) is reducible [Wi 1, Thm. 2.2.2] and there is an exact sequence of F_{𝔭}[G_{ℚ_{p}}]-modules

(1.3.5.1)  0 ⟶ F^{+}M_{𝔭} ⟶ M_{𝔭} ⟶ F^{−}M_{𝔭} ⟶ 0

with dim(F^{±}M_{𝔭}) = 1 and F^{+}M_{𝔭} unramified. The quadratic equation X^{2} − a_{p}X + χ(p)p^{k−1} = 0 has two distinct roots in F_{𝔭}: one of them, α_{p}, is a 𝔭-adic unit and the other is

β_{p} = \{ \begin{matrix}\frac{χ(p)p^{k−1}}{α_{p}} & if p ∤ N \\ 0 & if p | N. \end{matrix}

The geometric Frobenius Fr(p)_{geom} ∈ G_{ℚ_{p}}/I_{p} acts on F^{+}M_{𝔭} by the scalar α_{p}. The duality 1.3.4 gives

(1.3.5.1)  F^{−}M_{𝔭} ⟶^{∼} (F^{+}M_{𝔭})^{∗}(1 − k) ⊗ [χ]

(as F_{𝔭}[G_{ℚ_{p}}]-modules).

(1.3.6) If p ∤ N, then f^{0} = ∑_{n≥1} a_{n}q^{n} − β_{p} ∑_{n≥1} a_{n}q^{pn} is a normalized eigenform on Γ_{1}(Np) satisfying

T (ℓ)_{Alb}f^{0} = a_{ℓ}f^{0}  if ℓ ∤ p,  T (p)_{Alb}f^{0} = α_{p}f^{0}.

One says that f^{0} is the p-stabilization of f. By abuse of language, we shall call ρ_{f,𝔭} the Galois representation associated to f^{0}.

(1.3.7) Assume that p ∤ N. A normalized eigenform f of weight k ≥ 2 on Γ_{1}(Np^{n}) is said to be an ordinary p-stabilized newform of tame level N if f is ordinary, n ≥ 1, and f is new at N, i.e. the set of eigenvalues {a_{ℓ} : ℓ ∤ Np} does not occur for any newform of weight k on Γ_{1}(Mp^{m}) for any proper divisor M of N and m ≤ n. Equivalently, f is ordinary and either a newform on Γ_{1}(Np^{r}) for some r ≥ 1, or is equal to the p-stabilization of a newform on Γ_{1}(N).

(1.4) Hida’s theory

Assume that p > 3. Fix a finite extension F_{𝔭} of ℚ_{p} (in \overline{ℚ}_{p}); let O = O_{𝔭} be its ring of integers. Fix an integer N ≥ 1 not divisible by p.

(1.4.1) Hecke algebras For k ≥ 2, and r ≥ 1, let 𝔥_{k}(Γ_{1}(Np^{r})) be the subring of End (S_{k}(Γ_{1}(Np^{r}))) generated (over ℤ) by the Hecke operators T (n) = T (n)_{Alb}, n ≥ 1, and the diamond operators < a >=< a >_{k}, (a ∈ (ℤ/Nℤ)^{×}). Put 𝔥_{k,r} := 𝔥_{k}(Γ_{1}(Np^{r})) ⊗_{ℤ} O. This is free of finite rank over O; diamond operators give an O-algebra homomorphism

< >_{k}: O[(ℤ/Nℤ)^{×}] ⟶ 𝔥_{k,r}.

For fixed k ≥ 2 and s ≥ r ≥ 1, there are canonical homomorphisms 𝔥_{k,s} → 𝔥_{k,r}, given by T (n) ↦ T (n), < a >_{k}↦< a >_{k} (these are dual to the maps π_{1}^{∗} from 1.2.10 acting on cusp forms of weight k). The projective limit

𝔥_{k,∞} := \underleftarrow{lim}_{r} 𝔥_{k,r}

is equipped with morphisms

< >_{k}: O[[ℤ_{p}^{×}]] ↪ O[[Z_{N}]] ⟶ 𝔥_{k,∞},

where Z_{N} = \underleftarrow{lim}_{r}(ℤ/Np^{r}ℤ)^{×} = ℤ_{p}^{×} × (ℤ/Nℤ)^{×} and O[[Z_{N}]] = \underleftarrow{lim}_{r} O[(ℤ/Np^{r}ℤ)^{×}]. Put Γ = \{x ∈ ℤ_{p}^{×} : x ≡ 1

(mod p)} and denote the canonical inclusion Γ ↪ O[[Γ]] ↪ O[[Z_{N}]] by ı.

(1.4.2) Ordinary projector e For each 1 ≤ r ≤ ∞, there is a decomposition 𝔥_{k,r} = 𝔥_{k,r}^{ord} × 𝔥_{k,r}^{ss} such that 𝔥_{k,r}^{ord} (resp. 𝔥_{k,r}^{ss}) is the largest quotient of 𝔥_{k,r} on which T (p)_{Alb} is invertible (resp. T (p)_{Alb} is topologically nilpotent). More precisely,

𝔥_{k,r}^{ord} = e𝔥_{k,r}, 𝔥_{k,r}^{ss} = (1 − e)𝔥_{k,r},

for Hida’s ordinary projector

e = lim_{n→∞} T (p)_{Alb}^{n!} ∈ 𝔥_{k,r}.

One defines the ordinary part of any 𝔥_{k,r}-module M to be M^{ord} = eM = 𝔥_{k,r}^{ord} ⊗_{𝔥_{k,r}} M. There is a bijection between the set of ordinary normalized eigenforms of weight k ≥ 2 on Γ_{1}(Np^{r}) (r ≥ 1) and O-algebra homomorphisms 𝔥_{k,r}^{ord} ⟶ \overline{ℚ}_{p} (given by Hecke eigenvalues).

(1.4.3) Proposition. (i) [Hi 1, Thm. 1.1] For l ≥ k ≥ 2 there are canonical isomorphisms 𝔥_{l,∞}^{ord} ⟶^{∼} 𝔥_{k,∞}^{ord}. We use them to identify all 𝔥_{k,∞}^{ord} (k ≥ 2) with 𝔥_{∞}^{ord} = 𝔥_{2,∞}^{ord}. (ii) [Hi 2, Thm. 3.1] Consider 𝔥_{∞}^{ord} as an O[[Z_{N}]]-algebra via diamond operators < >_{2} acting on weight two cusp forms. Then 𝔥_{∞}^{ord} is finite and free over Λ = O[[Γ]]. (iii) [Hi 1, Thm. 1.2] The canonical maps 𝔥_{∞}^{ord} ⟵^{∼} 𝔥_{k,∞}^{ord} → 𝔥_{k,r}^{ord} induce isomorphisms

𝔥_{∞}^{ord}/ω_{k,r}𝔥_{∞}^{ord} ⟶^{∼} 𝔥_{k,r}^{ord}, (k ≥ 2, r ≥ 1)

where ω_{k,r} = ı(γ)^{p^{r−1}} − γ^{(k−2)p^{r−1}} for any fixed topological generator γ of Γ.

(1.4.4) Decomposition of 𝔥_{∞}^{ord} It follows from Prop. 1.4.3 (ii) that 𝔥_{∞}^{ord} = ∏ R is a product of local rings, finite and free over Λ. The local factors R are localizations of 𝔥_{∞}^{ord} at its maximal prime ideals. They are not necessarily integral domains; to get a further decomposition one must introduce denominators. We shall be interested only in the primitive part; let L be the fraction field of Λ. Hida [Hi 2, p. 250, 252] constructed an idempotent e_{prim} ∈ 𝔥_{∞}^{ord} ⊗_{Λ} L such that e_{prim} (𝔥_{∞}^{ord} ⊗_{Λ} L) = ∏ K is a product of fields (finite extensions of L). Making a finite extension of F_{𝔭} if necessary, one may assume the F_{𝔭} is equal to the algebraic closure of ℚ_{p} in K (i.e. “K is defined over F_{𝔭}” in the terminology of [Hi 2, p.252]) for each K. As in [Hi 1, p.554], fix one of the factors K, and put

𝔥(K) = the image of 𝔥_{∞}^{ord} in K.
∩
𝔥̃(K) = the free Λ-closure of 𝔥(K) in K.
∩
J (K) = the normalization of Λ in K.

Equivalently, 𝔥̃(K) is the intersection ⋂_{P} 𝔥(K)_{P} ⊂ 𝔥(K) ⊗_{Λ} L, where P runs through height one prime ideals of Λ. Denote

𝔥^{prim} := ∏_{K} 𝔥(K) ⊂ J = ∏_{K} J (K).

(1.4.5) Fix a topological generator γ of Γ. For an integer k ≥ 2 and a character ε : Γ → O^{×} of finite order, we put

P_{k,ε} = ı(γ) − ε(γ)γ^{k−2} ∈ Λ.

We define an arithmatic point of any finite Λ-algebra A to be a prime ideal 𝔭 ∈ Spec(A) lying above some prime ideal P = (P_{k,ε}) ∈ Spec(Λ). The set of arithmetic points of A will be denoted by 𝔛^{arith}(A).

(1.4.6) Proposition. (i) [Hi 1] The “restriction map” Spec(J) ⟶ Spec(𝔥^{prim}) gives a bijection 𝔛^{arith}(J)

⟶^{∼} 𝔛^{arith}(𝔥^{prim}).

(ii) [Hi 1, Cor. 1.4] For every P ∈ 𝔛^{arith}(𝔥^{prim}) and the corresponding P′ ∈ 𝔛^{arith}(J) above P = (P_{k,ε}) ∈ Spec(Λ), the localization (𝔥^{prim})_{P} = J_{P′} is a discrete valuation ring unramified over Λ_{P}. (iii) [Hi 1, Cor. 1.3] For every arithmetic point P ∈ 𝔛^{arith}(𝔥^{prim}), the map O ↪ 𝔥^{prim} ⟶ 𝔥^{prim}/P is an isomorphism and the homomorphism

𝔥_{∞}^{ord} ⟶ 𝔥^{prim} ⟶ 𝔥^{prim}/P ⟵^{∼} O

corresponds to an ordinary p-stabilized newform f of tame level N. Conversely, every f arises in this way, for a unique P ∈ 𝔛^{arith}(𝔥^{prim}).

Remark. (i) follows from the fact that

𝔥(K) ⊗_{Λ} Λ_{P} = 𝔥̃(K) ⊗_{Λ} Λ_{P} = J (K) ⊗_{Λ} Λ_{P}

for every P = (P_{k,ε}). The first equality holds because 𝔥̃(K)/𝔥(K) is a pseudo-null Λ-module. The second follows from the proof of [Hi 1, Cor.1.4].

(1.4.7) In the notation of 1.3, let f be an ordinary p-stabilized newform on Γ_{1}(Np^{r}) of weight k_{0} ≥ 2 and character χ : (ℤ/Np^{r}ℤ)^{×} → O^{×}. The homomorphism 𝔥_{∞}^{ord} → O corresponding to f factors through a unique local factor R of 𝔥_{∞}^{ord} and through a unique simple factor K of e_{prim}(R ⊗_{Λ} L). Replacing F_{𝔭} by a finite extension if necessary, we may assume that K is defined over F_{𝔭}, in the language of 1.4.4. Write e_{prim}(R ⊗_{Λ} L) = K × A, and define 𝔥(A) to be the image of R in A, and 𝔥̃(A) the free Λ-closure of 𝔥(A) in A. Hida [Hi 2, p.253] defines the congruence module C = C(K) by the exact sequence

0 ⟶ R ⟶ 𝔥̃(K) ⊕ 𝔥̃(A) ⟶ C ⟶ 0.

It is shown in [Hi 2, Thm. 3.6, Cor. 3.8] that C is a torsion Λ-module and C/P_{k,ε}C is finite (for all integers k ≥ 2 and characters of finite order ε : Γ → O^{×}). Denote by ω : (ℤ/pℤ)^{×} → ℤ_{p}^{×} the Teichmüller character and decompose χω^{k_{0}−2} = ψε into its tame, ψ : (ℤ/Npℤ)^{×} → O^{×}, and wild, ε : Γ → Γ/Γ^{p^{r−1}} → O^{×}, parts. Denote by \overline{P} ∈ 𝔛^{arith}(𝔥(K)) the arithmetic point corresponding to f (lying above P = (P_{k_{0},ε}) ∈ Spec(Λ)) and by P ∈ 𝔛^{arith}(R) its preimage in R. Localizing the exact sequence

0 ⟶ R ⟶ 𝔥(K) ⊕ 𝔥(A) ⟶ C′ ⟶ 0

(where C′ ⊂ C is a subgroup of finite index) at P, we get

R_{P} ⟶^{∼} 𝔥(K)\overline{P} ⊕ 𝔥(A)_{P}.

As 𝔥(K)\overline{P} ≠ 0, this implies that 𝔥(A)_{P} = 0 (which also follows from the multiplicity one statement of Prop. 1.4.6(iii)), hence R_{P} ⟶^{∼} 𝔥(K)\overline{P} is a discrete valuation ring, unramified over Λ_{P}. One obtains in this way an embedding

𝔥(K) ↪ F_{c} = { ∑_{i≥0} u_{i}(x − k_{0})^{i} : u_{i} ∈ F_{𝔭}, lim_{i→∞} ord_{p}(u_{i}) + ci = +∞} ⊆ F_{𝔭}[[x − k_{0}]]

for a suitable integer c ≥ 0 (cf. [Gr-St, 2.7]). The composite map

Z_{N} ↪ O[[Z_{N}]] ⟶ 𝔥(K) ↪ F_{𝔭}[[x − k_{0}]]

is equal to

t ⟼ ψ(t)ε(t)κ(t)^{x−2} = χ(t)ω(t)^{k_{0}−2}κ(t)^{x−2},

where κ is the projection to the group of principal units 1 + pℤ_{p}. For k ∈ k_{0} + p^{c}ℤ_{p}, the evaluation map “x = k” ev_{k} : F_{c} ⟶ F_{𝔭} is well defined. If k ≡ k′ (mod p^{n+c}) for n ≥ 0, then ev_{k} ≡ ev_{k′} (mod p^{n}). For every integer k ≥ 2 satisfying k ≡ k_{0} (mod p^{c}),

P_{k} := Ker (𝔥(K) ↪ F_{c}⟶^{ev_{k}}F_{𝔭})

is an arithmetic point of 𝔥(K) above (P_{k,ε}). It corresponds to a p-stabilized newform f_{k} on Γ_{1}(Np^{r}) of weight k and character χω^{k_{0}−k}. If k′ ≥ 2 is an integer satisfying k ≡ k′ (mod p^{n+c}) (for some n ≥ 0), then the two morphisms

𝔥(K) ⟶ 𝔥(K)/P_{k} = O, 𝔥(K) ⟶ 𝔥(K)/P_{k′} = O

are congruent (mod p^{n}), i.e. f_{k} ≡ f_{k′} (mod p^{n}).

(1.5) Big Galois representations The assumptions are as in 1.4 (but we do not assume that F_{𝔭} is “big enough” as in 1.4.4–7).

(1.5.1) Denote by J_{1}(M) = Pic^{0}(X_{1}(M)) the Jacobian of the modular curve X_{1}(M)_{/ℚ} (for M > 4). The degeneracy maps π_{1} : X_{1}(Np^{r+1}) → X_{1}(Np^{r}) induce maps on p-primary torsion

π_{1}^{∗} : J_{1}(Np^{r})(\overline{ℚ})_{p^{∞}} ⟶ J_{1}(Np^{r+1})(\overline{ℚ})_{p^{∞}}.

The inductive limit

J_{∞} = \underrightarrow{lim}_{π_{1}^{∗}} (J_{1}(Np^{r})(\overline{ℚ})_{p^{∞}}) ⊗_{ℤ_{p}} O

is an 𝔥_{2,∞}-module; denote by J_{∞}^{ord} = eJ_{∞} its ordinary part, which is a module over 𝔥_{∞}^{ord}. Fix one of the local factors R of 𝔥_{∞}^{ord} and denote by e_{R} ∈ 𝔥_{∞}^{ord} the corresponding idempotent. The “big Galois representation” we are most interested in is

T (R) = Hom_{O} (e_{R}J_{∞}^{ord}, μ_{p^{∞}} ⊗_{ℤ_{p}} O) .

It is an R-module of finite type with a continuous R-linear action of G_{ℚ,S} (with S = {ℓ : ℓ | Np} ∪ {∞}).

(1.5.2) Proposition. (i) (Eichler-Shimura relation) For every prime ℓ ∤ Np, the relation

T (ℓ)_{Alb} = Fr(ℓ)_{geom}^{−1} + ℓ < ℓ >_{Alb} Fr(ℓ)_{geom}

holds in End_{R}(T (R)). (ii) [Ma-Ti, Thm. 7] If T (R) has an irreducible residual representation, then R is a Gorenstein ring and T (R) is free of rank two over R. (iii) [Ti 1, Sect.4] Let a ∈ ℤ/(p−1)ℤ be the exponent such that (ℤ/pℤ)^{×} ⊂ Z_{N} acts on R by ω^{a}. If a ≠ 0, −1 (mod (p − 1)), then there is a canonical exact sequence of R[G_{ℚ_{p}}]-modules

0 ⟶ T (R)^{+} ⟶ T (R) ⟶ T (R)^{−} ⟶ 0

such that T (R)^{+} ⟶^{∼} R and T (R)^{−} ⟶^{∼} ω_{R} := Hom_{Λ}(R, Λ) as R-modules, and the inertia subgroup I_{p} ⊂ G_{ℚ_{p}} acts trivially on T (R)^{−} and by χ_{cycl} < χ_{cycl} >_{Alb} on T (R)^{+}, where

χ_{cycl} : G_{ℚ_{p}} → Gal(ℚ_{p}(μ_{p^{∞}})/ℚ_{p}) ⟶^{∼} ℤ_{p}^{×}

is the cyclotomic character.

(1.5.3) A few remarks are in order.

(1) As H^{1}(\overline{X_{1}(Np^{r})}_{et}, μ_{p^{∞}}) = J_{1}(Np^{r})(\overline{ℚ})_{p^{∞}}, the statement (i) follows from the Eichler-Shimura relation 1.2.11.

(2) A residual representation of T (R) is a representation ρ : G_{ℚ,S} → GL_{2}(R/𝔪), (where 𝔪 is the maximal ideal of R) such that the characteristic polynomial of \overline{ρ}(Fr(ℓ)_{geom}^{−1}) is equal to X^{2} − T (ℓ)_{Alb}X + ℓ < ℓ >_{Alb} (mod 𝔪) for all primes ℓ ∤ Np.

(3) \overline{ρ} is irreducible if and only if \overline{ρ} is absolutely irreducible, because \overline{ρ}(complex conjugation) has two distinct eigenvalues ±1 ∈ R/𝔪 (recall that char(R/𝔪) = p ≠ 2).

(4) As R is finite and flat over the regular ring Λ, it is Cohen-Macaulay and ω_{R} = Hom_{Λ}(R, Λ) is a dualizing module of R [Br-He, Thm. 3.3.7(b)]. In particular, R is Gorenstein if and only if ω_{R} is free of rank one over R.

(5) The statement (iii) of Proposition 1.5.2 probably holds also for a ≡ 0 (mod (p − 1)).

(1.5.4) Proposition. In the notation of Proposition 1.5.2 (iii), assume that a ≡ 0 (mod (p − 1)) and that T (R) admits an irreducible residual representation. Then the conclusions of Proposition 1.5.2 (iii) hold, with both T (R)^{±} free of rank one over R.

Proof. By Prop. 1.5.2 (ii), T (R) is free of rank two over R. By a version of 1.3.5.1 over R/𝔪 ([Ti 2, Thm. 3.2]), the inertia subgroup I_{p} ⊂ G_{ℚ_{p}} acts on \overline{ρ} by

( \begin{matrix}ω^{1+a} & ∗ \\ 0 & 1 \end{matrix} )

As a ≠ −1 (mod (p−1)), we have ω^{1+a} ≠ 1, which means that \overline{ρ} is “distinguished” in the sense of [Ti 2, Def. 3.3], [Wi 2, p. 481]. Fix r ≥ 1 and consider the quotient T (R)/ω_{2,r}T (R) as a module over R_{r} = R/ω_{2,r}R. It follows from [Ti 2, Theorem 3.4] and its proof and [Wi 2, Thm. 2.1, Cor. 1] that there is an exact sequence of R_{r}[G_{ℚ_{p}}]-modules

0 ⟶ F_{r}^{+} ⟶ T (R)/ω_{2,r}T (R) ⟶ F_{r}^{−} ⟶ 0

with F_{r}^{±} free of rank one over R_{r} and I_{p} acting on F_{r}^{±} as in Proposition 1.5.2 (iii) (freeness of F_{r}^{±} follows from Nakayama’s lemma, because it holds (mod 𝔪). As ω^{1+a} ≠ 1, F_{r+1}^{±} map to F_{r}^{±} under the canonical maps T (R)/ω_{2,r+1}T (R) → T (R)/ω_{2,r}T (R), and hence we get in the limit the desired sequence

0 ⟶ \underleftarrow{lim}_{r} F_{r}^{+} ⟶ T (R) ⟶ \underleftarrow{lim}_{r} F_{r}^{−} ⟶ 0.

(1.5.5) Assume that a ≠ −1 (mod (p − 1)) and that T (R) has an irreducible residual representation. Let P ∈ 𝔛^{arith}(R) be an arithmetic point corresponding to an ordinary eigenform g of weight k on Γ_{1}(Np^{r}). It is not necessarily a newform, but there is a unique newform f of weight k and character χ on Γ_{1}(M) for some M | Np^{r} with the same set of Hecke eigenvalues {a_{ℓ} : ℓ ∤ Np}. The quotient T (R)/PT (R) is free of rank two over O′ = R/P. Tensoring with the fraction field F′_{𝔭} of O′ we get a two-dimensional representation of G_{ℚ,S} over F′_{𝔭} such that Fr(ℓ)_{geom}^{−1} satisfies the equation

X^{2} − a_{ℓ}X + ℓ^{k−1}χ(ℓ) = 0

for all primes ℓ ∤ Np. By the Čebotarev density theorem and irreducibility of ρ_{f,𝔭}, it follows that the Galois representation (T (R)/PT (R)) ⊗_{O′} F′_{𝔭} is isomorphic to the base change − ⊗_{F_{𝔭}} F′_{𝔭} of the dual of ρ_{f,𝔭}, namely

ρ_{f,𝔭}^{∗} ≃ ρ_{f,𝔭}(k − 1) ⊗ [χ^{−1}].

In the notation of 1.3.5, (T (R)^{±}/PT (R)^{±}) ⊗_{O′} F′_{𝔭} corresponds to F^{±}M_{𝔭}(k − 1) ⊗ [χ^{−1}].

(1.5.6) If a ≠ 0, −1 (mod (p − 1)), but without any assumptions on a residual representation, let P ∈ 𝔛^{arith}(R) correspond to a p-stabilized newform f. As in 1.4.6–7, there is a unique simple component K of R ⊗_{Λ} L through which R → R/P factors. Denote by \overline{P} ∈ 𝔛^{arith}(𝔥(K)) the arithmetic point of 𝔥(K) corresponding to f. Then P = pr^{−1}(\overline{P}), where pr : R → 𝔥(K) is the canonical projection. For every pair of R-modules M, N one has

Tor_{i}^{R}(M, N)_{P} ≃ Tor_{i}^{R_{P}} (M_{P}, N_{P}).

In particular,

Supp(Tor_{i}^{R}(R/P, ω_{R})) ⊆ Supp(R/P) = {P, 𝔪}.

On the other hand, R_{P} is a discrete valuation ring (by 1.4.7), and hence (ω_{R})_{P} ⟶^{∼} ω_{R_{P}} = R_{P} and

Tor_{i}^{R}(R/P, ω_{R})_{P} = \{ \begin{matrix}k(P) & if i = 0 \\ 0 & if i > 0. \end{matrix}

It follows that Tor_{i}^{R}(R/P, ω_{R}) is finite for i > 0 and ω_{R}/Pω_{R} ⟶^{∼} R/P ⊕ (finite). In the exact sequence

Tor_{1}^{R}(R/P, T (R)^{−})⟶^{∂}T (R)^{+}/PT (R)^{+} ⟶ T (R)/PT (R) ⟶ T (R)^{−}/PT (R)^{−} ⟶ 0

the second term is free over R/P, hence flat over O. The first term being O-torsion by the previous discussion, we see that the map ∂ must be zero. As a result, we get an exact sequence

0 ⟶ T (R)^{+}/PT (R)^{+} ⟶ T (R)/PT (R) ⟶ T (R)^{−}/PT (R)^{−} ⟶ 0,

in which T (R)^{+}/PT (R)^{+} is free of rank one over R/P = O′ and T (R)^{−}/PT (R)^{−} is the sum of a free rank one R/P-module and a finite group. As in 1.5.5, the Galois representation (T (R)/PT (R)) ⊗_{O′} F′_{𝔭} is isomorphic to the base change of ρ_{f,𝔭}^{∗}, where f is the p-stabilized newform corresponding to P. Similarly, T (R)^{±}/PT (R)^{±} ⊗_{O′} F′_{𝔭} corresponds to F^{±}M_{𝔭}(k − 1) ⊗ [χ^{−1}]. (1.6) Self duality of T (R) The results of this section are well known, but we were unable to find a good reference that would cover all of them. The notation is as in 1.4. (1.6.1) For every integer r ≥ 1, put Δ_{r} = (ℤ/Np^{r}ℤ)^{×}, and

X_{r} := H^{1}(\overline{X_{1}(Np^{r})}_{et}, O) = H^{1}(\overline{X_{1}(Np^{r})}_{et}, ℤ_{p}) ⊗_{ℤ_{p}} O = T_{p}(J_{1}(Np^{r}))(−1) ⊗_{ℤ_{p}} O.

Poincaré duality gives an O-bilinear skew-symmetric perfect pairing

⟨ , ⟩_{P,r} : X_{r} × X_{r} ⟶ H_{et}^{2}(\overline{X_{1}(Np^{r})}_{et}, O) ∼^{⟶}^{Tr}O(−1),

which induces an isomorphism

\begin{matrix}X_{r} & ⟶^{∼} Hom_{O}(X_{r}, O)(−1) \\ x & ⟼ (y ↦ ⟨x, y⟩_{P,r}) \end{matrix}

of O[G_{ℚ}]-modules. In other words, the Galois action satisfies

⟨g(x), g(y)⟩_{P,r} = g(⟨x, y⟩_{P,r}) = χ_{cycl}^{−1}(g)⟨x, y⟩_{P,r},

where g ∈ G_{ℚ}, x, y ∈ X_{r}, and χ_{cycl} : G_{ℚ} → Gal(ℚ(μ_{p^{∞}})/ℚ) ⟶^{∼} ℤ_{p}^{×} is the cyclotomic character given by the action on μ_{p^{∞}}. In this section, we denote by

χ_{cycl,N} : G_{ℚ} ⟶ Gal(ℚ(μ_{Np^{∞}})/ℚ) ⟶^{∼} Δ_{∞} = \underleftarrow{lim}_{r} Δ_{r}

the character taking into account also the action on μ_{N}. (1.6.2) The pairing ⟨ , ⟩_{P,r} is functorial as follows: Hecke operators:

⟨T (n)_{Alb}(x), y⟩_{P,r} = ⟨x, T (n)_{Pic}(y)⟩_{P,r}  (n ≥ 1),
(1.6.2.1) ⟨< a >_{Alb} (x), y⟩_{P,r} = ⟨x, < a >_{Pic} (y)⟩_{P,r} (a ∈ Δ_{r}).

Degeneration maps: π_{1}, π_{p} : X_{1}(Np^{r+1}) → X_{1}(Np^{r}) satisfy ⟨π_{i∗}(x), y⟩_{P,r} = ⟨x, π_{i}^{∗}(y)⟩_{P,r+1}. Fricke involutions: Fix a compatible system of primitive Np^{r}-th roots of unity ζ_{Np^{r}} ∈ μ_{Np^{r}}, ζ_{Np^{r+1}}^{p} = ζ_{Np^{r}}, and put W_{r} := W_{ζ_{Np^{r}}}. Then ⟨W_{r}(x), W_{r}(y)⟩_{P,r} = ⟨x, y⟩_{P,r}. (1.6.3) We define a twisted pairing (depending on the choice of ζ_{Np^{r}})

\begin{matrix}( , )_{r} : X_{r} × X_{r} & ⟶ O(−1), \\ (x, y)_{r} & = ⟨x, W_{r}(y)⟩_{P,r} \end{matrix}

This is again skew-symmetric and O-bilinear. The formulas (1.6.2.1) and (1.2.8.1,3) imply

(T (n)_{Alb}(x), y)_{r} = ⟨x, T (n)_{Pic}W_{r}(y)⟩_{P,r} = ⟨x, W_{r}T (n)_{Alb}(y)⟩_{P,r}
(1.6.3.1) = (x, T (n)_{Alb}(y))_{r} (n ≥ 1)

and (< a >_{Alb} (x), y)_{r} = (x, < a >_{Alb} (y))_{r}, for a ∈ Δ_{r}. For g ∈ G_{ℚ}, we have (by 1.6.1 and (1.2.8.2))

χ_{cycl}^{−1}(g)(x, y)_{r} = χ_{cycl}^{−1}(g)⟨x, W_{r}(y)⟩_{P,r} = ⟨g(x), gW_{r}(y)⟩_{P,r}
= ⟨g(x), W_{r} < χ_{cycl,N}(g) >_{Alb} g(y)⟩_{P,r} = (g(x), < χ_{cycl,N}(g) >_{Alb} g(y))_{r}.

Hence,

(1.6.3.2)  (g(x), g(y))_{r} = χ_{cycl}^{−1}(g)(x, < χ_{cycl,N}(g) >_{Alb}^{−1} y)_{r}.

The Hecke algebra 𝔥_{2,r} = 𝔥(Γ_{1}(Np^{r}), O) acts on X_{r} by T (n)_{Alb} and < a >_{Alb}. The formulas (1.6.3.1–2) can be reformulated by saying that the map

\begin{matrix}α_{r} : X_{r} & ⟶^{∼} Hom_{O}(X_{r}, O)(−1) \\ x & ⟼ (y ↦ (x, y)_{r})) \end{matrix}

induces an isomorphism

(1.6.3.3)  α_{r} : X_{r} ⟶^{∼} Hom_{O}(X_{r}, O)(−1) < −1 >

of 𝔥_{2,r}[G_{ℚ}]-modules. Here, the notation Y < n >, for an 𝔥_{2,r}[G_{ℚ}]-module Y and n ∈ ℤ, means that the original Galois action by g ∈ G_{ℚ} on Y is multiplied by the action of < χ_{cycl,N}(g) >_{Alb}^{n}∈ 𝔥_{2,r}.

Taking the limit r → ∞

We know from 1.2.10 that the maps X_{r+1}⟶^{π_{i∗}}X_{r} and X_{r+1}⟵^{π_{i}^{∗}}X_{r} (i = 1, p) are compatible (for r ≥ 1) with the canonical maps 𝔥_{2,r+1} → 𝔥_{2,r}. Note that, by 1.2.9,

(π_{i∗}(x), y)_{r} = ⟨π_{i∗}(x), W_{r}(y)⟩_{P,r} = ⟨x, π_{i}^{∗}W_{r}(y)⟩_{P,r+1}
(1.6.3.4)  = ⟨x, W_{r+1}π_{p/i}^{∗}(y)⟩_{P,r+1} = (x, π_{p/i}^{∗}(y))_{r+1}.

This means that the isomorphisms α_{r} induce in the limit an isomorphism of 𝔥_{2,∞}[G_{ℚ}]-modules

α_{∞} : \underleftarrow{lim}_{π_{p∗}} X_{r} ⟶^{∼} Hom_{O}(\underrightarrow{lim}_{π_{1}^{∗}}(X_{r} ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p}), O ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p})(−1) < −1 >

(1.6.3.5)  = Hom_{O}(J_{∞}, O ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p}) < −1 >

(this is because the right hand side of (1.6.3.3) is canonically isomorphic to Hom_{O}(X_{r} ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p}, O ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p})(−1) < −1 >).

(1.6.4) Lemma. Let O be a commutative ring, Δ a finite group and X a left O[Δ]-module. Let f ∈ Hom_{O}(X, O), F ∈ Hom_{O[Δ]−left}(X, O[Δ]). Then

(i) The formulas (f ∗ a)(x) = f(ax), (F ∗ a)(x) = F (x)[a] (a ∈ Δ, x ∈ X) define right actions of Δ on Hom_{O}(X, O) and Hom_{O[Δ]−left}(X, O[Δ]). (ii) the formulas

(Φ(f))(x) = ∑_{a∈Δ} f(a(x))[a^{−1}], (Ψ(F))(x) = pr_{e}(F (x)),

define mutually inverse isomorphisms of right O[Δ]-modules

Hom_{O}(X, O) \underrightarrow{Φ} ; \overleftarrow{Ψ} Hom_{O[Δ]−left}(X, O[Δ])

(here, pr_{b}(∑_{a∈Δ} n_{a}[a]) = n_{b} for b ∈ Δ, and e ∈ Δ is the identity element).

Proof. Straightforward calculation.

If Δ is commutative, the formulas in (i) can be used to define a left action of Δ : a ∗ f := f ∗ a, a ∗ F := F ∗ a, and (ii) will be an isomorphism of left O[Δ]-modules.

(1.6.5) Applying Lemma 1.6.4 to X = X_{r} and Δ = Δ_{r}, the formula defining Φ gives yet another skewsymmetric pairing, this time with values in the group ring O[Δ_{r}] :

⟨ , ⟩_{Δ_{r}} : X_{r} × X_{r} ⟶ O[Δ_{r}]

⟨x, y⟩_{Δ_{r}} = ∑_{a∈Δ_{r}}(x, < a >_{Alb} y)_{r}[a]^{−1}.

It satisfies

⟨T (n)_{Alb}(x), y⟩_{Δ_{r}} = ⟨x, T (n)_{Alb}(y)⟩_{Δ_{r}}  (n ≥ 1)
⟨< a >_{Alb} (x), y⟩_{Δ_{r}} = ⟨x, < a >_{Alb} (y)⟩_{Δ_{r}} = [a]⟨x, y⟩_{Δ_{r}}  (a ∈ Δ_{r})
⟨g(x), g(y)⟩_{Δ_{r}} = χ_{cycl}(g)^{−1}[χ_{cycl,N}(g)^{−1}]⟨x, y⟩_{Δ_{r}}  (g ∈ G_{ℚ}).

the induced map

β_{r} : X_{r} ⟶^{∼} Hom_{O[Δ_{r}]}(X_{r}, O[Δ_{r}])(−1) < −1 >
(1.6.5.1)  x ⟼ (y ↦ ⟨x, y⟩_{Δ_{r}})

is again an isomorphism of 𝔥_{2,r}[G_{ℚ}]-modules, equal to Φ∘α_{r}. The same is true of the projection of β_{r} to the ordinary component:

β_{r}^{ord} : eX_{r} ⟶^{∼} Hom_{O[Δ_{r}]}(eX_{r}, O[Δ_{r}])(−1) < −1 >
(1.6.5.2)  x ⟼ (y ↦ ⟨x, y⟩_{Δ_{r}})

This is an isomorphism of 𝔥_{2,r}^{ord}[G_{ℚ}]-modules.

(1.6.6) Lemma. Fix r ≥ 1. Then

(i) π_{1∗}(eX_{r+1}) ⊆ p(eX_{r}). Denote by \frac{1}{p}π_{1∗} : eX_{r+1} → eX_{r} the unique map satisfying p · (\frac{1}{p}π_{1∗}) = π_{1∗}. (ii) (\frac{1}{p}π_{1∗})∘π_{1}^{∗} = p on eX_{r}.

(iii) π_{1}^{∗}∘(\frac{1}{p}π_{1∗}) = ∑_{a∈Γ_{r}/Γ_{r+1}} < a >_{Alb} on eX_{r+1}, where Γ_{r} = Γ^{p^{r−1}}. (iv) T (p)_{Alb}∘(\frac{1}{p}π_{1∗}) = π_{p∗} on eX_{r+1}.

Proof. See 1.6.11 below.

(1.6.7) Corollary. For x, y ∈ eX_{r+1}, (r ≥ 1), the canonical projection O[Δ_{r+1}] → O[Δ_{r}] maps ⟨x, y⟩_{Δ_{r+1}} to ⟨π_{p∗}(x), \frac{1}{p}π_{1∗}(y)⟩_{Δ_{r}}.

Proof. ⟨x, y⟩_{Δ_{r+1}} maps to

∑_{a∈Δ_{r}} ( ∑_{{b∈Δ_{r+1} \atop b↦a}} (x, < b >_{Alb} (y))_{r+1})[a^{−1}] = ∑_{a∈Δ_{r}}(x, π_{1}^{∗}∘(\frac{1}{p}π_{1∗}) < a >_{Alb} (y))_{r+1}[a^{−1}]  (by (1.6.6.iii))

= ∑_{a∈Δ_{r}}(π_{p∗}(x), (\frac{1}{p}π_{1∗}) < a >_{Alb} (y))_{r}[a^{−1}] (by (1.6.3.4))

= ⟨π_{p∗}(x), \frac{1}{p}π_{1∗}(y)⟩_{Δ_{r}}

as claimed.

(1.6.8) Corollary. The map

γ = (γ_{r}) : X = \underleftarrow{lim}_{π_{p∗}} eX_{r} ⟶ \underleftarrow{lim}_{\frac{1}{p} π_{1∗}} eX_{r}
(x_{r})_{r≥1} ⟼ (T (p)_{Alb}^{r−1}(x_{r}))_{r≥1}

is an isomorphism of 𝔥_{∞}^{ord}[G_{ℚ}]-modules.

Proof. Follows immediately from Lemma 1.6.6 (iv).

(1.6.9) Combining the two corollaries, we obtain a skew-symmetric pairing on X:

⟨ , ⟩_{X} : \underleftarrow{lim}_{π_{p∗}} eX_{r} × \underleftarrow{lim}_{π_{p∗}} eX_{r} ⟶ Λ(−1) < −1 >

given by ⟨(x_{r}), (y_{r})⟩_{X} = (⟨x_{r}, T (p)_{Alb}^{r−1}(y)_{r}⟩_{H_{r}}).

Theorem. ⟨ , ⟩_{X} induces an isomorphism of 𝔥_{∞}^{ord}[G_{ℚ}]-modules β_{∞} : X ⟶^{∼} Hom_{Λ}(X, Λ)(−1) < −1 >, (β_{∞}(x))(y) = ⟨x, y⟩_{X}. In particular, X is free of finite rank over Λ.

Proof. Firstly, the ordinary part of α_{∞} gives an isomorphism

α_{∞}^{ord} : X ⟶^{∼} Hom_{O}(eJ_{∞}, O ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p}) < −1 >,

and we know that the right hand side is of finite type over 𝔥_{∞}^{ord}. A fundamental result of Hida [Hi 1, Thm. 3.1] states that π_{1}^{∗} induces isomorphisms

eJ_{1}(Np^{r})(\overline{ℚ})_{p^{∞}} ⟶^{∼} (eJ_{∞})^{Γ_{r}} (r ≥ 1).

Dualizing – using (1.6.3.4) – we see that π_{p∗} induces isomorphisms X_{Γ_{r}} ⟶^{∼} eX_{r}. Together with (1.6.5.2), this implies that β_{∞} is an isomorphism.

(1.6.10) As in 1.5.1, fix a local component R of 𝔥_{∞}^{ord}. Let X(R) = e_{R}X be the R-part of X. Then α_{∞}^{ord} induces an isomorphism

X(R) ⟶^{∼} T (R)(−1) < −1 >

and ⟨ , ⟩_{X} defines a skew-symmetric pairing

⟨ , ⟩_{T (R)} : T (R) × T (R) ⟶ Λ(1) < 1 >

inducing an isomorphism of R[G_{ℚ}]-modules T (R) ⟶^{∼} Hom_{Λ}(T (R), Λ)(1) < 1 >. Whenever 1.5.2 (iii) or Proposition 1.5.4 apply, the pairing ⟨ , ⟩_{T (R)} induces isomorphisms of R[G_{ℚ}]-modules T (R)^{±} ⟶^{∼} Hom_{Λ}(T (R)^{∓}, Λ)(1) < 1 >. The canonical map of R-modules

Hom_{R}(T (R), Hom_{Λ}(R, Λ)) ⟶ Hom_{Λ}(T (R), Λ)

is an isomorphism, and hence the pairing ⟨ , ⟩_{T (R)} induces an isomorphism of R[G_{ℚ}]-modules

T (R) ⟶^{∼} Hom_{R}(T (R), ω_{R})(1) < 1 >

i.e. can be viewed as an R-bilinear pairing

T (R) × T (R) ⟶ ω_{R}(1) < 1 > .

(1.6.11) For the proof of Lemma 1.6.6, it will be convenient to use a group theoretic description of the Hecke operators. Let Γ ⊆ SL_{2}(ℝ) be a discrete subgroup of the form γΓ′γ^{−1}, where γ ∈ GL_{2}(ℚ) and Γ′ ⊆ SL_{2}(ℤ) is a congruence subgroup. Denote by X(Γ) = Γ\H^{∗} (where H^{∗} = H ∪ ℙ^{1}(ℚ)) the corresponding complex modular curve over ℂ. Let H be a reasonable cohomology theory, say H^{0}(X, Ω^{1}) or H^{1}(\overline{X}_{et}, ℤ_{p}); put H(Γ) = H(X(Γ)).

Functoriality: γ ∈ GL_{2}(ℚ) gives an isomorphism [γ] : X(Γ) → X(γΓγ^{−1}), which induces an isomorphism

[γ]^{∗} = [γ^{−1}]_{∗} : H(γΓγ^{−1}) ⟶^{∼} H(Γ).

Inclusion i : Γ ↪ Γ′ gives a finite map X(Γ) → X(Γ′), which induces i_{∗} : H(Γ) ⟶ H(Γ′), and i^{∗} : H(Γ′) ⟶ H(Γ).

Double cosets: For γ ∈ GL_{2}(ℚ) and Γ_{1}, Γ_{2} ⊆ GL_{2}(ℝ) of the above form, the maps

Γ_{1}↩^{i_{1}}Γ_{1} ∩ γΓ_{2}γ^{−1}⟶^{[γ^{−1}]}γ^{−1}Γ_{1}γ ∩ Γ_{2}↪^{i_{2}}Γ_{2}

induce the action of the double coset

[Γ_{1}γΓ_{2}] : H(Γ_{1})⟶^{i_{1}^{∗}}H(Γ_{1} ∩ γΓ_{2}γ^{−1})⟶^{[γ^{−1}]_{∗}}H(γ^{−1}Γ_{1}γ ∩ Γ_{2})⟶^{i_{2∗}}H(Γ_{2}).

In the coordinates on the upper half plane H, [Γ_{1}γΓ_{2}] corresponds to ∑_{i}(γα_{i})^{∗}, where α_{i} ∈ GL_{2}(ℚ) are representatives of (γ^{−1}Γ_{1}γ ∩ Γ_{2})\Γ_{2} (equivalently, Γ_{1}γΓ_{2} = ⋃ Γ_{1}γα_{i}).

Example: Γ_{1} = Γ_{2} = Γ = Γ_{1}(M), γ = ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ), with p prime. Then

Γ_{1}(M) ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) Γ_{1}(M) = ∐_{i∈ℤ/pℤ} Γ_{1}(M) ( \begin{matrix}1 & i \\ 0 & p \end{matrix} ) ( ∪ Γ_{1}(M) ( \begin{matrix}p & 0 \\ 0 & 1 \end{matrix} ) σ_{p} if p ∤ M),

where σ_{p} ∈ SL_{2}(ℤ), σ_{p} ≡ ( \begin{matrix}p^{−1} & 0 \\ 0 & p \end{matrix} ) (mod M). The action of [ΓγΓ] on H(Γ) = H^{0}(X(Γ), Ω^{1}) gives the

usual Hecke operator T (p) = T (p)_{Alb} on cusp forms of weight two on Γ_{1}(M).

Proof of Lemma 1.6.6. Let

γ = ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ) ,  Γ = Γ_{1}(Np^{r}) ⊃ Γ ∩ γΓγ^{−1} = Γ_{1}(Np^{r}) ∩ Γ_{0}(Np^{r+1}) ⊃ Γ′ = Γ_{1}(Np^{r+1})  (r ≥ 1).

Consider the maps

\begin{aligned}& & Γ \\ Γ′ & ^{_{⊂} \begin{matrix}i_{1} \\ \end{matrix}} & Γ ∩ γΓγ^{−1} & ^{i_{2}_{∪} } & \overline{\begin{matrix}[γ^{−1}] \\ \end{matrix}} & γ^{−1}Γγ ∩ Γ & ^{_{⊂} i_{3}} & Γ \end{aligned}

and their action on cohomology H(X) = H^{1}(X, ℤ) ⊗_{ℤ} ℤ_{p}. Each modular curve X corresponding to Γ, Γ′, or Γ ∩ γΓγ^{−1} has a model X_{/ℚ}, and hence H(X) = H^{1}((X_{/ℚ} ⊗_{ℚ} \overline{ℚ})_{et}, ℤ_{p}). By definition, we have

π_{1∗} = i_{2∗}i_{1∗} : H(Γ′) ⟶ H(Γ)
π_{p∗} = i_{3∗}[γ^{−1}]_{∗}i_{1∗} : H(Γ′) ⟶ H(Γ)
T (p)_{Alb} = [ΓγΓ] = i_{3∗}[γ^{−1}]_{∗}i_{2}^{∗} : H(Γ) ⟶ H(Γ)

Denote [(Γ ∩ γΓγ^{−1})γ(Γ ∩ γΓγ^{−1})] : H(Γ ∩ γΓγ^{−1}) ⟶ H(Γ ∩ γΓγ^{−1}) also by T (p)_{Alb}. It follows from the definitions that the operator ξ = [(Γ ∩ γΓγ^{−1})γΓ] = i_{3∗}[γ^{−1}]_{∗} : H(Γ ∩ γΓγ^{−1}) ⟶ H(Γ) satisfies

ξi_{2}^{∗} = T (p)_{Alb}  on H(Γ)
i_{2}^{∗}ξ = T (p)_{Alb}  on H(Γ ∩ γΓγ^{−1})

This fact – due to Shimura – is at the basis of all Hida’s theory [Hi 1, Thm. 4.4]. Applying the ordinary projector e = lim_{n→∞} T (p)_{Alb}^{n!}, we see that i_{2}^{∗} induces an isomorphism between the ordinary parts ([Hi 1, Cor. 4.5])

i_{2}^{∗} : eH(Γ) ⟶^{∼} eH(Γ ∩ γΓγ^{−1}),

with inverse

(i_{2}^{∗})^{−1} = T (p)_{Alb}^{−1}ξ = ξT (p)_{Alb}^{−1}.

As i_{2∗}i_{2}^{∗} = p, it follows that the map

(i_{2}^{∗})^{−1}i_{1∗} : eH(Γ′) ⟶ eH(Γ)

satisfies p(i_{2}^{∗})^{−1}i_{1∗} = i_{2∗}i_{1∗} = π_{1∗}, which proves Lemma 1.6.6 (i) and in fact shows that \frac{1}{p}π_{1∗} = (i_{2}^{∗})^{−1}i_{1∗}. The rest is easy:

(\frac{1}{p}π_{1∗}) π_{1}^{∗} = (i_{2}^{∗})^{−1}i_{1∗}i_{1}^{∗}i_{2}^{∗} = p(i_{2}^{∗})^{−1}i_{2}^{∗} = p on eH(Γ)

π_{1}^{∗}(\frac{1}{p}π_{1∗}) = i_{1}^{∗}i_{2}^{∗}(i_{2}^{∗})^{−1}i_{1∗} = i_{1}^{∗}i_{1∗} = ∑_{a∈Γ_{r}/Γ_{r+1}} < a >_{Alb} on eH(Γ′).

The last equality here is because Γ′ is a normal subgroup of Γ ∩ γΓγ^{−1}, with quotient equal to Γ_{r}/Γ_{r+1}. Finally,

T (p)_{Alb}(\frac{1}{p}π_{1∗}) = ξi_{2∗}(i_{2}^{∗})^{−1}i_{1∗} = ξi_{1∗} = i_{3∗}[γ^{−1}]_{∗}i_{1∗} = π_{p∗} on eH(Γ′).

This finishes the proof of Lemma 1.6.6.

(1.6.12) The arguments in the previous section show that all the statements of Lemma 1.6.6 hold if eX_{r} is replaced by the maximal direct summand of (X_{r} ⊗ ℚ) on which T (p)_{Alb} is invertible (i.e. the sum of generalized eigenspaces of T (p)_{Alb} for non-zero eigenvalues).

(1.6.13) Using the formula π_{p}∘W_{r+1} = W_{r}∘π_{1} (which follows from 1.2.9) and (1.2.8.3), (1.2.10), we see that the map

W = (W_{r}) : (x_{r})_{r≥1} ⟼ (W_{r}(x_{r}))_{r≥1}

induces an isomorphism of O-modules

\underleftarrow{lim}_{π_{1∗}} e^{∗}X_{r} ⟶^{∼} \underleftarrow{lim}_{π_{p∗}} eX_{r} = X,

where e^{∗} = lim_{n→∞} T (p)_{Pic}^{n!} is Hida’s dual projector.

2. Selmer groups in families

(2.1) Generalities on Selmer groups

(2.1.1) Notation. Let p be a rational prime, F_{𝔭} a finite extension of ℚ_{p} with ring of integers O = O_{𝔭}, and π ∈ O a prime element. Let K be a number field, S a finite set of primes of K containing all archimedean primes and all primes dividing p, and K_{S} the maximal extension of K unramified outside S. Let S_{f} ⊂ S be the subset of non-archimedean primes. For v ∈ S_{f}, fix embeddings \overline{K} ↪ \overline{K}_{v}. They induce maps G_{v} = Gal(\overline{K}_{v}/K_{v}) → G_{K} = Gal(\overline{K}/K) → G_{K,S} = Gal(K_{S}/K).

(2.1.2) Let T be an O-adic representation of G_{K,S} (i.e. a free O-module of finite rank equipped with a continuous O-linear action of G_{K,S}). There is a tautological exact sequence

0 → T ⟶^{i}V ⟶^{pr}A ⟶ 0

with V = T ⊗_{O} F_{𝔭}, A = V/T. All Galois cohomology groups with values in T or V will be continuous cohomology ([Ta], [Ja]). Recall that H^{i}(G, V) = H^{i}(G, T)⊗_{O} F_{𝔭} and that H^{i}(G, T) coincides with the naive cohomology group

\underleftarrow{lim}_{n} H^{i}(G, T/π^{n}T)

for i ≤ 1 (resp. for all i if G = G_{v} or G_{K,S}). For G = G_{v} or G_{K,S}, the O-modules H^{i}(G, T) (resp. H^{i}(G, A)) are of finite (resp. co-finite) type.

(2.1.3) Recall Flach’s abstract treatment of Selmer groups [Fl]. Given F_{𝔭}-subspaces W_{v} ⊆ H^{1}(G_{v}, V) for all v ∈ S, the Selmer groups associated to W = (W_{v}) are defined by

S(K, A; W) = Ker [H^{1}(G_{K,S}, A) → ⨁_{v∈S} H^{1}(G_{v}, A)/pr_{∗}(W_{v})]

S(K, V; W) = Ker [H^{1}(G_{K,S}, V) → ⨁_{v∈S} H^{1}(G_{v}, V)/W_{v}]

They do not change if S is replaced by S′ ⊃ S with W_{v} = H_{ur}^{1}(G_{v}, V) = H^{1}(G_{v}/I_{v}, V) for v ∈ S′ − S (as pr_{∗}(H_{ur}^{1}(G_{v}, V)) = H_{ur}^{1}(G_{v}, A) for such v). The O-module S(K, A; W) is of co-finite type and pr_{∗}(S(K, V; W)) coincides with its maximal 𝔭-divisible subgroup S(K, A; W)_{div}, i.e. S(K, A; W)_{div} ⟶^{∼} (F_{𝔭}/O)^{r}, where r = dim_{F_{𝔭}}S(K, V; W), and the quotient � (K, A; W) := S(K, A; W)/S(K, A; W)_{div} is finite.

(2.1.4) Duality. Put ℤ_{p}(1) = \underleftarrow{lim}_{n} μ_{p^{n}}, O(1) = O ⊗_{ℤ_{p}} ℤ_{p}(1), T^{∗}(1) = Hom_{O}(T, O(1)), V^{∗}(1) = T^{∗}(1)⊗_{O} F_{𝔭},

A^{∗}(1) = V^{∗}(1)/T^{∗}(1). For v ∈ S_{f}, let W_{v}^{⊥} ⊆ H^{1}(G_{v}, V^{∗}(1)) be the annihilator of W_{v} under Tate’s local duality

H^{1}(G_{v}, V) × H^{1}(G_{v}, V^{∗}(1))⟶^{∪}H^{2}(G_{v}, F_{𝔭}(1)) ⟶^{∼} F_{𝔭}.

For v ∤ p, H_{ur}^{1}(G_{v}, V)^{⊥} = H_{ur}^{1}(G_{v}, V^{∗}(1)). Flach [Fl] constructed a pairing^{(1)}

S(K, A; W) × S(K, A^{∗}(1); W^{⊥}) ⟶ F_{𝔭}/O

and showed that its left (resp. right) kernel is equal to S(K, A; W)_{div} (resp. S(K, A; W^{⊥})_{div}). In other words, the induced pairing

� (K, A; W) × � (K, A^{∗}(1); W^{⊥}) ⟶ F_{𝔭}/O

is non-degenerate.

(2.1.5) Let

⟨ , ⟩_{V} : V × V ⟶ F_{𝔭}(1)

be a Galois-invariant non-degenerate skew-symmetric bilinear form such that T × T is mapped to O(1). The formula α(x)(y) = ⟨x, y⟩_{V} defines an isomorphism of F_{𝔭}[G_{K,S}]-modules α : V ⟶^{∼} V^{∗}(1) which is skewsymmetric (i.e. α^{∗}(1) = −α) and such that α(T) ⊆ T^{∗}(1). If the local conditions W are self dual in the sense that α_{∗}(W_{v}) = W_{v}^{⊥} for all v ∈ S, then α induces a map

α_{∗} : S(K, A; W) ⟶ S(K, A^{∗}(1); W^{⊥})

(1) Flach considered only F_{𝔭} = ℚ_{p}, but the generalization to F_{𝔭}-representations is straightforward. with finite kernel and cokernel. The induced pairing

S(K, A; W) × S(K, A; W) ⟶^{(id,α_{∗})} S(K, A; W) × S(K, A^{∗}(1); W^{⊥}) ⟶ F_{𝔭}/O

is skew-symmetric [Fl].

(2.1.6) For v | p, Bloch-Kato [Bl-Ka, Sect.3] introduced subgroups H_{f}^{1}(G_{v}, V) ⊆ H_{g}^{1}(G_{v}, V) ⊆ H^{1}(G_{v}, V) of a crystalline nature;

\begin{aligned}H_{f}^{1}(G_{v}, V) & = Ker [H^{1}(G_{v}, V) ⟶ H^{1}(G_{v}, V ⊗_{ℚ_{p}} B_{cris})] \\ H_{g}^{1}(G_{v}, V) & = Ker [H^{1}(G_{v}, V) ⟶ H^{1}(G_{v}, V ⊗_{ℚ_{p}} B_{dR})] \end{aligned}

(see [Bures, Exp. II] for more details on Fontaine’s rings B_{cris} ⊂ B_{dR}). Write D_{∗}(V) = H^{0}(G_{v}, V ⊗_{ℚ_{p}} B_{∗}), for ∗ ∈ {cris, dR}. If we define

W_{v} = \{ \begin{matrix}H_{ur}^{1}(G_{v}, V) & v ∈ S_{f}, v ∤ p \\ H_{f}^{1}(G_{v}, V) & v | p \end{matrix}

then the Selmer group S(K, V; W) is usually denoted by H_{f}^{1}(K, V). The basic properties of H_{f}^{1} ⊆ H_{g}^{1} are as follows:

(i) If V is a de Rham representation of G_{v}, then H_{g}^{1}(G_{v}, V)/H_{f}^{1}(G_{v}, V) is dual to D_{cris}(V^{∗}(1))/(f − 1) ([Bl-Ka, Prop. 3.8, Cor. 3.8.4]). (ii) If V satisfies Pančiškin’s condition at v ([Ne 1, 6.7]), i.e. if there is an exact sequence of F_{𝔭}[G_{v}]-modules

0 ⟶ V_{v}^{+} ⟶ V ⟶ V_{v}^{−} ⟶ 0

such that F^{0}D_{dR}(V_{v}^{+}) = D_{dR}(V_{v}^{−})/F^{0} = 0 (where F^{i}D_{dR}(W) = H^{0}(G_{v}, W ⊗_{ℚ_{p}} Fil^{i}D_{dR})), then

H^{1}(G_{v}, V_{v}^{+}) ⟶ H_{g}^{1}(G_{v}, V) ⟶ H_{g}^{1}(G_{v}, V_{v}^{−})

is exact. If V_{v}^{−} is a semi-stable representation of G_{v}, then

H_{g}^{1}(G_{v}, V_{v}^{−}) = H_{f}^{1}(G_{v}, V_{v}^{−}) = D_{cris}(V_{v}^{−})/(f − 1)

(by weak admissibility and [Bl-Ka, Cor. 3.8.4]), which has dimension equal to that of

D_{cris}(V_{v}^{−})^{f=1} = D_{cris}(V_{v}^{−}) ∩ F^{0}D_{dR}(V_{v}^{−}) = H^{0}(G_{v}, V_{v}^{−}).

In particular, if H^{0}(G_{v}, V_{v}^{−}) = 0, then

H_{g}^{1}(G_{v}, V) = Im [H^{1}(G_{v}, V_{v}^{+}) ⟶ H^{1}(G_{v}, V)] .

(2.2) Selmer complexes

As observed in [Ne 3], a more satisfactory theory of Selmer groups is obtained if one imposes local conditions on the level of complexes, rather than on cohomology (as in 2.1.3). In this section we recall the basic setup of [Ne 3] for the coefficient ring O. In what follows, if p = 2 and K is not totally imaginary, then most statements are valid only modulo 2-torsion.

(2.2.1) Let G be a profinite group and M a topological G-module (i.e. such that the action G × M → M is continuous). The complex of (non-homogeneous) continuous cochains on G with values in M will be denoted Ccont^{•}(G, M); in degree q, C_{cont}^{q}(G, M) consists of continuous maps G^{q} → M. The cohomology of Ccont^{•}(G, M) will be denoted by H^{i}(G, M). If M has discrete topology, then

Ccont^{•}(G, M) = \underrightarrow{lim}_{U} Ccont^{•}(G/U, M^{U}),

with U running through all open normal subgroups of G.

(2.2.2) In the notation of 2.1.2, we consider M = T (with the 𝔭-adic topology) and M = V = ⋃_{n≥0} π^{−n}T (with the inductive limit topology) as topological modules over G = G_{K}, G_{K,S}, or G_{v}. In this case,

Ccont^{•}(G, T) = \underleftarrow{lim}_{n} Ccont^{•}(G, T/π^{n}T) and Ccont^{•}(G, V) = Ccont^{•}(G, T) ⊗_{O} F_{𝔭} ([Ja, Thm. 5.15]) and

0 ⟶ Ccont^{•}(G, T)⟶^{i_{∗}}Ccont^{•}(G, V)⟶^{pr_{∗}}Ccont^{•}(G, A) ⟶ 0

is an exact sequence of complexes (as i is strict and pr admits a continuous section).

(2.2.3) In order to define Selmer complexes we need the following data: for each v ∈ S_{f}, complexes U_{v}^{±}(T) of O-modules together with maps of complexes

Δ_{v} : U_{v}^{+}(T) ⟶ Ccont^{•}(G, T) ⟶ U_{v}^{−}(T)

such that Δ_{v} defines a distinguished triangle in D_{ft}^{b}(O-mod) (the derived category of cohomologically bounded complexes of O-modules, with cohomology of finite type over O). Putting U_{v}^{±}(V) = U_{v}^{±}(T) ⊗_{O} F_{𝔭} and U_{v}^{±}(A) = Cone (U_{v}^{±}(T) ⟶ U_{v}^{±}(V)) (isomorphic to U_{v}^{±}(T) ⊗_{O}^{𝕃} (F_{𝔭}/O) in D^{b}(O-mod)), we get distinguished triangles in D^{b}(O-mod)

U_{v}^{+}(X) ⟶ Ccont^{•}(G, X) ⟶ U_{v}^{−}(X)

for X = T, V, A. The Selmer complex with values in X ∈ {T, V, A} associated to local conditions Δ = (Δ_{v}) is defined as

Cf^{•}(G_{K,S}, X; Δ) = Cone(Ccont^{•}(G_{K,S}, X) ⟶ ⨁_{v∈S_{f}} U_{v}^{−}(X))[−1].

The corresponding object of D^{b}(O-mod) will be denoted by ℝΓ̃_{f}(G_{K,S}, X; Δ) and its cohomology by H̃_{f}^{i}(G_{K,S}, X; Δ).

(2.2.4) Properties of Selmer complexes

(i) ℝΓ̃_{f}(G_{K,S}, X; Δ) for X = T (resp. X = A) is an object of D_{ft}^{b}(O-mod) (resp. D_{coft}^{b}(O-mod)) i.e. the cohomology groups H̃_{f}^{i}(G_{K,S}, X; Δ) are of finite type (resp. cofinite type) over O. (ii) There is a distinguished triangle

ℝΓ̃_{f}(G_{K,S}, T; Δ)⟶^{i_{∗}}ℝΓ̃_{f}(G_{K,S}, V; Δ)⟶^{pr_{∗}}ℝΓ̃_{f}(G_{K,S}, A; Δ)

with

ℝΓ̃_{f}(G_{K,S}, V; Δ) ⟶^{∼} ℝΓ̃_{f}(G_{K,S}, T; Δ) ⊗_{O} F_{𝔭}
ℝΓ̃_{f}(G_{K,S}, A; Δ) ⟶^{∼} ℝΓ̃_{f}(G_{K,S}, T; Δ) ⊗^{𝕃}_{O} (F_{𝔭}/O)

(iii) The cohomology sequence of the triangle in (ii) induces isomorphisms

H̃_{f}^{q}(G_{K,S}, T; Δ)_{tors} ⟶^{∼}Im [H̃_{f}^{q−1}(G_{K,S}, A; Δ) ⟶ H̃_{f}^{q}(G_{K,S}, T; Δ)]
H̃_{f}^{q}(G_{K,S}, A; Δ)_{div} ⟶^{∼}Im [pr_{∗} : H̃_{f}^{q}(G_{K,S}, V; Δ) ⟶ H̃_{f}^{q}(G_{K,S}, A; Δ)] .

(iv) There is an exact sequence

⋯ ⟶ ⨁_{v∈S_{f}} H^{q−1}(G_{v}, U_{v}^{−}(X)) ⟶ H̃_{f}^{q}(G_{K,S}, X; Δ) ⟶ H^{q}(G_{K,S}, X) ⟶ ⨁_{v∈S_{f}} H^{q}(G_{v}, U_{v}^{−}(X)) ⟶ ⋯

(2.2.5) Duality Local Tate duality can be reformulated as an isomorphism

ℝΓ(G_{v}, T^{∗}(1)) ⟶^{∼} ℝHom_{O} (ℝΓ(G_{v}, T), O) [−2] (v ∤ ∞)

in D_{ft}^{b}(O-mod), where ℝΓ(G_{v}, T) denotes the image of Ccont^{•}(G_{v}, T) in the derived category. Assume that, for each v ∈ S_{f}, we are given local conditions

Δ_{v}^{∗}(1) : U_{v}^{+}(T^{∗}(1)) ⟶ Ccont^{•}(G_{v}, T^{∗}(1)) ⟶ U_{v}^{−}(T^{∗}(1))

and isomorphisms in D_{ft}^{b}(O-mod),

U_{v}^{±}(T^{∗}(1)) ⟶^{∼} ℝHom_{O}(U_{v}^{∓}(T))[−2],

compatible with local Tate duality. One defines Selmer conditions for T^{∗}(1), V^{∗}(1), A^{∗}(1) using local conditions Δ^{∗}(1) = (Δ_{v}^{∗}(1)).

Examples: (i) If U_{v}^{+}(T) = 0, then U_{v}^{+}(T^{∗}(1)) = Ccont^{•}(G_{v}, T^{∗}(1)). (ii) If 0 → T_{v}^{+} → T → T_{v}^{−} → 0 is an exact sequence of O[G_{v}]-modules with T_{v}^{−} free over O and U_{v}^{±}(T) =

Ccont^{•}(G_{v}, T_{v}^{±}), then U_{v}^{±}(T) = Ccont^{•}(G_{v}, T^{∗}(1)_{v}^{±}), where T^{∗}(1)_{v}^{±} = (T_{v}^{∓})^{∗}(1).

It is proven in [Ne 3] that there is a pairing

H̃_{f}^{1}(G_{K,S}, A; Δ) × H̃_{f}^{1}(G_{K,S}, A^{∗}(1); Δ^{∗}(1)) ⟶ F_{𝔭}/O

with left (resp. right) kernel equal to H̃_{f}^{1}(G_{K,S}, A; Δ)_{div} (resp. H̃_{f}^{1}(G_{K,S}, A^{∗}(1); Δ^{∗}(1))_{div}).

(2.3) Big Galois representations

(2.3.1) Let R be a complete local noetherian ring containing O, with maximal ideal 𝔪 and finite residue field R/𝔪. Let T be an R-module of finite type equipped with a continuous (with respect to the 𝔪-adic topology) R-linear action of G_{K,S}. Assume that, for each v | p, there is an exact sequence of R[G_{v}]-modules

0 ⟶ T_{v}^{+} ⟶ T ⟶ T_{v}^{−} ⟶ 0

with T_{v}^{+} free over R.

(2.3.2) Specializations of T Let X ⊆ Spec(R) be the set of prime ideals I ⊂ R such that

(i) R/I is free of finite rank over O. (ii) For all v | p, T_{v}^{−} ⊗_{R} R_{I} is free over R_{I}. The same argument as in 1.5.6 shows that, for every I ∈ X, Tor_{1}^{R}(T_{v}^{−}, R/I) is finite and

0 ⟶ T_{v}^{+}/IT_{v}^{+} ⟶ T/IT ⟶ T_{v}^{−}/IT_{v}^{−} ⟶ 0

is an exact sequence of (R/I)[G_{v}]-modules, with T_{v}^{+}/IT_{v}^{+} free over O (for all v | p). Fix, once and for all, I ∈ X. Let c_{0} ≥ 0 be the smallest integer such that π^{c_{0}} kills the O-torsion of T_{v}^{−}/IT_{v}^{−}, for all v | p. Put

X(I) = {J ∈ X : rk_{O}(T_{v}^{−}/JT_{v}^{−}) = rk_{O}(T_{v}^{−}/IT_{v}^{−}) for all v | p, and (J, π^{c_{0}+1}) = (I, π^{c_{0}+1})}
X(I)_{n} = {J ∈ X(I) : (J, π^{c_{0}+n}) = (I, π^{c_{0}+n})}  (n ≥ 1)

(here, rk_{O}(M) = dim_{F_{𝔭}}(M ⊗_{O} F_{𝔭})).

(2.3.3) Lemma. For every J ∈ X(I), (i) rk_{O}(R/J) = rk_{O}(R/I). (ii) rk_{O}(T/JT) = rk_{O}(T/IT). (iii) There are canonical isomorphisms of O/π^{c_{0}}-modules

(T_{v}^{−}/JT_{v}^{−})_{tors} ⟶^{∼} (T_{v}^{−}/IT_{v}^{−})_{tors}  (for all v | p)
(T/JT)_{tors} ⟶^{∼} (T/IT)_{tors}

Proof. (i) This follows from the isomorphism R/(J, π) ⟶^{∼} R/(I, π). (ii) By (i) and freeness of T_{v}^{+}, rk_{O}(T_{v}^{+}/JT_{v}^{+}) = rk_{O}(T_{v}^{+}/IT_{v}^{+}) for all v | p. For T_{v}^{−}, the analogous equality hold by definition, proving (ii). (iii) The exact sequence (for every v | p)

T_{v}^{+}/(I, π^{c_{0}+1})T_{v}^{+} ⟶ T/(I, π^{c_{0}+1})T ⟶ T_{v}^{−}/(I, π^{c_{0}+1})T_{v}^{−} ⟶ 0

is isomorphic to the corresponding sequence for J. The free parts of T_{v}^{−}/IT_{v}^{−} and T_{v}^{−}/JT_{v}^{−} are isomorphic by definition and the torsion part of T_{v}^{−}/IT_{v}^{−} is killed by π^{c_{0}}. This implies that the torsion parts of T_{v}^{−}/IT_{v}^{−} and T_{v}^{−}/JT_{v}^{−} are again isomorphic. As (T/IT)_{tors} injects into (T_{v}^{−}/IT_{v}^{−})_{tors} (and similarly for J), it is killed by π^{c_{0}}. The isomorphism T/(I, π^{c_{0}+1})T ⟶^{∼} T/(J, π^{c_{0}+1})T together with (ii) imply that (T/JT)_{tors} ⟶^{∼} (T/IT)_{tors}.

(2.3.4) For J ∈ X(I) and v | p, put

V_{J} = (T/JT) ⊗_{O} F_{𝔭},  T_{J} = Im (T/JT → V_{J}) ,  A_{J} = V_{J}/T_{J},  (V_{J})_{v}^{±} = (T_{v}^{±}/JT_{v}^{±}) ⊗_{O} F_{𝔭},
(T_{J})_{v}^{+} = T_{J} ∩ (V_{J})_{v}^{+} ,  (T_{J})_{v}^{−} = Im (T_{J} → (V_{J})_{v}^{−}) ,  (A_{J})_{v}^{±} = (V_{J})_{v}^{±} / (T_{J})_{v}^{±} .

There are exact sequences of O[G_{v}]-modules

0 ⟶ (X_{J})_{v}^{+} ⟶ X_{J} ⟶ (X_{J})_{v}^{−} ⟶ 0

for all v | p and X ∈ {T, V, A}.

(2.3.5) Lemma. If J ∈ X(J)_{n} for n ≥ 1, then there are canonical isomorphisms (for all v | p)

(A_{J})_{π^{n}} ⟶^{∼} (A_{I})_{π^{n}} , ((A_{J})_{v}^{±})_{π^{n}} ⟶^{∼} ((A_{I})_{v}^{±})_{π^{n}} .

Proof. This follows from the definitions and Lemma 2.3.3.

(2.3.6) Consider the following conditions on V_{J}, T_{J}, and A_{J} : (C1) H^{0}(G_{K}, V_{J}) = 0 and (for all v ∈ S_{f}, v ∤ p) H^{0}(G_{v}, V_{J}) = 0. (C1′) There exists c_{1} ≥ 0 such that π^{c_{1}}H^{0}(G_{K}, A_{J}) = 0 and (for all v ∈ S_{f}, v ∤ p) π^{c_{1}}H^{0}(G_{v}, A_{J}) = 0. (C2) V_{J} is an absolutely irreducible F_{𝔭}[G_{K}]-module. (C2′) There exists c_{2} ≥ 0 such that π^{c_{2}}End_{O}(T_{J}) ⊆ Im (O[G_{K}] → End_{O} (T_{J})). (C3) There exists α_{J} : V_{J} ⟶^{∼} V_{J}^{∗}(1), a skew-symmetric (α_{J}^{∗}(1) = −α_{J}) isomorphism of F_{𝔭}[G_{K}]-modules.

(C4) α_{J} from (C3) induces isomorphisms (V_{J})_{v}^{±} ⟶^{∼} ((V_{J})_{v}^{∓})^{∗} (1) for all v | p.

We have (C1) ⟺ (C1′) and (C2) ⟺ (C2′). Of course, the condition H^{0}(G_{K}, V_{J}) = 0 in (C1) is superfluous if the set {v ∈ S_{f} : v ∤ p} is non-empty. Note that (C1) implies that H_{ur}^{1}(G_{v}, V_{J}) = 0 for all v ∈ S_{f}, v ∤ p. We shall consider Selmer groups associated to local conditions

W_{v} = \{ \begin{matrix}0 & v ∈ S_{f}, v ∤ p \\ Ker (H^{1}(G_{v}, V_{J}) → H^{1}(G_{v}, (V_{J})_{v}^{−})) & v | p \end{matrix}

and Selmer complexes corresponding to

\begin{matrix}U_{v}^{+}(T_{J}) = & \{ \begin{matrix}0 & v ∈ S_{f}, v ∤ p \\ Ccont^{•}(G_{v}, (T_{J})_{v}^{+}) & v | p \end{matrix} \\ U_{v}^{−}(T_{J}) = & \{ \begin{matrix}Ccont^{•}(G_{v}, T_{J}) & v ∈ S_{f}, v ∤ p \\ Ccont^{•}(G_{v}, (T_{J})_{v}^{−}) & v | p \end{matrix} \end{matrix}

The exact sequence 2.2.4 (iv) then becomes

(2.3.6.1)  0 ⟶ ⨁_{v|p} H^{0}(G_{v}, (V_{J})_{v}^{−}) ⟶ H̃_{f}^{1}(G_{K,S}, V_{J}; Δ) ⟶ S(K, V_{J}; W) ⟶ 0.

(2.3.7) Theorem (“Continuity Principle”). Assume that V_{I} satisfies (C1) and let c_{1} ≥ 0 be as in (C1′). If J ∈ X(I)_{n+c_{1}} with n ≥ 1, then

(1) A_{J} satisfies (C1′) with the same value of c_{1} (hence V_{J} satisfies (C1)).

(2) There is a canonical isomorphism

H̃_{f}^{1}(G_{K,S}, A_{J}; Δ)_{π^{n}} ⟶^{∼} H̃_{f}^{1}(G_{K,S}, A_{I}; Δ)_{π^{n}}.

Proof. (1) This follows from Lemma 2.3.5.

(2) For every m ≥ 1, we define Selmer complexes for the finite G_{K,S}-modules (A_{J})_{π^{m}} by local conditions

\begin{aligned}U_{v}^{+} & = 0,  U_{v}^{−} = Ccont^{•}(G_{v}, (A_{J})_{π^{m}}),  v ∈ S_{f}, v ∤ p, \\ U_{v}^{±} & = Ccont^{•}(G_{v}, ((A_{J})_{v}^{±})_{π^{m}}),  v | p. \end{aligned}

The local conditions are compatible with the isomorphisms of Lemma 2.3.5, which means that the induced maps

H̃_{f}^{1}(G_{K,S}, (A_{J})_{π^{n+c_{1}}} ; Δ) ⟶^{∼} H̃_{f}^{1}(G_{K,S}, (A_{I})_{π^{n+c_{1}}} ; Δ)

are again isomorphisms. Applying ℝΓ̃_{f}(G_{K,S}, − ; Δ) to all entries of the commutative diagram

[figure: 0; (A_{J})_{π^{n}}; A_{J}; π^{n}; A_{J}; 0]

∥

[figure: 0; (A_{J})_{π^{n}}; (A_{J})_{π^{n+c_{1}}}; ∪; “π^{n}”; (A_{J})_{π^{c_{1}}}; ∪; 0]

[figure: 0; (A_{I})_{π^{n}}; ≀; (A_{I})_{π^{n+c_{1}}}; ≀; “π^{n}”; (A_{I})_{π^{c_{1}}}; ≀; 0]

∥

[figure: 0; (A_{I})_{π^{n}}; A_{I}; ∩; π^{n}; A_{I}; ∩; 0]

we get a commutative diagram with exact rows (dropping G_{K,S} and Δ from the notation):

[figure: 0; H̃_{f}^{0} (A_{J}) /π^{n}; H̃_{f}^{1} ((A_{J})_{π^{n}}); H̃_{f}^{1} (A_{J})_{π^{n}}; 0]

∥

[figure: 0; H̃_{f}^{0} ((A_{J})_{π^{c_{1}}} ) /Im(“π_{∗}^{n}”); β_{J}; H̃_{f}^{1} ((A_{J})_{π^{n}}); H̃_{f}^{1} ((A_{J})_{π^{n+c_{1}}} ); 0]

[figure: 0; H̃_{f}^{0} ((A_{I})_{π^{c_{1}}} ) /Im(“π_{∗}^{n}”); ≀; H̃_{f}^{1} ((A_{I})_{π^{n}}); ≀; H̃_{f}^{1} ((A_{I})_{π^{n+c_{1}}} ); ≀; 0]

∥

[figure: 0; H̃_{f}^{0} (A_{I}) /π^{n}; β_{I}; H̃_{f}^{1} ((A_{I})_{π^{n}}); H̃_{f}^{1} (A_{I})_{π^{n}}; 0]

It follows from the exact sequence 2.2.4(iv) that both arrows β_{I} and β_{J} in the diagram above are isomorphisms, proving that

H̃_{f}^{1} (A_{J})_{π^{n}} ⟶^{∼} H̃_{f}^{1} (A_{I})_{π^{n}}

as claimed.

(2.3.8) Lemma. Assume that V_{I} satisfies (C2) and let c_{2} ≥ 0 be as in (C2′). If J ∈ X(I)_{n+c_{2}} with n ≥ 1, then

(1) T_{J} satisfies (C2′) with the same value of c_{2} (hence V_{J} satisfies (C2)).

(2) If n ≥ 1 + ord_{π}(2) and if V_{I} satisfies (C3), then V_{J} also satisfies (C3). If the groups

Hom_{F_{𝔭}[G_{v}]} ((V_{J})_{v}^{±} , ((V_{J})_{v}^{±})^{∗} (1)) = 0 vanish for all v | p, then V_{J} satisfies (C4).

Proof. (1) Put C_{J} := Im (O[G_{K}] → End_{O} (T_{J})). Under the isomorphism

End_{O}(T_{J})/π^{c_{2}+1} = End_{O}(T_{J}/π^{c_{2}+1}) ⟶^{∼} End_{O}(T_{I}/π^{c_{2}+1}) = End_{O}(T_{I})/π^{c_{2}+1}

C_{J} (mod π^{c_{2}+1}) corresponds to C_{I} (mod π^{c_{2}+1}). As π^{c_{2}}End_{O}(T_{I}) ⊆ C_{J} by assumption, we have

π^{c_{2}}End_{O}(T_{J}) ⊆ C_{J} + π^{c_{2}+1}End_{O}(T_{J}).

Nakayama’s Lemma then implies π^{c_{2}}End_{O}(T_{J}) ⊆ C_{J}.

(2) As V_{I} satisfies (C2) and (C3), Hom_{O[G_{v}]}(T_{I}, T_{I}^{∗}(1)) is a free O-module of rank one. In the exact sequence

0 → Hom_{O[G_{v}]}(T_{I}, T_{I}^{∗}(1))/π^{n+c_{2}} → Hom_{O[G_{v}]}(T_{I}/π^{n+c_{2}}, T_{I}^{∗}(1)/π^{n+c_{2}}) → Ext_{O[G_{v}]}^{1}(T_{I}, T_{I}^{∗}(1)) →^{π^{n+c_{2}}} ⋯

the Ext^{1} term is killed by π^{c_{2}} ([Cu-Re, Thm. 29.4]), and the same is true if T_{I} is replaced by T_{J} (by (1)). Using the isomorphism

γ : Hom_{O[G_{v}]}(T_{I}/π^{n+c_{2}}, T_{I}^{∗}(1)/π^{n+c_{2}}) ⟶^{∼} Hom_{O[G_{v}]}(T_{J}/π^{n+c_{2}}, T_{J}^{∗}(1)/π^{n+c_{2}})

we see that there is a non-zero element α ∈ Hom_{O[G_{v}]}(T_{J}, T_{J}^{∗}(1)) such that π^{c_{2}}(α_{I} (mod π^{n+c_{2}}) corresponds to π^{c_{2}}(α (mod (π^{n+c_{2}}) under the isomorphism γ. As V_{J} satisfies (C2), α induces an isomorphism α_{J} : V_{J} ⟶^{∼} V_{J}^{∗}(1) such that α_{J}^{∗}(1) = εα_{J} for ε = ±1. Since α_{I}^{∗}(1) = −α_{I}, the definition of α implies that ε ≡ −1 (mod π^{n}) ≡ −1 (mod 2π), hence ε = −1. If Hom_{F_{𝔭}[G_{v}]}((V_{J})_{v}^{±}, ((V_{J})_{v}^{±})^{∗}(1)) = 0 for all v | p (for example, this is always true if (V_{J})_{v}^{±} are of the form described in 2.1.6(ii) (“Pančiškin’s condition”)), then α_{J} induces injective maps

α_{J,v}^{±} : (V_{J})_{v}^{±} ↪ ((V_{J})_{v}^{∓})^{∗}(1).

Counting dimensions, we see that all maps α_{J,v}^{±} are isomorphisms, which means that V_{J} satisfies (C4).

(2.3.9) Lemma. Assume that V_{J} satisfies (C1′), (C2′) and (C3). Then

π^{c_{1}d}H_{ur}^{1}(G_{v}, A_{J}) = 0, π^{c_{1}+c_{2}} (H^{1}(G_{v}, A_{J})/H_{ur}^{1}(G_{v}, A_{J})) = 0

for all v ∈ S_{f}, v ∤ p (where d = dim_{F_{𝔭}}(V_{J})).

Proof. H^{1}(G_{v}, A_{J})/H_{ur}^{1}(G_{v}, A_{J}) is dual to

H_{ur}^{1}(G_{v}, T_{J}^{∗}(1)) = (T_{J}^{∗}(1)^{I_{v}}) /(Fr(v) − 1) ⟶^{∼} (A_{J}^{∗}(1)^{I_{v}})^{Fr(v)=1} = H^{0}(G_{v}, A_{J}^{∗}(1))

(as Fr(v) − 1 acting on V_{J}^{I_{v}} ⟶^{∼} V_{J}^{∗}(1)^{I_{v}} is an isomorphism). Multiplying α_{J}(T_{J}) by a constant, we may assume that α_{J}(T_{J}) ⊆ T_{J}^{∗}(1) and α_{J}(T_{J}) ⊄ πT_{J}^{∗}(1), which implies that π^{c_{2}}T_{J}^{∗}(1) ⊆ α_{J}(T_{J}). It follows that the module M defined by

0 ⟶ M ⟶ A_{J} ⟶^{α_{J}}A_{J}^{∗}(1) ⟶ 0

is killed by π^{c_{2}}. As π^{c_{1}}H^{0}(G_{v}, A_{J}) = 0, π^{c_{1}+c_{2}} kills H^{0}(G_{V} , A_{J}^{∗}(1)). Writing

0 ⟶ V_{J}^{I_{v}}/T_{J}^{I_{v}} ⟶ A_{J}^{I_{v}} ⟶ N_{v} ⟶ 0

(with N_{v} finite) and (again) using the fact that Fr(v) − 1 acting on V_{J}^{I_{v}} ⟶^{∼} V_{J}^{∗}(1)^{I_{v}} is an isomorphism, we get

H_{ur}^{1}(G_{v}, A_{J}) = A_{J}^{I_{v}}/(Fr(v) − 1) = N_{v}/(Fr(v) − 1).

This is an O-module of length equal to

ℓ_{O} (N_{v}/(Fr(v) − 1)) = ℓ_{O} (N_{v}^{Fr(v)=1}) = ℓ_{O} (H^{0}(G_{v}, A_{J})) ≤ c_{1}d,

hence killed by π^{c_{1}d} as claimed.

(2.3.10) Theorem. Assume that V_{I} satisfies (C1)–(C4) with constants c_{1}, c_{2} in (C1′),(C2′). Put d = dim_{F_{𝔭}}(V_{I}) and let c_{3} ≥ 0 be an integer such that π^{c_{3}} kills

� (G_{K,S}, A_{I}; Δ) := H̃_{f}^{1}(G_{K,S}, A_{I}; Δ)/pr_{∗}H̃_{f}^{1}(G_{K,S}, V_{I}; Δ).

Let J ∈ X(I)_{n} with n ≥ (d + 3)c_{1} + 4c_{2} + 2ord_{π}(2) + 1 + c_{3} and assume that V_{J} satisfies (C4). Then

dim_{F_{𝔭}} H̃_{f}^{1}(G_{K,S}, V_{I}; Δ) − H̃_{f}^{1}(G_{K,S}, V_{J}; Δ) ≥ 0

is an even integer.

Proof. By Theorem 2.3.7 and Lemma 2.3.8, V_{J} satisfies (C1)–(C3), with the same constants c_{1}, c_{2} in (C1′),(C2′). Adjusting α_{J} as in the proof of Lemma 2.3.9, it follows from the condition (C4) for V_{J} that

(2.3.10.1)  π^{c_{2}} ((T_{J})_{v}^{∓})^{∗} (1) ⊆ α_{J} ((T_{J})_{v}^{±})

for all v | p. The local conditions for M = Ker (α_{J} : A_{J} → A_{J}^{∗}(1))

U_{v}^{+}(M) = \{ \begin{matrix}0 & v ∈ S_{f}, v ∤ p \\ Ccont^{•}(G_{v}, M ∩ (A_{J})_{v}^{+}) & v | p. \end{matrix}

define a Selmer complex ℝΓ̃_{f}(G_{K,S}, M; Δ) sitting in a distinguished triangle

ℝΓ̃_{f}(G_{K,S}, M; Δ) ⟶ ℝΓ̃_{f}(G_{K,S}, A_{J}; Δ)⟶^{α_{J}}ℝΓ̃_{f}(G_{K,S}, A_{J}^{∗}(1); Δ)

The dual local conditions Δ^{∗}(1) for T_{J}^{∗}(1) are given by

U_{v}^{+}(T_{J}^{∗}(1)) = \{ \begin{matrix}Ccont^{•}(G_{v}, T_{J}) & v ∈ S_{f}, v ∤ p \\ Ccont^{•}(G_{v}, ((T_{J}^{−})_{v})^{∗}(1)) & v | p. \end{matrix}

It follows from 2.3.10.1 that there is a distinguished triangle

ℝΓ̃_{f}(G_{K,S}, A_{J}^{∗}(1); Δ) ⟶ ℝΓ̃_{f}(G_{K,S}, A_{J}^{∗}(1); Δ^{∗}(1)) ⟶ ⨁_{{v∈S_{f} \atop v∤p}} Ccont^{•}(G_{v}, A_{J}) ⊕ C^{•}

with π^{c_{2}}H^{q}(C^{•}) = 0 for all q. Combining the two triangles and using Lemma 2.3.9, we deduce that the kernel (resp. cokernel) of the map

(α_{J})_{∗} : H̃_{f}^{1}(G_{K,S}, A_{J}; Δ) → H̃_{f}^{1}(G_{K,S}, A_{J}^{∗}(1); Δ^{∗}(1))

is killed by π^{c_{1}+c_{2}} (resp. by π^{(d+1)c_{1}+c_{2}}). The pairing 2.2.5 (non-degenerate up to 2-torsion)

� (G_{K,S}, A_{J}; Δ) × � (G_{K,S}, A_{J}^{∗}(1); Δ^{∗}(1)) → F_{𝔭}/O

then induces via id×(α_{J})_{∗} a skew-symmetric pairing on � (G_{K,S}, A_{J}; Δ), with kernel killed by 2π^{(d+2)c_{1}+4c_{2}}. It follows from the theory of symplectic spaces that there is an exact sequence of finite O-modules

0 ⟶ Y (J) ⟶ Z(J) ⊕ Z(J) ⟶ � (G_{K,S}, A_{J}; Δ) ⟶ 0

with 4π^{(d+2)c_{1}+4c_{2}}Y (J) = 0. Put r(J) := dim_{F_{𝔭}} H̃_{f}^{1}(G_{K,S}, V_{J}; Δ). Then

\begin{aligned}H̃_{f}^{1}(G_{K,S}, A_{J}; Δ) & ≅ (F_{𝔭}/O)^{r(J)} ⊕ (Z(J) ⊕ Z(J)) /Y (J) \\ H̃_{f}^{1}(G_{K,S}, A_{I}; Δ) & ≅ (F_{𝔭}/O)^{r(I)} ⊕ (Z(I) ⊕ Z(I)) /Y (I) \end{aligned}

In the isomorphism of Theorem 2.3.7,

H̃_{f}^{1}(G_{K,S}, A_{J}; Δ)_{π^{n−c_{1}}} ⟶^{∼} H̃_{f}^{1}(G_{K,S}, A_{I}; Δ)_{π^{n−c_{1}}} ,

the right hand side is isomorphic to

(O/π^{n−c_{1}}O)^{r(I)} ⊕ K, π^{c_{3}}K = 0

and the left hand side is isomorphic to

(O/π^{n−c_{1}}O)^{r(J)} ⊕ ((Z(J) ⊕ Z(J))/Y (J))_{π^{n−c_{1}}} .

This implies that r(I) − r(J) = 2 × (number of generators of the O-module 4π^{(d+2)c_{1}+4c_{2}+c_{3}}Z(J)) is even as claimed.

3. Selmer groups in Hida families

(3.1) The notation is as in 1.3; in particular, i_{p} and i_{∞} are fixed.

(3.1.1) Let f be a normalized newform on Γ_{1}(N) of weight k ≥ 2 and character χ. We assume that

• f is ordinary at p.

• k ≥ 2 is even.

• p ∤ cond(χ).

Under these assumptions, the two cases (i), (ii) of 1.3.5 boil down to Case (I): p ∤ N. Case (II): p ∥ N, k = 2, a_{p}^{2} = χ(p). We say that we are in the exceptional case (a subcase of (II)) if p ∥ N, k = 2, a_{p} = χ(p) = 1.

(3.1.2) Let V be the two-dimensional F_{𝔭}-representation V = M_{𝔭}(k/2) = V (f)(k/2) of G_{ℚ,S} (where S consists of primes dividing pN and ∞). As in 1.3.4, we have

Λ^{2}V ⟶^{∼} F_{𝔭}(1) ⊗ [χ], V ⊗ [χ^{−1}] ⟶^{∼} V^{∗}(1).

(3.1.3) Lemma. For every finite extension K/ℚ and every non-archimedian prime v ∤ p of K,

H_{ur}^{1}(G_{v}, V) = 0,  H^{i}(G_{v}, V) = H^{i}(G_{v}, V^{∗}(1)) = 0  for i = 0, 1, 2.

Proof. V is pure of weight −1. This means that, for every v ∤ N, all eigenvalues of Fr(v)_{geom} on V have absolute values (Nv)^{−1/2}, hence H^{0}(G_{v}, V) = H^{0}(G_{v}, V^{∗}(1)) = 0. If v | N, then results of [La], [Ca] imply that both spaces V^{I_{v}}, V^{∗}(1)^{I_{v}} are either zero or Fr(v)_{geom} acts on them with weights less than zero, showing that H^{0}(G_{v}, V) = H^{0}(G_{v}, V^{∗}(1)) = 0. The rest follows from Tate’s local duality H^{i}(G_{v}, V^{∗}(1)) ⟶^{∼} H^{2−i}(G_{v}, V)^{∗} and Euler characteristic formula ∑_{i=0}^{2}(−1)^{i}dim H^{i}(G_{v}, V) = 0 (and dim H_{ur}^{1}(G_{v}, V) = dim H^{0}(G_{v}, V)).

(3.1.4) V is reducible as a representation of G_{ℚ_{p}}; we have a short exact sequence

(3.1.4.1)  0 ⟶ V_{p}^{+} ⟶ V ⟶ V_{p}^{−} ⟶ 0

with V_{p}^{±} = F^{±}M_{𝔭}(k/2) in the notation of 1.3.5. Both V_{p}^{±} are crystalline representations of G_{ℚ_{p}}. The filtered module D_{cris}(V_{p}^{±}) = H^{0}(G_{ℚ_{p}}, V_{p}^{±} ⊗_{ℚ_{p}} B_{cris}) is one-dimensional over F_{𝔭}, with crystalline Frobenius f acting by the scalar α_{p}p^{−k/2} on V^{+} and β_{p}p^{−k/2} = χ(p)α_{p}^{−1}p^{k/2−1} on V^{−}. We have gr_{F}^{i} D_{dR}(V_{p}^{±}) ≠ 0 if and only if i = −k/2 for V_{p}^{+} and i = k/2 − 1 for V_{p}^{−}, so the exact sequence 3.1.4.1 satisfies Pančiškin’s condition (2.1.6 (ii)).

(3.1.5) The subspace H_{f}^{1}(G_{ℚ_{p}}, V) ⊂ H^{1}(G_{ℚ_{p}}, V) can be described fairly explicitly. Let us first analyse Case (I): p ∤ N. This implies that V is a crystalline representation of G_{ℚ_{p}}. In fact, for k > 2, every extension of A by B(k−1) with A, B unramified (such as M_{𝔭} = V (−k/2)) is automatically crystalline ([Bures, Exp. IV, Prop. 3.1]). If k = 2, then V ⊂ V_{p}(J_{1}(N)) and J_{1}(N) has good reduction at p. Both α_{p}p^{−k/2}, β_{p}p^{−k/2} have absolute values p^{−1/2}, hence

(3.1.5.1)  D_{cris}(W)^{f=1} = D_{cris}(W^{∗}(1))^{f=1} = 0,  W = V, V_{p}^{±}.

This is still true if G_{ℚ_{p}} is replaced by G_{K_{v}}, for a finite extension K_{v}/ℚ_{p}. As a result, we get from 2.1.6

H_{f}^{1}(G_{K_{v}}, V) = H_{g}^{1}(G_{K_{v}}, V) = Im[H^{1}(G_{K_{v}}, V_{p}^{+}) → H^{1}(G_{K_{v}}, V)]
H^{0}(G_{K_{v}}, W) = H^{0}(G_{K_{v}}, W^{∗}(1)) = 0, W = V, V_{p}^{±}.

(3.1.6) Case (II): p ∥ N, k = 2, a_{p}^{2} = χ(p). In this case, V ⊂ V_{p}(J_{1}(N/p; p)^{p−new}), where J_{1}(N/p; p) is the Jacobian of X_{1}(N/p; p) and J_{1}(N/p; p)^{p−new} is its quotient by the image of

(s^{∗}, t^{∗}) : J_{1}(N/p) × J_{1}(N/p) → J_{1}(N/p; p).

It is known that J_{1}(N/p; p)^{p−new} has completely toric reduction at p ([De-Ra]), which implies that V is not crystalline. The quotient V_{p}^{−} is unramified, with Fr(v)_{geom} acting by the scalar a_{p}, and V_{p}^{+} ⟶^{∼} V_{p}^{−}(1) : in the exceptional case, G_{ℚ_{p}} acts trivially on V_{p}^{−} and V is a Kummer extension

0 ⟶ F_{𝔭}(1) ⟶ V ⟶ F_{𝔭} ⟶ 0

with extension class q ∈ H^{1}(G_{ℚ_{p}}, F_{𝔭}(1)) = ℚ_{p}^{×} ˆ⊗F_{𝔭} such that q ∉ ℤ_{p}^{×} ˆ⊗F_{𝔭}. For example, if χ = 1 and f has coefficients in ℚ, it corresponds to (the isogeny class of) an elliptic curve E/ℚ with ordinary reduction at p. The exceptional case occurs iff E has split multiplicative reduction at p, in which case q = q_{E} ⊗ 1, where q_{E} ∈ ℚ_{p}^{×} is Tate’s multiplicative period of E/ℚ_{p} : E(\overline{ℚ}_{p}) = \overline{ℚ}_{p}^{×}/q_{E}^{ℤ}. In the exceptional case (V_{p}^{−} = V_{p}^{+}(−1) = F_{𝔭}), there is an exact sequence

0 ⟶ H^{0}(G_{ℚ_{p}}, F_{𝔭})⟶^{∂}ℚ_{p}^{×} ˆ⊗F_{𝔭}⟶^{α}H_{g}^{1}(G_{ℚ_{p}}, V)⟶^{β}H_{g}^{1}(G_{ℚ_{p}}, F_{𝔭})⟶^{∂′}H_{g}^{2}(G_{ℚ_{p}}, F_{𝔭}(1)) ⟶ H_{g}^{2}(G_{ℚ_{p}}, V)

with ∂(1) = q. The groups H_{g}^{i} are defined in [Fo-PR, I.3.3.3]; for a semistable representation W this H_{g}^{1} coincides with that of [Bl-Ka] and H_{g}^{2}(G_{ℚ_{p}}, W) is dual to D_{cris}(W^{∗}(1))^{f=1}. The dual extension

0 ⟶ F_{𝔭}(1) ⟶ V^{∗}(1) ⟶ F_{𝔭} ⟶ 0

has extension class equal to q^{−1}. It follows that the map D_{cris}(F_{𝔭}(1)) ⟶ D_{cris}(V^{∗}(1)) is an isomorphism, hence H_{g}^{1}(G_{ℚ_{p}}, V) = H_{f}^{1}(G_{ℚ_{p}}, V) by 2.1.6 (i). The group H_{g}^{2}(G_{ℚ_{p}}, V) vanishes, as D_{cris}(V^{∗}(1))^{f=1} = D_{cris}(F_{𝔭}(1))^{f=1} = 0. Both groups H_{g}^{1}(G_{ℚ_{p}}, F_{𝔭}), H_{g}^{2}(G_{ℚ_{p}}, F_{𝔭}(1)) = H^{2}(G_{ℚ_{p}}, F_{𝔭}(1)) are isomorphic to F_{𝔭} ([Bl-Ka, Ex. 3.9]), which implies that ∂′ is an isomorphism and β = 0. Putting everything together, we obtain an exact sequence

0 ⟶ F_{𝔭} ⟶ H^{1}(G_{ℚ_{p}}, V_{p}^{+}) ⟶ H_{f}^{1}(G_{ℚ_{p}}, V) ⟶ 0.

If not in the exceptional case, then 3.1.5.1 holds again, which implies that

H_{f}^{1}(G_{ℚ_{p}}, V) = H_{g}^{1}(G_{ℚ_{p}}, V) = Im[H^{1}(G_{ℚ_{p}}, V_{p}^{+}) → H^{1}(G_{ℚ_{p}}, V)]
H^{0}(G_{ℚ_{p}}, W) = H^{0}(G_{ℚ_{p}}, W^{∗}(1)) = 0, W = V, V_{p}^{±}.

(3.1.7) The local calculations from 3.1.3–6 can be summed up as follows; for K/ℚ finite, consider local conditions

W_{v} = \{ \begin{matrix}H_{ur}^{1}(G_{v}, V) & v ∈ S_{f}, v ∤ p \\ H_{f}^{1}(G_{v}, V) & v | p \end{matrix}

corresponding to H_{f}^{1}(K, V) := S(K, V; W), resp.

U_{v}^{+}(V) = \{ \begin{matrix}0 & v ∈ S_{f}, v ∤ p \\ Ccont^{•}(G_{v}, V_{p}^{+}) & v | p \end{matrix}

U_{v}^{−}(V) = \{ \begin{matrix}Ccont^{•}(G_{v}, V) & v ∈ S_{f}, v ∤ p \\ Ccont^{•}(G_{v}, V_{p}^{−}) & v | p \end{matrix}

(S contains primes dividing pN and ∞). Then W_{v} = 0 for v ∤ p. It follows from (2.3.6.1) that,

In Case (I), H̃_{f}^{1}(G_{K,S}, V; Δ) = H_{f}^{1}(K, V). In Case (II), in the exceptional case, there is an exact sequence

0 ⟶ F_{𝔭} ⟶ H̃_{f}^{1}(G_{ℚ,S}, V; Δ) ⟶ H_{f}^{1}(ℚ, V) ⟶ 0,

otherwise, H̃_{f}^{1}(G_{ℚ,S}, V; Δ) = H_{f}^{1}(ℚ, V).

(3.2) For the rest of Section 3, the notation is as in 1.4. In particular, p > 3 (with the exception of 3.4.1–3).

(3.2.1) Let g be a p-stabilized ordinary newform of tame level N, with trivial character χ = 1 and even weight k_{0} ≥ 2. Then either Case (I): g = f^{0} is the p-stabilization of a newform f on Γ_{1}(N). Case (II): g = f is a newform on Γ_{1}(Np), k_{0} = 2, a_{p} = ±1.

(3.2.2) As in 1.4.7, g = g_{k_{0}} is an element of a p-adic family of p-stabilized newforms g_{k} of weight k on Γ_{1}(Np) for all k ≥ 2, k ≡ k_{0} (mod p^{c}). Assuming that F_{𝔭} is big enough in the sense of 1.4.4, each g_{k} corresponds to an arithmetic point P_{k} ∈ 𝔛^{arith}(R) of a fixed local factor R of 𝔥_{∞}^{ord}. The character of g_{k} is equal to χ_{k} = χω^{k_{0}}ω^{−k} = ω^{k_{0}−k}. In particular, for k ≡ k_{0} (mod (p − 1)p^{c}), k ≥ 2, we have again either g_{k} = (f_{k})^{0} or g_{k} = f_{k} for a newform f_{k} (the latter possible only if k = 2). The tame part ψ (resp. the wild part ε) of χω^{k_{0}−2} is equal to ψ = ω^{k_{0}−2} (resp. ε = 1). this implies that each P_{k} lies above (P_{k}) = (ı(γ) − γ^{k−2}) ∈ Spec(Λ) and that the invariant a ∈ ℤ/(p − 1)ℤ from 1.5.2 (iii) is equal to a ≡ k_{0} − 2 (mod (p − 1)).

(3.2.3) Let T (R) be the big Galois representation of G_{ℚ,S} from 1.5.1. In order to apply 1.5.2,4 we need the following

Assumption. If k_{0} ≡ 2 (mod (p − 1)), then ρ_{f,𝔭} has an irreducible residual representation.

The action of a ∈ ℤ_{p}^{×} on T (R) satisfies < a >_{Alb}= ω(a)^{k_{0}−2} < κ(a) >_{Alb}, where κ denotes the projection of ℤ_{p}^{×} to 1 + pℤ_{p}. Define a twisted Galois representation

T = “T (R) < −1/2 >”

as follows: as an R-module, T = T (R). The action of g ∈ G_{ℚ} such that χ_{cycl}(g) = a ∈ ℤ_{p}^{×} is given by the action of g on T (R) followed by ω(a)^{1−k_{0}/2} < κ(a)^{−1/2} >_{Alb} (morally, this is “< a >_{Alb}^{−1/2}”). The skew-symmetric bilinear form

⟨ , ⟩_{T (R)} : T (R) × T (R) ⟶ Λ < 1 > (1)

defines a skew-symmetric pairing

⟨ , ⟩_{T} : T × T ⟶ Λ(1)

which induces an isomorphism

T ⟶^{∼} T^{∗}(1) := Hom_{Λ} (T, Λ) (1) ⟶^{∼} Hom_{R} (T, ω_{R}) (1).

It follows from 1.5.2,4 that there is an exact sequence of R[G_{ℚ_{p}}]-modules

0 ⟶ T^{+} ⟶ T ⟶ T^{−} ⟶ 0

with T^{±} = “T (R)^{±} < −1/2 >”.

(3.2.4) For k ≡ k_{0} (mod (p − 1)p^{c}), k ≥ 2, let V (f_{k}) be the Galois representation associated to f_{k}. We know from 1.5.5 that the specialization of T (R) at P_{k} is isomorphic to

(T (R)/P_{k}T (R)) ⊗_{O} F_{𝔭} ⟶^{∼} V (f_{k})^{∗} ⟶^{∼} V (f_{k})(k − 1).

Recall that we transform Dirichlet characters into characters of G_{ℚ} by using geometric Frobenius elements (1.3.4). This forces us to adopt the same convention for p-adic characters, namely [κ](Fr(ℓ)_{geom}) = κ(ℓ). This implies that χ_{cycl} = [κ]^{−1}[ω]^{−1}, since χ_{cycl}(Fr(ℓ)_{geom}) = ℓ^{−1} = κ(ℓ)^{−1}ω(ℓ)^{−1}.

As < κ(a) >_{Alb}≡ κ(a)^{k−2} (mod P_{k}) for all a ∈ ℤ_{p}^{×}, it follows that the specialization of T at P_{k} is isomorphic to

\begin{matrix}(T/P_{k}T) ⊗_{O} F_{𝔭} ⟶^{∼} & V (f_{k})(k − 1) ⊗ [ω^{k_{0}/2−1}] ⊗ [κ^{k/2−1}] = V (f_{k}) ⊗ [ω^{k_{0}/2−k}] ⊗ [κ^{−k/2}] \\ = & V (f_{k})(k/2) ⊗ [ω^{(k_{0}−k)/2}] = V (f_{k})(k/2) ⊗ [(\frac{·}{p})^{(k_{0}−k)/2(p−1)}]. \end{matrix}

In particular, for k ≡ k_{0} (mod 2(p − 1)p^{c}),

(T/P_{k}T) ⊗_{O} F_{𝔭} ⟶^{∼} V (f_{k})(k/2).

Similarly, the specializations (T^{±}/P_{k}T^{±}) ⊗_{O} F_{𝔭} are then isomorphic to the F_{𝔭}[G_{ℚ_{p}}]-modules V_{p}^{±} from 3.1.4 (for f_{k} instead of f).

(3.3) Theorem A

(3.3.1) Theorem A. Let f_{k} be the family of newforms as in 3.2.2. If k_{0} ≡ 2 (mod (p − 1)), assume in addition that V (f_{k_{0}}) has an irreducible residual representation. Then there is an integer n ≥ c such that for every k ≡ k_{0} (mod 2(p − 1)p^{n}),

dim_{F_{𝔭}} H̃_{f}^{1}(G_{ℚ,S}, V (f_{k_{0}})(k_{0}/2); Δ) − dim_{F_{𝔭}} H̃_{f}^{1}(G_{ℚ,S}, V (f_{k})(k/2); Δ) ≥ 0

is even (where the local conditions are as in 3.1.7).

Proof. We apply Theorem 2.3.10 to the big representation T of G_{ℚ,S} and to its specializations at I = P_{k_{0}}, J = P_{k}. We must check the assumptions: First of all, T^{+} is free over R by Prop. 1.5.2(iii) and 1.5.4. The representation V_{J} = V (f_{k})(k/2) satisfies (C1) by Lemma 3.1.3, (C2) by [Ri 1, Thm. 2.3], (C3) by 1.3.4. The condition (C4) follows from Lemma 2.3.8 (2) (or from (1.3.5.2)).

(3.3.2) Corollary. Put ε_{k} = 1 if f_{k} is in the exceptional case, ε_{k} = 0 otherwise. Then

ε_{k_{0}} + dim_{F_{𝔭}}H_{f}^{1}(ℚ, V (f_{k_{0}})(k_{0}/2)) ≡ dim_{F_{𝔭}}H_{f}^{1}(ℚ, V (f_{k})(k/2)) + ε_{k} (mod 2)

whenever k ≡ k_{0} (mod (2(p − 1)p^{n})).

Proof. Use 3.1.7.

(3.4) Theorem B and Theorem C

(3.4.1) p-adic L-functions

Recall the basic properties of p-adic L-functions of modular forms, after Mazur-Tate-Teitelbaum [Ma-Ta-Te]: let f = ∑_{n≥1} a_{n}q^{n} be a newform of weight k ≥ 2 and character ε on Γ_{1}(N). Fix a root α of X^{2} − a_{p}X + ε(p)p^{k−1} = 0 satisfying ord_{p}(α) < k − 1 (with the convention that ε(p) = 0 if p | N). The p-adic L-function

χ ↦ L_{p,α}^{MT T} (f, χ)

constructed in [Ma-Ta-Te] is a function of continuous characters χ : ℤ_{p}^{×} ⟶ ℂ_{p}^{×} satisfying the following interpolation property:

L_{p,α}^{MT T} (f, x^{j}ψ) = (\frac{p^{j+1}}{α} )^{ν} \frac{j!}{(−2πi)^{j}G(\overline{ψ})} (1 − \frac{\overline{ψ}(p)ε(p)p^{k−2−j}}{α} ) (1 − \frac{ψ(p)p^{j}}{α} ) L_{∞}(f, \overline{ψ}, j + 1),

where j = 0, 1, …, k − 2, ψ is a Dirichlet character of conductor p^{ν}, G(\overline{ψ}) is the Gauss sum associated to \overline{ψ}, L_{∞}(f, \overline{ψ}, s) = ∑_{n≥1} a_{n}\overline{ψ}(n)n^{−s} and x : ℤ_{p}^{×} ↪ ℂ_{p}^{×} is the inclusion map. Rather confusingly, this p-adic L-function has values in a certain module of periods, rather than in ℂ_{p}, as the value of the complex L-function is not divided by a complex period. For s ∈ ℤ_{p} one often writes L_{p,α}^{MT T} (f, χ, s) for L_{p,α}^{MT T} (f, χκ^{s}). The case p = 2 is allowed in [Ma-Ta-Te]; κ then denotes the projection from ℤ_{2}^{∗} to 1 + 4ℤ_{2}. If the form f is ordinary, there is only one choice of α, namely α = α_{p} in the notation of 1.3.5; it is often omitted from the notation.

(3.4.2) Assume that k is even and ε = 1. In this case the p-adic L-function satisfies a particularly simple functional equation [Ma-Ta-Te, Sect. 17]:

(3.4.2.1)  L_{p,α}^{MT T} (f, x^{k/2−1}ψ, s) = w_{p}(f)ψ^{−1}(−Q)κ(Q)^{−s}L_{p,α}^{MT T} (f, x^{k/2−1}ψ^{−1}, −s),

where ψ is as above, w_{p}(f) = ±1 and Q denotes the largest positive divisor of N prime to p. Putting

Λ_{p,α}(f, s) = κ(Q)^{s/2}L_{p,α}^{MT T} (f, x^{k/2−1}, s − k/2),

the functional equation implies that

Λ_{p,α}(f, s) = w_{p}(f)Λ_{p,α}(f, k − s).

The value at the centre of symmetry of the functional equation is equal to

L_{p,α}^{MT T} (f, x^{k/2−1}) = \frac{(k/2 − 1)!}{(−2πi)^{k/2−1}} (1 − \frac{ε(p)p^{k/2−1}}{α} ) (1 − \frac{p^{k/2−1}}{α} ) L_{∞}(f, k/2).

(3.4.3) We now restrict our attention to ordinary forms. Changing notation, assume that p ∤ N and let g be an ordinary p-stabilized newform of weight k ≥ 2 and character ε on Γ_{1}(Np^{r}), r ≥ 1. Then either g = f is a newform on Γ_{1}(Np^{r}), r ≥ 1, or g = f^{0} is equal to the p-stabilization of a newform f on Γ_{1}(N). In either case, a_{p}(g) = α_{p}(f). Greenberg-Stevens [Gr-St] use a slightly different normalization of p-adic L-functions, namely

L_{p}^{GS}(g, ψκ^{s}) = \frac{L_{p}^{MT T} (f, ψκ^{s−1})}{Ω_{f}^{sgn(ψ)}},

with ψ as in 3.4.1 and suitable periods Ω_{f}^{±} ∈ ℂ^{×} of f. As before, the function on the L.H.S. will be denoted by L_{p}^{GS}(g, ψ, s). If k is even and ε = 1, put

L_{p}(f, s) = L_{p}^{GS}(g, ω^{k/2−1}, s).

(3.4.4) Two-variable p-adic L-functions

Assume now that p > 3 and that we are in the situation of 3.2.2. There is a two-variable p-adic L-function interpolating the p-adic L-functions of the forms f_{k} ([Gr-St], [Ki]). In the notation of [Gr-St] (omitting the variables Φ, κ), this function L_{p}^{GS}(k, ψ, s) depends on k ∈ k_{0} + p^{c}ℤ_{p} and ψ, s as in 3.4.3. Its main properties are the following ([Gr-St, Thm. 5.15, Cor. 5.17]):

(i) L_{p}^{GS}(k, ψ, s) is analytic in (k, s) ∈ (k_{0} + p^{c}ℤ_{p}) × ℤ_{p}.

(ii) If k ∈ k_{0} + p^{c}ℤ_{p} is an integer, k ≥ 2, then L_{p}^{GS}(k, ψ, s) = C_{k}L_{p}^{GS}(g_{k}, ψ, s) for some C_{k} ∈ ℂ_{p}^{×}.

(iii) L_{p}^{GS}(k, ψ, s) = −uψ^{−1}(−N)κ(−N)^{k/2−s}L_{p}^{GS}(k, ω^{k_{0}−2}ψ^{−1}, k − s), where u^{2} = ω^{k_{0}−2}(−N).

Put

\begin{aligned}L_{p}(k, s) & = L_{p}^{GS}(k, ω^{k_{0}/2−1}, s) \\ Λ_{p}(k, s) & = κ(N)^{s/2}L_{p}(k, s) \end{aligned}

Then (iii) implies that

Λ_{p}(k, s) = w_{p}Λ_{p}(k, k − s)

with w_{p} = ±1. Comparing with (3.4.2.1) – with N playing the rôle of Q – we see that, for every integer k ≥ 2, k ≡ k_{0} (mod (2(p − 1)p^{c})), we have

w_{p}(f_{k}) = w_{p}, L_{p}(k, s) = C_{k}L_{p}(f_{k}, s)

and

L_{p}(k, k/2) = C_{k} \frac{(k/2 − 1)!}{(−2πi)^{k/2−1}} (Eul_{k})\frac{L_{∞}(f_{k}, k/2)}{Ω_{f_{k}}^{±}}, (± = (−1)^{k/2−1})

where the Euler factor is equal to

(Eul_{k}) = (1 − \frac{p^{k/2−1}}{α_{p}(f_{k})})^{b_{k}} , b_{k} = \{ \begin{matrix}1 & if g_{k} = f_{k} \\ 2 & if g_{k} = (f_{k})^{0} \end{matrix}

It follows from [Ma-Ta-Te, Prop. 15] and 1.3.5 that the Euler factor Eul_{k} vanishes iff we are in the exceptional case (in the language of 3.1.1). The vanishing of Eul_{k} is sometimes referred to as a “trivial zero ”of L_{p}(f_{k}, s). The value of w_{p} is related to the sign in the archimedean functional equation

Λ_{∞}(f_{k}, s) = w_{∞}(f_{k})Λ_{∞}(f_{k}, k − s)

by

w_{p}(f_{k}) = w_{∞}(f_{k}) × \{ \begin{matrix}−1 & in the exceptional case \\ 1 & otherwise \end{matrix}

i.e. w_{p}(f_{k}) = w_{∞}(f_{k})(−1)^{ε_{k}} ([Ma-Ta-Te, Sect. 18).

(3.4.5) Write w_{p} = (−1)^{e_{p}} with e_{p} = 0 or 1. It follows that the function

F (k) := \frac{L_{p}(k, s)}{(s − k/2)^{e_{p}}} |_{s=k/2}

is analytic for k ∈ k_{0} + p^{c}ℤ_{p}. A variant of conjectures formulated in [Gr 2] says the following:

Greenberg’s Conjecture. The function F (k) is not identically zero.

Equivalently, for all but finitely many k ∈ ℤ, k ≥ 2, k ≡ k_{0} (mod (p−1)p^{c}), we have ord_{s=k/2}L_{p}(f_{k}, s) = e_{p}. This conjecture is known when f_{k_{0}} is a modular form of weight 2 corresponding to an elliptic curve E over ℚ with complex multiplication for which e_{p} = 0 ([Gr 1], [Ro]).

(3.4.6) As before, let k ≡ k_{0} (mod 2(p − 1)p^{c}), k ≥ 2. We recall some of the known results relating behaviour of L_{p}(f_{k}, s) at the central point s = k/2 and Selmer groups of Galois representations V_{k} := V (f_{k})(k/2).

(1) [Ne 2, Thm. C,D] Assume k > 2 (hence f_{k} is a newform on Γ_{0}(N), p ∤ N).

(i) If ord_{s=k/2}L_{p}(f_{k}, s) = 1 then dim_{F_{𝔭}}H_{f}^{1}(ℚ, V_{k}) = 1.

(ii) If L_{p}(f_{k}, k/2) ≠ 0 and if there is an imaginary quadratic field K = ℚ(\sqrt{D}) with odd discriminant D < 0 such that all primes dividing pN split in K and

ord_{s=k/2}L_{p}(f_{k} ⊗ (\frac{D}{.} ), s) = 1

then H_{f}^{1}(ℚ, V_{k}) = 0.

(2) [Ka] If L_{p}(f_{k}, k/2) ≠ 0, then H_{f}^{1}(ℚ, V_{k}) = 0. In fact, Kato has announced a proof of the inequality

dim_{F_{𝔭}}H_{f}^{1}(ℚ, V_{k}) ≤ ord_{s=k/2}L_{p}(f_{k}, s)

even without the ordinarity assumption.

(3.4.7) Theorem B. Under the assumptions of Theorem A, Greenberg’s Conjecture for L_{p}(k, s) implies that

dim_{F_{𝔭}}H_{f}^{1}(ℚ, V_{k_{0}}) ≡ e_{∞} (mod 2),

where e_{∞} ≡ e_{p} + ε_{k_{0}} (mod 2) is such that w_{∞}(f_{k_{0}}) = (−1)^{e_{∞}}.

Proof. Choose k ≡ k_{0} (mod 2(p − 1)p^{n}), k > 2, k ∈ ℤ for big enough n. Greenberg’s Conjecture together with 3.4.3 imply that

dim_{F_{𝔭}}H_{f}^{1}(ℚ, V_{k}) = ord_{s=k/2}L_{p}(f_{k}, s) = e_{p}.

By Theorem A,

dim_{F_{𝔭}}H_{f}^{1}(ℚ, V_{k_{0}}) ≡ dim_{F_{𝔭}}H_{f}^{1}(ℚ, V_{k}) + ε_{k_{0}} ≡ e_{p} + ε_{k_{0}} (mod 2)

(we have ε_{k} = 0, since k > 2).

(3.4.8) Theorem C. Let E be a modular elliptic curve over ℚ with ordinary reduction at a prime p > 3. Assume that the p-torsion E_{p}(\overline{ℚ}) is an irreducible 𝔽_{p}[G_{ℚ}]-module and that Greenberg’s conjecture holds for the two-variable p-adic L-function of E. Then

dim_{ℚ} (E(ℚ) ⊗ ℚ) + cork_{ℤ_{p}}� (E/ℚ) ≡ ord_{s=1}L_{∞}(E, s) (mod 2).

Proof. By modularity, E corresponds to a cusp form f of weight k_{0} = 2 on Γ_{0}(N) with rational coefficients. The form f is ordinary iff E has ordinary reduction (possibly bad) at p. The p-torsion E_{p}(\overline{ℚ}) is a residual representation of V_{k_{0}} = V_{p}(E). Apply Theorem B.

(3.4.9) The parity statement of Theorem C has been proved in the following cases:

Unconditional results:

(1) E : y^{2} = x^{3} − Dx (resp. E : x^{3} + y^{3} = A), p = 2 (resp. p = 3) (Birch-Stephens [Bi-St]).

(2) E is modular and ord_{s=1}L_{∞}(E, s) ≤ 1; in this case dim_{ℚ} (E(ℚ) ⊗ ℚ) = ord_{s=1}L_{∞}(E, s) and � (E/ℚ) is finite (Kolyvagin [Ko 1]).

(3) E has complex multiplication and ordinary reduction at p > 3 (Guo [Gu], who also proves Theorem A for some modular forms of higher weight with complex multiplication).

(4) E is modular and either p = 2, or E_{p}(\overline{ℚ}) is a reducible 𝔽_{p}[G_{ℚ}]-module (Monsky [Mo]).

Conditional results:

(5) E is modular, p > 2, the Galois representation G_{ℚ} ⟶ Aut_{O⊗ℤ_{p}}(T_{p}(E)) is surjective (where O = End(E)) and the higher Heegner points satisfy a suitable non-degeneracy conjecture (Kolyvagin [Ko 2]).

(6) E is modular with split multiplicative reduction at p > 3, w_{∞} = −1 and a variant of Greenberg’s conjecture holds (Greenberg [Gr 3]).

(3.4.10) The results of Greenberg [Gr 1] and [Ro] on Greenberg’s Conjecture for elliptic curves with complex multiplication and e_{p} = 0 give one instance when Theorem C gives an unconditional result. However, this case is already covered by (3) above.

(3.5) Theorem A′ and Theorem B′

(3.5.1) In the situation of 3.2, we can also consider specializations of T at more general arithmetic points. Using the notation of 1.4.7, fix an arithmetic point \overline{P} ∈ 𝔛^{arith}(𝔥(K)) and denote by P = pr^{−1}(\overline{P}) ∈ 𝔛^{arith}(R) its preimage in R. Alternatively, we can forget \overline{P} and simply fix P ∈ 𝔛^{arith}(R) containing the minimal prime ideal 𝔮 = Ker [R ⟶ 𝔥(K)] of R.

(3.5.2) Let us compute the Galois representation T/PT. First of all, P lies above (P_{k,ε}) ∈ Spec(Λ) for some integer k ≥ 2 and a character of finite order ε : Γ ⟶ O^{×}. By 1.4.7, P corresponds to a p-stabilized ordinary newform f_{P} of weight k and character χ_{P} = εω^{k_{0}−k}. As < κ(a) >_{Alb}≡ ε(a)κ(a)^{k−2} (mod P), the same calculation as in 3.2.4 shows that

\begin{matrix}V_{[P]} := (T/PT) ⊗_{O} F_{𝔭} ⟶^{∼} & V (f_{P})^{∗} ⊗ [κ^{k_{0}/2−1}] ⊗ [ε^{1/2}κ^{k/2−1}] \\ ⟶^{∼} & V (f_{P})(k − 1) ⊗ [χ_{P}^{−1}ω^{k_{0}/2−1}ε^{1/2}κ^{k/2−1}] = V (f_{P}) ⊗ [ω^{−k_{0}/2}κ^{−k/2}ε^{−1/2}] \end{matrix}

(as p ≠ 2, ε^{1/2} : Γ ⟶ O^{×} is well defined). If k is even, we have

V_{[P]} ⟶^{∼} V (f_{P})(k/2) ⊗ [ω^{(k−k_{0})/2}ε^{−1/2}].

As (ω^{(k−k_{0})/2}ε^{−1/2})^{2} = χ_{P}^{−1}, it makes sense to denote ω^{(k−k_{0})/2}ε^{−1/2} by “χ_{P}^{−1/2} ”, hence

(3.5.2.1)  V_{[P]} ⟶^{∼} V (f_{P})(k/2) ⊗ [“χ_{P}^{−1/2} ”].

If the Fourier expansion of f_{P} is ∑_{n≥1} a_{n}(f_{P})q^{n}, then ∑_{n≥1} “χ_{P}^{−1/2} ”(n)a_{n}(f_{P})q^{n} is the Fourier expansion of a Hecke eigenform f̃_{P} on some Γ_{0}(Np^{r}) (i.e. f̃_{P} has trivial character). The formula (3.5.2.1) then becomes

V_{[P]} ⟶^{∼} V (f̃_{P})(k/2).

This representation is self-dual in the usual way: V_{[P]} ⟶^{∼} V_{[P]}^{∗}(1).

(3.5.3) From now on, P is fixed and we assume that the weight k of f_{P} is even. We have the following Galois representations associated to T:

T^{∗} = Hom_{Λ}(T, Λ),  (T^{∗})^{±} = Hom_{Λ}(T^{∓}, Λ),  A^{∗} = Hom_{O,cont}(T, F_{𝔭}/O),
A = Hom_{O,cont}(T^{∗}, F_{𝔭}/O), A^{±} = Hom_{O,cont}((T^{∗})^{∓}, F_{𝔭}/O)

and also various specializations

T_{[P]} = Im (T/PT ⟶ V_{[P]}) ,  T_{[P]}^{+} = T^{+}/PT^{+} ⊂ T_{[P]},  V_{[P]}^{+} = T_{[P]}^{+} ⊗_{O} F_{𝔭} ⊂ V_{[P]},
A_{[P]} = V_{[P]}/T_{[P]}, A_{[P]}^{+} = V_{[P]}^{+}/T_{[P]}^{+}.

The isomorphism T ⟶^{∼} T^{∗}(1) defined by the pairing ⟨ , ⟩_{T} : T × T ⟶ Λ(1) induces isomorphisms

(3.5.3.1)  T^{±} ⟶^{∼} (T^{∗})^{±}(1),  A ⟶^{∼} A^{∗}(1).

As in 2.2, we consider Selmer complexes, this time for “big Galois representations” X = T, A of G_{ℚ,S}, where S = {ℓ | pN} ∪ {∞}. All cochains will be continuous with respect to the 𝔪-adic topology on T, T^{∗}, T^{±} and the discrete topology on A, A^{∗}, A^{±}. The local conditions will be given by

Δ_{v} : U_{v}^{+}(X) = \{ \begin{matrix}0 & v ≠ p, v ∈ S_{f} \\ Ccont^{•}(G_{ℚ_{p}}, X^{+}) & v = p \end{matrix}

Identifying X^{∗}(1) with X as in (3.5.3.1), the dual local conditions become

Δ_{v}^{∗}(1) : U_{v}^{+}(X) = \{ \begin{matrix}Ccont^{•}(G_{ℚ_{v}}, X) & v ≠ p, v ∈ S_{f} \\ Ccont^{•}(G_{ℚ_{p}}, X^{+}) & v = p \end{matrix}

(one can make the local conditions at v ≠ p also self dual, but it is irrelevant for our purposes). For the specializations X_{[P]} = T_{[P]}, A_{[P]}, V_{[P]} we use the same local conditions

Δ_{v} : U_{v}^{+}(X_{[P]}) = \{ \begin{matrix}0 & v ≠ p, v ∈ S_{f} \\ Ccont^{•}(G_{ℚ_{p}}, X_{[P]}^{+}) & v = p \end{matrix}

In this case, the above formula for Δ_{v}^{∗}(1) will be true up to a finite error term.

(3.5.4) Recall that an R-module M of finite type is pseudo-null if Supp(M) has codimension ≥ 2 in Spec(R). As dim(R) = 2 and R/𝔪 is finite, this is equivalent to M being finite. We shall ignore pseudo-null modules and work in the category (R-mod)/(ps-null) which is obtained from (R-mod) by inverting all morphisms with pseudo-null kernel and cokernel. Recall that, for every R-module of finite type with P ∈ Supp(M), there is a non-zero homomorphism M/PM ⟶ R/P.

(3.5.5) Lemma. The canonical maps

T^{+}⊗^{𝕃}_{R}R/P ⟶ T^{+}/PT^{+} ⟶ T_{[P]}^{+} (i)

T ⊗^{𝕃}_{R}R/P ⟶ T/PT ⟶ T_{[P]} (ii)

ℝΓ̃_{f}(G_{ℚ,S}, T; −)⊗^{𝕃}_{R}R/P ⟶ ℝΓ̃_{f}(G_{ℚ,S}, T_{[P]}; −) (− = Δ, Δ^{∗}(1)) (iii)

are isomorphisms in (i) D^{b}(R-mod); (ii), (iii) D^{b}(R-mod)/(ps-null).

Proof. (i) T^{+} is free over R. (ii) This was proved in 1.5.6. (iii) Taking Ccont^{•}(G, −) commutes with tensor products with flat R-modules, for a suitable definition of “cont” ([Ne3]). The statement then follows from (i), (ii) and the fact that H^{i}(G, M) is finite (and zero for i ≥ 2) for G = G_{ℚ,S}, G_{ℚ_{v}} and every finite G-module M.

(3.5.6) Duality ([Ne 3])

Local duality: H^{i}(G_{ℚ_{v}}, T) ⟶^{∼} Hom_{O,cont}(H^{2−i}(G_{ℚ_{v}}, A^{∗}(1)), F_{𝔭}/O)

Global duality: (a) Hom_{O,cont}(H̃_{f}^{i}(G_{ℚ,S}, T; Δ), F_{𝔭}/O) ⟶^{∼} H̃_{f}^{3−i}(G_{ℚ,S}, A^{∗}(1); Δ^{∗}(1))

(b) ℝΓ̃_{f}(G_{ℚ,S}, T; Δ) ⟶^{∼} ℝHom_{R}(ℝΓ̃_{f}(G_{ℚ,S}, T^{∗}(1); Δ^{∗}(1)), ω_{R})[−3] in D_{ft}^{b}(R-mod) (this isomorphism being suitably skew-symmetric).

(3.5.7) Proposition. The canonical map

H̃_{f}^{2}(G_{ℚ,S}, T; −)/P ⟶ H̃_{f}^{2}(G_{ℚ,S}, T_{[P]}; −) (− = Δ, Δ^{∗}(1))

is an isomorphism in (R-mod)/(ps-null).

Proof. Lemma 3.5.5 (iii) boils down to a spectral sequence

E_{i,j}^{2} = Tor_{i}^{R}(R/P, H̃_{f}^{−j}(G_{ℚ,S}, T; −)) ⇒ H̃_{f}^{−i−j}(G_{ℚ,S}, T_{[P]}; −)

in (R-mod)/(ps-null) which gives an exact sequence

(3.5.7.1)  0 ⟶ E_{0,−2}^{2} ⟶ H̃_{f}^{2}(G_{ℚ,S}, T_{[P]}; −) ⟶ E_{1,−3}^{2} ⟶ 0.

However, global duality 3.5.6 implies that

Hom_{O,cont}(H̃_{f}^{3}(G_{ℚ,S}, T; −), F_{𝔭}/O) ⟶^{∼} H̃_{f}^{0}(G_{ℚ,S}, A^{∗}(1); (−)^{∗}(1))

is a subgroup of H^{0}(G_{ℚ}, A^{∗}(1)) = H^{0}(G_{ℚ}, A), which is dual to H_{0}(G_{ℚ}, T). The representation V_{[P]} has weight −1, hence H_{0}(G_{ℚ}, T)/P is finite and (E_{1,−3}^{2})_{P} = 0. As Supp(E_{1,−3}^{2}) ⊆ {P, 𝔪}, E_{1,−3}^{2} is pseudo-null. We conclude by (3.5.7.1).

(3.5.8) Proposition. For every prime v ∈ S_{f}, v ≠ p,

H^{0}(G_{ℚ_{v}}, T) = 0, H^{1}(G_{ℚ_{v}}, T)_{P} = H^{2}(G_{ℚ_{v}}, T)_{P} = 0.

Proof. Lemma 3.1.3 applies to V_{[P]} (as k is even), hence H^{0}(G_{ℚ_{v}}, T_{[P]}) = 0, and H^{0}(G_{ℚ_{v}}, T) = 0. By the same argument, the P-torsion in H^{0}(G_{ℚ_{v}}, A^{∗}(1)) = H^{0}(G_{ℚ_{v}}, A) is finite, hence H^{2}(G_{ℚ_{v}}, T)/P is finite by local duality. This implies that H^{2}(G_{ℚ_{v}}, T)_{P} = 0 (cf. the remark at the end of 3.5.4). For H^{1}, consider the spectral sequence in (R-mod)/(ps-null)

E_{i,j}^{2} = Tor_{i}^{R}(R/P, H^{−j}(G_{ℚ_{v}}, T)) ⇒ H_{f}^{−i−j}(G_{ℚ_{v}}, T_{[P]})).

In the exact sequence

0 ⟶ E_{0,−1}^{2} ⟶ H^{1}(G_{ℚ_{v}}, T_{[P]}) ⟶ E_{1,−2}^{2} ⟶ 0

the middle term is finite (again by Lemma 3.1.3), and so E_{0,−1}^{2} = H^{1}(G_{ℚ_{v}}, T)/P must be finite, too.

(3.5.9) Corollary. The canonical map ℝΓ̃_{f}(G_{ℚ,S}, T; Δ) ⟶ ℝΓ̃_{f}(G_{ℚ,S}, T; Δ^{∗}(1)) induces isomorphisms on localizations H̃_{f}^{i}(G_{ℚ,S}, T; Δ)_{P} ⟶^{∼} H̃_{f}^{i}(G_{ℚ,S}, T; Δ^{∗}(1))_{P}.

Proof. This follows from Prop. 3.5.8 and a distinguished triangle

ℝΓ̃_{f}(G_{ℚ,S}, T; Δ) ⟶ ℝΓ̃_{f}(G_{ℚ,S}, T; Δ^{∗}(1)) ⟶ ⨁_{{v∈S_{f} \atop v≠p}} ℝΓ(G_{ℚ_{v}}, T).

(3.5.10) The global duality 3.5.6 gives a spectral sequence in (R-mod) (with all E_{2}^{i,j} of finite type over R)

E_{2}^{i,j} = Ext_{R}^{i}(H̃_{f}^{3−j}(G_{ℚ,S}, T; Δ^{∗}(1)), ω_{R}) ⇒ H̃_{f}^{i+j}(G_{ℚ,S}, T; Δ).

We shall localize this sequence at P. As R_{P} is a discrete valuation ring, there is an isomorphism (ω_{R})_{P} = ω_{R_{P}} ⟶^{∼} R_{P}, well-defined up to a unit in R_{P} (the exact normalization is irrelevant). The localization gives an exact sequence (using Corollary 3.5.9)

0 ⟶ Ext_{R_{P}}^{1} (N_{P}, R_{P})⟶^{α}N_{P} ⟶ Hom_{R_{P}} (H̃_{f}^{1}(G_{ℚ,S}, T; Δ)_{P}, R_{P}) ⟶ 0,

where N = H̃_{f}^{2}(G_{ℚ,S}, T; Δ). This is an R-module of finite type; denote its torsion submodule by M = Tors_{R}(N) (x ∈ N is torsion iff rx = 0 for some r ∈ R which is not a zero-divisor) and put Q = N/M. As M_{P} = Tors_{R_{P}} (N_{P}), the exact sequence of Ext’s associated to

0 ⟶ M_{P} ⟶ N_{P} ⟶ Q_{P} ⟶ 0

implies that the canonical map

Ext_{R_{P}}^{1} (N_{P}, R_{P}) ⟶^{∼} Ext_{R_{P}}^{1} (M_{P}, R_{P})

is an isomorphism (Q_{P} is torsion free, hence free over R_{P}). This implies that α induces an isomorphism

α′ : Ext_{R_{P}}^{1} (M_{P}, R_{P}) ⟶^{∼} Tors_{R_{P}} (N_{P}) = M_{P}.

The R_{P}-module M_{P} is killed by some r ∈ R_{P}, r ≠ 0. The sequence

0 ⟶ R_{P} ⟶^{r}R_{P} ⟶ R_{P}/rR_{P} ⟶ 0

gives an isomorphism

Hom_{R_{P}} (M_{P}, R_{P}/rR_{P}) ⟶^{∼} Ext_{R_{P}}^{1} (M_{P}, R_{P})

which, combined with α′, yields an isomorphism

Hom_{R_{P}} (M_{P}, R_{P}/rR_{P}) ⟶^{∼} M_{P}

which is skew-symmetric, i.e. comes from a skew-symmetric bilinear form

M_{P} × M_{P} ⟶ R_{P}/rR_{P}

(by skew-symmetry of the global duality isomorphism). The residue field of R_{P} has characteristic zero, which implies that the form is alternating. Standard structure theory of symplectic modules of finite length over discrete valuation rings implies that

M_{P} ⟶^{∼} X ⊕ X

for an R_{P}-module X of finite length.

(3.5.11) Lemma. Let A be an integral domain and I ⊂ A a prime ideal such that the localization A_{I} is a discrete valuation ring. If Y is a torsion-free A-module of finite type, then

rk_{A/I}(Y/IY) = rk_{A}(Y)

(recall that rk_{A}(Y) = dim_{K}(Y ⊗_{A} K), where K is the fraction field of A).

Proof. The localization Y_{I} is a torsion-free A_{I}-module of finite type, hence free of rank n. Then

rk_{A/I}(Y/IY) = dim_{A_{I}/IA_{I}} (Y_{I}/IY_{I}) = n
dim_{K}(Y ⊗_{A} K) = dim_{K}(Y_{I} ⊗_{A_{I}} K) = n.

(3.5.12) Proposition.

rk_{R/P}(N/PN) ≡ rk_{R/𝔮}(Q/𝔮Q) (mod 2).

Proof. Recall that 𝔮 = Ker [R ⟶ 𝔥(K)], R/𝔮 = 𝔥(K). We have

rk_{R/P}(N/PN) = rk_{R/P}(M/PM) + rk_{R/P}(Q/PQ),

as Q_{P} is free over R_{P} (of course, R/P = O). However,

rk_{R/P}(M/PM) = dim_{R_{P} /PR_{P}} (M_{P}/PM_{P}) = 2 dim_{R_{P} /PR_{P}} (X/PX)

is even and

rk_{R/P}(Q/PQ) = rk_{R/𝔮}(Q/𝔮Q)

by Lemma 3.5.11 applied to A = R/𝔮, I = P/𝔮 = \overline{P} and Y = Q/𝔮Q.

(3.5.13) Theorem A′. In the notation of (3.5.1–2) – in particular, if k_{0} ≡ 2 (mod (p − 1)), assume that V (f_{k_{0}}) has an irreducible residual representation – for every P ∈ 𝔛^{arith}(R) containing 𝔮 and such that f_{P} has even weight,

dim_{F_{𝔭}} H̃_{f}^{1}(G_{ℚ,S}, V_{[P]}; Δ) ≡ rk_{R/𝔮}(Q/𝔮Q) (mod 2),

hence the parity of the L.H.S. does not depend on P.

Proof. the dimension on the L.H.S. is equal to

cork_{O}H̃_{f}^{1}(G_{ℚ,S}, A_{[P]}; Δ) = cork_{O}H̃_{f}^{1}(G_{ℚ,S}, A^{∗}(1)_{[P]}; Δ^{∗}(1))

(by almost self duality of the local conditions; cf. proof of 2.3.10)

= rk_{O}H̃_{f}^{2}(G_{ℚ,S}, T_{[P]}; Δ) (global duality 3.5.6)
= rk_{O}(N/PN) (Prop. 3.5.7)

Now apply Proposition 3.5.12.

(3.5.14) Let us compare the group H̃_{f}^{1}(G_{ℚ,S}, V_{[P]}; Δ) and the Bloch-Kato Selmer group H_{f}^{1}(ℚ, V_{[P]}). First of all, Lemma 3.1.3 still holds for V = V_{[P]}, thus the local conditions for both groups vanish at all primes v ∈ S_{f}, v ≠ p. As a representation of G_{ℚ_{p}}, there is an exact sequence

0 ⟶ V_{p}^{+} ⟶ V ⟶ V_{p}^{−} ⟶ 0,

with (V_{p}^{±})^{∗} (1) ⟶^{∼} V_{p}^{∓} by self duality of V. Each V_{p}^{±} is a twist of a crystalline representation by “χ_{P}^{−1/2} ”. Note that “χ_{P}^{−1/2} ” = ω^{(k−k_{0})/2}ε^{−1/2} is unramified iff “χ_{P}^{−1/2} ” = 1 (which is equivalent to k ≡ k_{0} (mod 2(p − 1)) and ε = 1). In this case, results of 3.1.6–7 apply. Assume that “χ_{P}^{−1/2} ” ≠ 1. Then neither of V_{p}^{±} is a crystalline representation of G_{ℚ_{p}}, hence

D_{cris}(V_{p}^{±}) = D_{cris}(V) = H^{0}(G_{ℚ_{p}}, V_{p}^{±}) = H^{0}(G_{ℚ_{p}}, V) = 0.

The representation V is potentially semistable (in fact is semistable over ℚ_{p}(μ_{p^{r}}) for suitable r), therefore de Rham. It follows from 2.1.6 and self-duality V ⟶^{∼} V^{∗}(1) that

H_{f}^{1}(G_{ℚ_{p}}, V) = H_{g}^{1}(G_{ℚ_{p}}, V) = Im[H^{1}(G_{ℚ_{p}}, V^{+}) ⟶ H^{1}(G_{ℚ_{p}}, V)].

Combining with the results of 3.1.7, we obtain an exact sequence

(3.5.14.1)  0 ⟶ F_{𝔭}^{ε(P)} ⟶ H̃_{f}^{1}(G_{ℚ,S}, V_{[P]}; Δ) ⟶ H_{f}^{1}(ℚ, V) ⟶ 0,

in which ε(P) = 1 in the exceptional case, and ε(P) = 0 otherwise. The exceptional case occurs iff k = 2, ε = 1, k ≡ k_{0} (mod 2(p − 1)), and a_{p}(f_{P}) = 1.

(3.5.15) Theorem B′. Let p > 3 be a prime not dividing N and f = ∑_{n≥1} a_{n}q^{n} an ordinary newform of even weight k ≥ 2 and character χ^{−2} on Γ_{1}(Np^{r}), where cond(χ) = p^{s} (necessarily with s = r, if χ ≠ 1). Then f̃ = ∑_{n≥1} a_{n}χ(n)q^{n} is a newform of weight k on Γ_{0}(Np^{2r}) (resp. Γ_{0}(Np^{r})) if χ ≠ 1 (resp. χ = 1). If the tame part of χ^{2} is equal to ω^{k−2}, where ω is the Teichmüller character, assume that V (f) has an irreducible residual representation. If Greenberg’s Conjecture holds for the two-variable p-adic L-function of f, then

dim_{F_{𝔭}}H_{f}^{1}(ℚ, V (f̃)(k/2)) ≡ ord_{s=k/2}L_{∞}(f̃, s) (mod 2).

Proof. Replacing F̃_{𝔭} by a finite extension (and V (f̃) by its base change) we can assume that F̃_{𝔭} contains all values of χ. The form f – or its p-stabilization, if r = 0 – is then equal to f_{P} and f̃ to f̃_{P}, for a suitable Hida family of the type considered in 3.5.1–2 and P ∈ 𝔛^{arith}(R), P ⊃ 𝔮, “χ_{P}^{−1/2} ” = χ. If χ = 1, then f̃ is ordinary and Theorem B applies. Assume that χ ≠ 1 (hence χ is ramified at p). In this case

H_{f}^{1}(ℚ, V (f̃)(k/2)) = H̃_{f}^{1}(G_{ℚ,S}, V (f̃)(k/2); Δ)

by (3.5.14.1). Choose a p-stabilized newform f_{k′} corresponding to some P′ ∈ 𝔛^{arith}(R) containing 𝔮, of even weight k′ ≥ 2, with trivial character and such that e_{p} = ord_{s=k′/2}L_{p}(f_{k′}, s) ≤ 1. It follows from Theorem A′ and 3.4.6 that

dim_{F_{𝔭}}H_{f}^{1}(ℚ, V (f̃)(k/2)) ≡ e_{p} (mod 2).

It remains to show that w_{p}(f_{k′}) = (−1)^{e_{p}} (the sign in the functional equation of L_{p}(f_{k′}, s)) coincides with w_{∞}(f̃), the sign in the functional equation of L_{∞}(f̃, s). First of all, w_{p}(f_{k′}) = w_{p} = w_{p}(f), by 3.4.4. The formula [Ma-Ta-Te, Sect. 17, Cor.2] implies that

L_{p,α}^{MT T} (f, x^{k/2−1}, χ^{−1}, s) = (−1)^{k/2}c_{N}χ^{−1}(N)κ(N)^{−s}L_{p,α}^{MT T} (f, x^{k/2−1}, χ^{−1}, −s)

where c_{N} is the eigenvalue of the Atkin-Lehner involution acting on f: f|W_{N} = c_{N}χ^{−1}(−N)f. Hence

w_{p}(f) = (−1)^{k/2}c_{N}χ^{−1}(−N).

The archimedean sign w_{∞}(f̃) is also equal to

w_{∞}(f̃) = (−1)^{k/2}c_{N}χ^{−1}(−N)

by [Ma-Ta-Te, Sect.18, Rmk] (in their notation, q = N, q′ = p^{r}, ψ = χ, ε = χ^{−2}, ϕ = χ, f_{ψ} = f̃, g = c_{N}f). Theorem follows.

4. Λ-adic Selmer groups

In this section, we give an elementary approach to an important special case of theorem A′, when the local component R of the Hecke algebra is equal to the Iwasawa algebra Λ = O[[X]], using the Λ-adic Selmer groups and Tate-Šafarevič groups introduced in [Pl]. To a self dual Λ-adic Galois representation T, we shall associate a Λ-adic Tate-Šafarevič group whose Pontryagin dual ˆ� is a finitely generated torsion Λ-module, and prove the following result.

(4.0.1) Proposition. Let p > 2. If T satisfies the assumptions of 4.1.1 below, then there is a finitely generated torsion Λ-module X such that ˆ� is pseudo-isomorphic to X ⊕ X.

The corresponding result for specializations of T is a consequence of a generalized Cassels-Tate pairing. We deduce the Λ-adic result by specializing at enough primes.

(4.1) Λ-adic Galois representations

(4.1.1) Let p be a rational prime, F_{𝔭} a finite extension of ℚ_{p} with ring of integers O and prime element π ∈ O. Let Λ = O[[X]] with maximal ideal 𝔪 = (X, π). Let T be a free rank two Λ-module with a continuous Λ-linear G_{ℚ,S}-action, where S is some finite set of primes including p and infinity. We assume that T is ordinary at p, by which we mean that T sits in an exact sequence of Λ[G_{ℚ_{p}}]-modules

(4.1.1.1)  0 ⟶ T^{+} ⟶ T ⟶ T^{−} ⟶ 0

with T^{+} and T^{−} free rank one Λ-modules, and assume there is a skew-symmetric Galois-invariant bilinear form

(4.1.1.2)  ⟨ , ⟩_{T} : T × T ⟶ Λ(1)

inducing an isomorphism

T ⟶^{∼} T^{∗}(1) := Hom_{Λ}(T, Λ)(1), t ↦ ⟨t, . ⟩_{T}

and isomorphisms T^{±} ⟶^{∼} (T^{∗})^{±}(1), where (T^{∗})^{±} := Hom_{Λ}(T^{∓}, Λ).

We also assume that the following cohomology groups are trivial:

H^{0}(ℚ_{v}, T) = 0 for all v ≠ p, (4.1.1.3) H^{0}(ℚ_{p}, T^{+}) = H^{0}(ℚ_{p}, T^{−}) = 0. (4.1.1.4)

(4.1.2) Specializations The height one primes of Λ are the prime (π) and primes (λ), where λ ∈ Λ is an irreducible distinguished polynomial, i.e. λ is irreducible and of the form λ = X^{n} + a_{n−1}X^{n−1} + ⋯ + a_{0} with π | a_{i} for 0 ≤ i ≤ n − 1. Let X = Spec(Λ) − {(π)}. This agrees with the definition of X in 2.3.2 in this special case. For (λ) ∈ X, we write O_{λ} = Λ/(λ) and T_{λ} = T/λT = T ⊗_{Λ} O_{λ}, T_{λ}^{±} = T^{±}/λT^{±}. So O_{λ} is free of finite rank over O and T_{λ} (resp. T_{λ}^{±}) is a free O_{λ}-module of rank two (resp. one). We call T_{λ} the specialization of T at λ.

(4.1.3) Let G = G_{ℚ,S} or G_{ℚ_{v}}. The continuous cohomology groups H^{n}(G, T) are finitely generated Λmodules and the H^{n}(G, T_{λ}) are finitely generated O_{λ}-modules. Let M = T, T^{+} or T^{−}. Taking cohomology of the short exact sequence

(4.1.3.1)  0 ⟶ M⟶^{λ}M ⟶ M_{λ} ⟶ 0

as G-modules gives a short exact sequence of O_{λ}-modules

(4.1.3.2)  0 ⟶ \frac{H^{n}(G, M)}{λH^{n}(G, M)} ⟶ H^{n}(G, M_{λ}) ⟶ H^{n+1}(G, M)[λ] ⟶ 0.

(4.1.4) In the special case when R = Λ = O[[Γ]], and assuming the Assumption of 3.2.3, the twisted Galois representation T of 3.2.3 satisfies the conditions of 4.1.1. If we identify O[[Γ]] with O[[X]] via γ = 1+p ↦ 1+X (choosing γ = 1+p as a topological generator for Γ ⟶^{∼} 1+pℤ_{p}) then the arithmetic point (P_{k,ε}) ∈ Spec(Λ) corresponds to the height one prime ideal generated by λ_{k,ε} = 1 + X − ε(1 + p)(1 + p)^{k−2} ∈ O[[X]]. We just need to check conditions (4.1.1.3) and (4.1.1.4); To check (4.1.1.3), it is enough to specialize at any arithmetic point λ, where H^{0}(ℚ_{v}, T_{λ}) = 0 by Lemma 3.1.3. Then 4.1.3.2 gives

0 ⟶ \frac{H^{0}(ℚ_{v}, T)}{λH^{n}(ℚ_{v}, T)} ⟶ H^{0}(ℚ_{v}, T_{λ}) = 0

and so H^{0}(ℚ_{v}, T) = 0 by Nakayama’s lemma. For (4.1.1.4) we do the same thing, using 3.1.4, for any non-exceptional arithmetic point.

(4.1.5) The exact sequence (4.1.3.2) allows us to compare specializations of Λ-adic cohomology groups (the left hand term) with cohomology groups of the specializations (the middle term). The right hand term is an error term which we handle using the following simple lemma:

Lemma. Let M be a finitely generated Λ-module. For λ not in the support of M, the groups M[λ] are finite of order bounded independently of λ.

Proof. Without loss of generality we may take M to be Λ-torsion. By the structure theory of finitely generated Λ-modules, M sits in an exact sequence of the form

0 ⟶ A ⟶ M ⟶ ⨁_{i=1}^{r} \frac{Λ}{P_{i}^{n_{i}}} ⟶ B ⟶ 0,

where A and B are finite and the P_{i} are the height one primes in the support of M. So there is an exact sequence

0 ⟶ A[λ] ⟶ M[λ] ⟶ ⨁_{i=1}^{r} \frac{Λ}{P_{i}^{n_{i}}} [λ].

For λ not in the support of M, the last term of this sequence is zero, and so M[λ] is finite, of order bounded by the order of A.

(4.1.6) For v ∈ S_{f} and M = T, T^{+} or T^{−}, the Λ-rank of H^{n}(ℚ_{v}, M) is equal to the O_{λ}-rank of H^{n}(ℚ_{v}, M_{λ}), for any λ not in the support of Tors_{Λ}H^{n}(ℚ_{v}, M) or Tors_{Λ}H^{n+1}(ℚ_{v}, M) (this is immediate from (4.1.3.2)). So the O_{λ}-ranks of the H^{n}(ℚ_{v}, M_{λ}) are constant and equal to this “generic”rank outside of this explicit bad set (which depends on v). Staying away from this bad set, we obtain Λ-adic Euler characteristic formulas:

∑_{n=0}^{2}(−1)^{n}rk_{Λ}H^{n}(ℚ_{v}, M) = \{ \begin{matrix}0 & v ≠ p \\ − rk_{Λ}(M) & v = p. \end{matrix}

Using (4.1.1.2–4) we deduce the following:

v ∈ S_{f}, v ≠ p : H^{0}(ℚ_{v}, T) = 0 rk_{Λ}H^{1}(ℚ_{v}, T) = 0 rk_{Λ}H^{2}(ℚ_{v}, T) = 0
v = p :  H^{0}(ℚ_{p}, T) = 0 rk_{Λ}H^{1}(ℚ_{p}, T) = 2 rk_{Λ}H^{2}(ℚ_{p}, T) = 0
H^{0}(ℚ_{p}, T^{+}) = 0 rk_{Λ}H^{1}(ℚ_{p}, T^{+}) = 1 rk_{Λ}H^{2}(ℚ_{p}, T^{+}) = 0
H^{0}(ℚ_{p}, T^{−}) = 0 rk_{Λ}H^{1}(ℚ_{p}, T^{−}) = 1 rk_{Λ}H^{2}(ℚ_{p}, T^{−}) = 0

(4.1.7) Write X_{bad} for the finite set of (λ) ∈ X in the support of Tors_{Λ}H^{1}(ℚ_{p}, T^{+}), Tors_{Λ}H^{1}(ℚ_{p}, T^{−}), or H^{1}(ℚ_{v}, T) for some v ∈ S_{f}, v ≠ p, and let X_{good} = X − X_{bad}. For any (λ) ∈ X_{good} we have H^{0}(ℚ_{v}, T_{λ}) = 0 for v ∈ S_{f}, v ≠ p, and H^{0}(ℚ_{p}, T_{λ}^{±}) = 0. Euler characteristic calculations then show that the O_{λ}-ranks of the H^{i}(ℚ_{v}, M_{λ}), i = 0, 1, 2, M = T, T^{±}, are equal to the “generic”Λ-adic ranks of the H^{i}(ℚ_{v}, M) in 4.1.6. Combining this with (4.1.3.2), it follows that X_{bad} already contains those λ in the support of Tors_{Λ}H^{n}(ℚ_{v}, M), for all v ∈ S_{f}, n = 0, 1, 2, and M = T, T^{+}, T^{−}.

(4.1.8) Local conditions

For a local ring R and a finitely generated R-module M and an R-submodule N ⊆ M, we define the Rsaturation of N in M to be the set of m ∈ M such that rm ∈ N for some non zero-divisor r ∈ R (when R = O_{λ}, the O_{λ}-saturation is equal to the ℤ_{p}-saturation, because O_{λ} is finite over ℤ_{p}).

For each prime v ∈ S_{f}, we define a Λ-submodule H_{f}^{1}(ℚ_{v}, T) ⊆ H^{1}(ℚ_{v}, T) as follows.

For v ∈ S_{f}, v ≠ p: define H_{f}^{1}(ℚ_{v}, T) = H^{1}(ℚ_{v}, T).

For v = p: define

H_{f}^{1}(ℚ_{p}, T)^{0} = Ker [H^{1}(ℚ_{p}, T) → H^{1}(I_{p}, T^{−})] ,

and define H_{f}^{1}(ℚ_{p}, T) to be the Λ-saturation of H_{f}^{1}(ℚ_{p}, T)^{0} in H^{1}(ℚ_{p}, T). Condition (4.1.1.4) for T^{−} implies that

Ker [H^{1}(ℚ_{p}, T^{−}) → H^{1}(I_{p}, T^{−})]

is Λ-torsion, hence H_{f}^{1}(ℚ_{p}, T) is also the Λ-saturation of

Ker [H^{1}(ℚ_{p}, T) → H^{1}(ℚ_{p}, T^{−})]

in H^{1}(ℚ_{p}, T). The Λ-rank of H_{f}^{1}(ℚ_{p}, T) is one.

We use the same local conditions to define O_{λ}-submodules H_{f}^{1}(ℚ_{v}, T_{λ}) ⊆ H^{1}(ℚ_{v}, T_{λ}) for the specializations at any (λ) ∈ X_{good}. For v ∈ S_{f}, v ≠ p: define H_{f}^{1}(ℚ_{v}, T_{λ}) = H^{1}(ℚ_{v}, T_{λ}).

For v = p: define

H_{f}^{1}(ℚ_{p}, T_{λ})^{0} = Ker [H^{1}(ℚ_{p}, T_{λ}) → H^{1}(I_{p}, T_{λ}^{−})] ,

and define H_{f}^{1}(ℚ_{p}, T_{λ}) to be the O_{λ}-saturation of H_{f}^{1}(ℚ_{p}, T_{λ})^{0} in H^{1}(ℚ_{p}, T_{λ}). As in the Λ-adic case, H_{f}^{1}(ℚ_{p}, T_{λ}) is also equal to the O_{λ}-saturation of

Ker [H^{1}(ℚ_{p}, T_{λ}) → H^{1}(ℚ_{p}, T_{λ}^{−})]

in H^{1}(ℚ_{p}, T_{λ}) and H_{f}^{1}(ℚ_{p}, T) has O_{λ}-rank one.

(4.1.9) Lemma. For (λ) ∈ X_{good}, v ∈ S_{f}, specialization gives a homomorphism

\frac{H^{1}(ℚ_{v}, T)}{λH^{1}(ℚ_{v}, T)} ⟶ H^{1}(ℚ_{v}, T_{λ})

which is injective, with cokernel that is finite and bounded independently of λ.

Proof. This follows from (4.1.3.2) and Lemma 4.1.5, if we take λ not in the support of H^{2}(ℚ_{v}, T). However, for (λ) ∈ X_{good}, both sides have O_{λ}-rank two (resp. zero) for v = p (resp. v ≠ p), and so H^{2}(ℚ_{v}, T)[λ] is finite. So the support of H^{2}(ℚ_{v}, T) is already included in X_{bad}.

(4.1.10) We now need the analogues of Lemma 4.1.9, both for the subgroups H_{f}^{1}(ℚ_{p}, T) and for the quotient groups H^{1}(ℚ_{p}, T)/H_{f}^{1}(ℚ_{p}, T).

Lemma. For (λ) ∈ X_{good}, the image of H_{f}^{1}(ℚ_{p}, T) in H^{1}(ℚ_{p}, T_{λ}) lies in H_{f}^{1}(ℚ_{p}, T_{λ}).

Proof. It is clear from the definition that the image of H_{f}^{1}(ℚ_{p}, T)^{0} lies in H_{f}^{1}(ℚ_{p}, T_{λ})^{0}. If (λ) ∈ X_{good} , then the image of H_{f}^{1}(ℚ_{p}, T) contains a subgroup of finite index in H_{f}^{1}(ℚ_{p}, T_{λ})^{0} and has the same O_{λ}-rank. So it is contained in the O_{λ}-saturation of H_{f}^{1}(ℚ_{p}, T_{λ})^{0} in H^{1}(ℚ_{p}, T_{λ}), which is H_{f}^{1}(ℚ_{p}, T_{λ}).

(4.1.11) The kernel of the map H_{f}^{1}(ℚ_{p}, T) → H_{f}^{1}(ℚ_{p}, T_{λ}) is H_{f}^{1}(ℚ_{p}, T) ∩ λH^{1}(ℚ_{p}, T) = λH_{f}^{1}(ℚ_{p}, T), so specialization gives an injection

\frac{H_{f}^{1}(ℚ_{p}, T)}{λH_{f}^{1}(ℚ_{p}, T)} ↪ H_{f}^{1}(ℚ_{p}, T_{λ}).

Write Y for H^{1}(ℚ_{p}, T)/H_{f}^{1}(ℚ_{p}, T). So Y has Λ-rank one, for (λ) ∈ X_{good}. Because H_{f}^{1}(ℚ_{p}, T) is Λ-saturated in H^{1}(ℚ_{p}, T), Y is Λ-torsion free, and so we have a short exact sequence

0 ⟶ \frac{H_{f}^{1}(ℚ_{p}, T)}{λH_{f}^{1}(ℚ_{p}, T)} ⟶ \frac{H^{1}(ℚ_{p}, T)}{λH^{1}(ℚ_{p}, T)} ⟶ \frac{Y}{λY} ⟶ 0.

The index of H_{f}^{1}(ℚ_{p}, T)/λH_{f}^{1}(ℚ_{p}, T) in its O_{λ}-saturation in H^{1}(ℚ_{p}, T)/λH^{1}(ℚ_{p}, T) is equal to the order of the maximal finite O_{λ}-submodule of Y/λY. By the structure theory, Y sits in an exact sequence

0 ⟶ Y ⟶ Λ ⟶ B ⟶ 0

where B is finite, so the order of this finite submodule is the order of B[λ], which is bounded independently of λ. Let Z be the inverse image of H_{f}^{1}(ℚ_{p}, T_{λ}) in H^{1}(ℚ_{p}, T)/λH^{1}(ℚ_{p}, T) under the inclusion

\frac{H^{1}(ℚ_{p}, T)}{λH^{1}(ℚ_{p}, T)} ↪ H^{1}(ℚ_{p}, T_{λ}).

So Z is equal to the O_{λ}-saturation of H_{f}^{1}(ℚ_{p}, T)/λH_{f}^{1}(ℚ_{p}, T) in H^{1}(ℚ_{p}, T)/λH^{1}(ℚ_{p}, T). By the above, the index of H_{f}^{1}(ℚ_{p}, T)/λH_{f}^{1}(ℚ_{p}, T) in Z is finite and bounded for λ ∈ X_{good}. This implies the following lemma.

(4.1.12) Lemma. For (λ) ∈ X_{good}, the specialization map

\frac{H^{1}(ℚ_{p}, T)/H_{f}^{1}(ℚ_{p}, T)}{λ (H^{1}(ℚ_{p}, T)/H_{f}^{1}(ℚ_{p}, T))} ⟶ \frac{H^{1}(ℚ_{p}, T_{λ})}{H_{f}^{1}(ℚ_{p}, T_{λ})}

is injective, with cokernel that is finite and bounded independently of λ.

(4.1.13) Lemma. For (λ) ∈ X_{good}, the specialization map

\frac{H_{f}^{1}(ℚ_{p}, T)}{λH_{f}^{1}(ℚ_{p}, T)} ⟶ H_{f}^{1}(ℚ_{p}, T_{λ})

is injective, with cokernel that is finite and bounded independently of λ.

Proof. We already know that it is injective. We have a commutative diagram with exact rows

0 ⟶ \frac{H_{f}^{1}(T)}{λH_{f}^{1}(T)} ⟶ \frac{H^{1}(T)}{λH^{1}(T)} ⟶ \frac{H^{1}(T)/H_{f}^{1}(T)}{λH^{1}(T)/H_{f}^{1}(T)} ⟶ 0
↓  ↓  ↓
0 ⟶ H_{f}^{1}(T_{λ}) ⟶ H^{1}(T_{λ}) ⟶ H^{1}(T_{λ})/H_{f}^{1}(T_{λ}) ⟶ 0

(dropping ℚ_{p} from the notation). By Lemmas 4.1.9 and 4.1.12, the kernels and cokernels of the vertical arrows are all finite and bounded for (λ) ∈ X_{good}, except possibly the cokernel of the first vertical arrow. This now follows from the snake lemma.

(4.1.14) Discrete modules

Write M̂ := Hom_{cont}(M, ℚ_{p}/ℤ_{p}) for the Pontryagin dual of a Λ-module M. We define a discrete dual A of T by A = Hom_{cont}(T, ℚ_{p}/ℤ_{p}(1)). Since T is a free Λ-module, we have a canonical isomorphism A ⟶^{∼} T^{∗}(1) ⊗_{Λ} Λ̂. Since T ⟶^{∼} T^{∗}(1), this is also isomorphic to T ⊗_{Λ} Λ̂ (so there is no discrepancy with the notation of 3.5.3). A is a (discrete) Λ-module with Λ-action (λϕ)(t) := ϕ(λt) for ϕ ∈ A, t ∈ T. We have a perfect pairing

A × T → ℚ_{p}/ℤ_{p}(1)

which induces perfect pairings (Tate local duality) for any v ∈ S_{f}

H^{i}(ℚ_{v}, A) × H^{2−i}(ℚ_{v}, T) → ℚ_{p}/ℤ_{p}

for i = 0, 1, 2.

For the specializations at (λ) ∈ X_{good}, we define A_{λ} = Hom(T_{λ}, ℚ_{p}/ℤ_{p}(1)). As in the Λ-adic case, because T_{λ} is a free O_{λ}-module, A_{λ} is canonically isomorphic to T_{λ}^{∗}(1) ⊗_{O_{λ}} Ô_{λ}, where T_{λ}^{∗} := Hom_{O_{λ}}(T_{λ}, O_{λ}) (and also to T_{λ} ⊗_{O_{λ}} Ô_{λ} via T_{λ}^{∗}(1) ⟶^{∼} T_{λ}). Note that O_{λ}, being a complete intersection, is a Gorenstein ring. Fixing a generator for Hom_{ℤ_{p}}(O_{λ}, ℤ_{p}) as a free rank one O_{λ}-module gives an isomorphism

Ô_{λ} = Hom_{ℤ_{p}}(O_{λ}, ℤ_{p}) ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p} ⟶^{∼} O_{λ} ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p} ⟶^{∼} F_{𝔭,λ}/O_{λ},

where F_{𝔭,λ} = O_{λ} ⊗_{ℤ_{p}} ℚ_{p} is the fraction field of O_{λ}. So we also have

A_{λ} ⟶^{∼} T_{λ}^{∗}(1) ⊗_{O_{λ}} F_{𝔭,λ}/O_{λ}

(in the case where O_{λ} is the ring of integers of F_{𝔭,λ}, this would have been called A^{∗}(1)_{[P]} in 3.5.3). We can also canonically identify A_{λ} with A[λ]. With this identification, we have an exact sequence

(4.1.14.1)  0 ⟶ A_{λ} ⟶ A⟶^{λ}A ⟶ 0.

Local duality gives a perfect pairing

(4.1.14.2)  H^{i}(ℚ_{v}, A_{λ}) × H^{2−i}(ℚ_{v}, T_{λ}) → ℚ_{p}/ℤ_{p}

for i = 0, 1, 2.

(4.1.15) Local conditions for discrete modules

For any v ∈ S_{f}, we define H_{f}^{1}(ℚ_{v}, A) to be the orthogonal complement to H_{f}^{1}(ℚ_{v}, T) under the pairing (4.1.14.2), for i = 1. In particular, for v ≠ p, we get H_{f}^{1}(ℚ_{v}, A) = 0.

Similarly for the specializations at (λ) ∈ X_{good}, we define H_{f}^{1}(ℚ_{v}, A_{λ}) to be the orthogonal complement to H_{f}^{1}(ℚ_{v}, T_{λ}). For v ≠ p, H_{f}^{1}(ℚ_{v}, A_{λ}) = 0.

(4.1.16) Lemma. Let G = G_{ℚ,S} or G_{ℚ_{v}} for v ∈ S_{f}. For (λ) ∈ X_{good}, the natural map

(4.1.16.1)  H^{1}(G, A_{λ}) → H^{1}(G, A)[λ]

is surjective with kernel that is finite and bounded independently of λ.

Proof. Taking cohomology of (4.1.14.1) as G-modules gives an exact sequence

(4.1.16.1)  H^{0}(G, A)⟶^{λ}H^{0}(G, A) ⟶ H^{1}(G, A_{λ}) ⟶ H^{1}(ℚ_{v}, A)[λ] ⟶ 0.

The lemma follows, for λ not in the support of H^{0}(G, A). For G = G_{ℚ_{v}}, this module is equal to H^{2}(ℚ_{v}, T) whose support is already included in X_{bad} by Lemma 4.1.9 (alternatively for ℚ_{v}, the lemma follows immediately from Lemma 4.1.9 and local duality). For G = G_{ℚ,S}, λ is in the support of H^{0}(G, A) iff H^{0}(G, A)[λ] is infinite which is true iff H^{0}(G, A_{λ}) is infinite, which again implies that λ is already included in X_{bad}.

(4.1.17) Lemma. For v ∈ S_{f}, (λ) ∈ X_{good}, the image of H_{f}^{1}(ℚ_{v}, A_{λ}) in H^{1}(ℚ_{v}, A)[λ] lies in H_{f}^{1}(ℚ_{v}, A)[λ]. The map

H_{f}^{1}(ℚ_{v}, A_{λ}) → H_{f}^{1}(ℚ_{v}, A)[λ]

has kernel and cokernel that are finite and bounded independently of λ.

Proof. For v ≠ p, both H_{f}^{1}(ℚ_{v}, A_{λ}) and H_{f}^{1}(ℚ_{v}, A) are zero, so the result is trivial. For v = p, this is the dual of Lemma 4.1.12 (using Lemma 4.1.13).

(4.1.18) Selmer groups

We define the Selmer group 𝕊 ⊆ H^{1}(G_{ℚ,S}, A) by

𝕊 = Ker [H^{1}(G_{ℚ,S}, A) → ⨁_{v∈S} \frac{H^{1}(ℚ_{v}, A)}{H_{f}^{1}(ℚ_{v}, A)}] .

Similarly for the specializations, define

𝕊_{λ} = Ker [H^{1}(G_{ℚ,S}, A_{λ}) → ⨁_{v∈S} \frac{H^{1}(ℚ_{v}, A_{λ})}{H_{f}^{1}(ℚ_{v}, A_{λ})}] .

(4.1.19) Lemma. For (λ) ∈ X_{good}, the image of 𝕊_{λ} in H^{1}(G_{ℚ,S}, A)[λ] under the map (4.1.16.1) lies in 𝕊, and the map

(4.1.19.1)  𝕊_{λ} ⟶ 𝕊[λ]

has kernel and cokernel that are finite and bounded independently of λ.

Proof. Apply the snake lemma to the diagram

0 ⟶ 𝕊_{λ} ⟶ H^{1}(G_{ℚ,S}, A_{λ}) ⟶ ⨁_{v∈S} H^{1}(ℚ_{v}, A_{λ})/H_{f}^{1}(ℚ_{v}, A_{λ})
↓  ↓  ↓
0 ⟶ 𝕊[λ] ⟶ H^{1}(G_{ℚ,S}, A)[λ] ⟶ ⨁_{v∈S} H^{1}(ℚ_{v}, A)/H_{f}^{1}(ℚ_{v}, A)[λ].

We just need to know that the kernel of the maps

\frac{H^{1}(ℚ_{v}, A_{λ})}{H_{f}^{1}(ℚ_{v}, A_{λ})} ⟶ \frac{H^{1}(ℚ_{v}, A)}{H_{f}^{1}(ℚ_{v}, A)}[λ]

are finite and bounded, and this follows from dualizing Lemma 4.1.13.

(4.1.20) Tate-Šafarevič groups

An O_{λ}-module M is called π-divisible if πM = M. A Λ-module M is said to be 𝔪-divisible if 𝔪M = M. A co-finitely generated Λ-module M is 𝔪-divisible if and only if Hom(M, ℚ_{p}/ℤ_{p}) is Λ-torsion free. For any specialization 𝕊_{λ}, we define a Tate-Šafarevič group by � _{λ} = 𝕊_{λ}/div(𝕊_{λ}), where div(𝕊_{λ}) is the maximal π-divisible submodule of 𝕊_{λ}. So � _{λ} is finite and ˆ�_{λ} is the torsion subgroup of 𝕊̂_{λ}. By analogy, for 𝕊 itself, we define � = 𝕊/𝔪-div(𝕊), where 𝔪-div(𝕊) is the maximal 𝔪-divisible submodule of 𝕊. Equivalently, ˆ� = Tors_{Λ}𝕊̂ is the maximal Λ-torsion submodule of 𝕊̂. Dualizing (4.1.19) gives a map 𝕊̂/λ𝕊̂ ⟶ 𝕊̂_{λ} which restricts to a map

(4.1.20.1)  ˆ�/λˆ� ⟶ ˆ�_{λ}.

(4.1.21) Lemma. The map (4.1.20.1) has kernel and cokernel which are finite and bounded independently of λ, for (λ) ∈ X′_{good}, where we define X′_{bad} = X_{bad} ∪ {λ ∈ Supp(Tors_{Λ}𝕊̂)} and X′_{good} = X − X′_{bad}.

Proof. If M is any finitely generated Λ module, then, for λ not in the support of Tors_{Λ}M, the map Tors_{Λ}M/λTors_{Λ}M → Tors_{ℤ_{p}}(M/λM) has kernel and cokernel that are finite and bounded independently of λ. Combining this fact with Corollary 4.1.19 proves the lemma.

(4.1.22) We shall now study the structure of 𝕊̂ via its specializations. We fix an irreducible distinguished polynomial λ and write λ_{k} = λ+π^{k}. For k sufficiently large, λ_{k} is also an irreducible distinguished polynomial.

Also, for k sufficiently large, λ_{k} is not in any finite bad set that we might need to avoid and, in particular, is not in X′_{bad}. We shall write T_{k} for T_{λ_{k}}, A_{k} for A_{λ_{k}}, 𝕊_{k} for 𝕊_{λ_{k}}, and � _{k} for � _{λ_{k}}.

By the structure theory of finitely generated Λ-modules, ˆ� = Tors_{Λ}𝕊̂ sits in an exact sequence

(4.1.22.1)  0 ⟶ A ⟶ ⨁_{i=1}^{r} \frac{Λ}{P_{i}^{n_{i}}} ⟶ ˆ� ⟶ B ⟶ 0,

where A and B are finite. Decompose the second term in this sequence as

⨁_{i=1}^{r} \frac{Λ}{P_{i}^{n_{i}}} = M ⊕ \overline{M}

where M consists of those direct summands with P_{i} = (λ) for our fixed λ, and consider the composite map

\frac{M}{λ_{k}M} ⟶ \frac{ˆ�}{λˆ�} ⟶ \frac{𝕊̂}{λ_{k}𝕊̂} ⟶ 𝕊̂_{k}.

For k sufficiently large, λ_{k} is not in the support of Tors_{Λ}𝕊̂ and so the image of this composite map is finite and therefore lies in ˆ�_{k}. Lemma 4.1.21 now implies the following lemma.

(4.1.23) Lemma. For k sufficiently large, the map

\frac{M}{λ_{k}M} ⟶ ˆ�_{k}

has kernel and cokernel that are finite of order bounded independently of k.

Remark. We do not need to assume that λ itself is not in X′_{bad}, so there is no restriction on λ, since we study T_{λ} via the specializations at λ + π^{k}, which are not in X′_{bad} for k sufficiently large.

(4.2) Alternating pairings

Recall the following classical result.

Proposition. Suppose A is a finite abelian group equipped with a non-degenerate alternating pairing

⟨ , ⟩ : A × A ⟶ ℚ/ℤ.

Then there is a group B such that A ⟶^{∼} B ⊕ B.

Applying this proposition when ⟨ , ⟩ is the Cassels pairing allows one to deduce that the order of the Tate-Šafarevič group of an elliptic curve divided by its maximal divisible subgroup is a square. In what follows, we shall consider certain Tate-Šafarevič groups as Λ-modules rather than just ℤ-modules, and so we shall now prove a version of this result in the context of Λ/(g)-modules, where g ∈ Λ is an irreducible distinguished polynomial.

(4.2.1) Fix an irreducible distinguished polynomial f ∈ Λ, and a Λ-module

M = (\frac{Λ}{(f^{n_{1}})})^{α_{1}} ⊕ ⋯ ⊕ (\frac{Λ}{(f^{n_{r}})})^{α_{r}}  with  n_{1} > ⋯ > n_{r}.

For k sufficiently large, f + π^{k} is also an irreducible distinguished polynomial. Let M_{k} = M/(f + π^{k}). So M_{k} is a finite O_{k}-module, where O_{k} = Λ/(f + π^{k}).

(4.2.2) Theorem. Suppose we have a non-degenerate alternating pairing

⟨ , ⟩ : M_{k} × M_{k} ⟶ ℚ_{p}/ℤ_{p}

satisfying ⟨rx, y⟩ = ⟨x, ry⟩ for all r in O_{k} and all x and y in M_{k}. Then the α_{i} are all even.

Note that we only need the existence of ⟨ , ⟩ for any one fixed k to deduce the result for M, and nondegenerate just means that the left and right kernels are both zero. Also, the condition ⟨rx, y⟩ = ⟨x, ry⟩ means that the isomorphism of abelian groups

M_{k} ⟶^{∼} Hom(M_{k}, ℚ_{p}/ℤ_{p})

x ⟼ ⟨x, . ⟩

is an isomorphism of O_{k}-modules, where here the O_{k}-module structure on Hom(M_{k}, ℚ_{p}/ℤ_{p}) is given by defining (rϕ) (x) = ϕ(rx).

(4.2.3) The proof of Theorem 4.2.2 consists of a series of reduction steps. Write L_{n} = Λ/(f^{n}, f + π^{k}) (recall that k is fixed throughout). So we have

M_{k} ⟶^{∼} (L_{n_{1}})^{α_{1}} ⊕ (L_{n_{2}})^{α_{2}} ⊕ ⋯ ⊕ (L_{n_{r}})^{α_{r}} = A ⊕ B

where A = (L_{n_{1}})^{α_{1}} and B = (L_{n_{2}})^{α_{2}}⊕ ⋯ ⊕(L_{n_{r}})^{α_{r}}. We have f = −π^{k} in O_{k}, so L_{n} ≅ O_{k}/(f^{n}) ≅ O_{k}/(π^{nk}).

(4.2.4) Lemma. ⟨ , ⟩|_{A} is non-degenerate.

Proof. Suppose a ∈ A is such that π^{k}a = 0. Then ⟨a, b⟩ = 0 for all b ∈ B. To see this, notice that we can find a′ ∈ A such that a = π^{(n_{1}−1)k}a′ (this is easy to check). So

⟨a, b⟩ = ⟨π^{(n_{1}−1)k}a′, b⟩ = ⟨a′, π^{(n_{1}−1)k}b⟩ = ⟨a′, 0⟩ = 0.

Now suppose that a ∈ A is non-zero and such that ⟨a, x⟩ = 0 for all x ∈ A. Multiplying a by some power of π we may assume that πa = 0 but a ≠ 0. So

⟨a, x⟩ = 0  for all x ∈ A
and  ⟨a, b⟩ = 0  for all b ∈ B (since π^{k}a = 0).

So ⟨a, m⟩ = 0 for all m ∈ M_{k}, giving a contradiction.

(4.2.5) Lemma 4.2.4 tells us that

⟨ , ⟩ : A × A ⟶ ℚ_{p}/ℤ_{p}

is non-degenerate, and so gives an isomorphism

A ⟶^{∼} Hom(A, ℚ_{p}/ℤ_{p})

a ⟼ ⟨a, . ⟩

of O_{k}-modules. In particular, any map A ⟶ ℚ_{p}/ℤ_{p} can be written as ⟨a, . ⟩ for some a ∈ A.

(4.2.6) Lemma. ⟨ , ⟩|_{B} is non-degenerate.

Proof. Suppose b ∈ B is non-zero and such that ⟨b, y⟩ = 0, for all y ∈ B. Multiplying by a power of π if necessary, we may assume that b ≠ 0 but πb = 0. Now consider the map

A ⟶ ℚ_{p}/ℤ_{p}
a ⟼ ⟨b, a⟩

By Lemma 4.2.4, there is some α ∈ A such that this map is given by ⟨α, . ⟩, i.e. such that ⟨b, a⟩ = ⟨α, a⟩ for all a ∈ A. Also, πb = 0. So

⟨πα, a⟩ = ⟨α, πa⟩ = ⟨b, πa⟩ = ⟨πb, a⟩ = 0

for all a ∈ A. Therefore πα = 0, which implies that ⟨α, y⟩ = 0 for all y ∈ B. So we have α ∈ A and b ∈ B such that ⟨α, y⟩ = 0 for all y ∈ B, ⟨b, y⟩ = 0 for all y ∈ B, and ⟨α, a⟩ = ⟨b, a⟩ for all a ∈ A, So ⟨α − b, m⟩ = 0 for all m ∈ M_{k}. So α − b = 0 which implies that α = b = 0. This proves Lemma 4.2.6.

(4.2.7) Lemmas 4.2.4 and 4.2.6 allow us to reduce Theorem 4.2.2 to the case where M_{k} ⟶^{∼} (L_{n})^{α}. The next step is to reduce to the case n = 1. For this, note that we have isomorphisms of O_{k}-modules

L_{1} ⟶^{∼} π^{(n−1)k}L_{n}  and  L_{n}/π^{k}L_{n} ⟶^{∼} L_{1}.

Also, the orthogonal complement to π^{(n−1)k}M_{k} is π^{k}M_{k}, since

⟨π^{(n−1)k}x, y⟩ = 0 for all x ⟺ ⟨x, π^{(n−1)k}y⟩ = 0 for all x
⟺ π^{(n−1)k}y = 0
⟺ there exists y′ such that y = π^{k}y′
⟺ y ∈ π^{k}M_{k}.

So we get a non-degenerate pairing

π^{(n−1)k}M_{k} × \frac{M_{k}}{π^{k}M_{k}} ⟶ ℚ_{p}/ℤ_{p}.

So we have reduced Theorem 4.2.2 to proving the following result.

(4.2.8) Lemma. Let \overline{O}_{k} = Λ/(f + π^{k}, π^{k}). Let M be a free \overline{O}_{k}-module of rank α and

⟨ , ⟩ : M × M ⟶ ℚ_{p}/ℤ_{p}

a non-degenerate alternating pairing satisfying ⟨rx, y⟩ = ⟨x, ry⟩ for all r in \overline{O}_{k} and all x and y in M. Then α is even.

Proof. One can easily reduce to the case of fields where the result is well known, but we give a direct proof by induction on α. Let X be any free rank one direct summand of M and let N be any complementary direct summand. So M = X ⊕ N. Let N^{⊥} = {x ∈ M : ⟨n, x⟩ = 0 for all n ∈ N}. It is easy to see that ⟨ , ⟩ : X ×N^{⊥} ⟶ ℚ_{p}/ℤ_{p} is non-degenerate. Now N^{⊥} is a rank one direct summand of M, (in fact M = N^{⊥}⊕X^{⊥}) and so ⟨ , ⟩ : N^{⊥}×N^{⊥} ⟶ ℚ_{p}/ℤ_{p} is zero (since ⟨ , ⟩ is alternating). So N^{⊥} ⊆ (N^{⊥})^{⊥} = N. Let L = N ∩X^{⊥}. We claim that N = N^{⊥} ⊕ L. To prove this, let n ∈ N and consider the map ⟨ . , n⟩ : X ⟶ ℚ_{p}/ℤ_{p}. There is some y ∈ N^{⊥} such that this map is given by ⟨ . , y⟩, ie such that ⟨x, n⟩ = ⟨x, y⟩ for all x ∈ X. So ⟨x, n−y⟩ = 0. So n−y ∈ X^{⊥}. So n = (n−y)+y with y ∈ N^{⊥} and n−y ∈ X^{⊥} ∩N = L. Also, if y ∈ N^{⊥} ∩L then y ∈ N^{⊥} and y ∈ X^{⊥}, so y ∈ M^{⊥} = 0. This proves the claim. So we have M = X ⊕ N^{⊥} ⊕ L. Finally, we claim that ⟨ , ⟩ : L × L ⟶ ℚ_{p}/ℤ_{p} is non-degenerate. This will be enough to finish the inductive proof of the lemma since the rank of L is α − 2. For this, suppose that l ∈ L is such that ⟨l, z⟩ = 0 for all z ∈ L. Consider the map ⟨l, . ⟩ : N^{⊥} ⟶ ℚ_{p}/ℤ_{p}. There is some x ∈ X such that this map is given by ⟨x, . ⟩, i.e. such that ⟨l, w⟩ = ⟨x, w⟩ for all w ∈ N^{⊥}. So ⟨l − x, w⟩ = 0 for all w ∈ N^{⊥}. So l − x ∈ (N^{⊥})^{⊥} = N. So we have l ∈ L = X^{⊥} ∩ N, x ∈ X, and l − x ∈ N. So x ∈ N ∩ X = 0. So x = 0. So ⟨l, w⟩ = 0 for all w ∈ N^{⊥}. Also we have l ∈ X^{⊥} so we have ⟨l, x⟩ = 0 for all x ∈ X, ⟨l, z⟩ = 0 for all z ∈ L, and ⟨l, w⟩ = 0 for all w ∈ N^{⊥}. So ⟨l, m⟩ = 0 for all m ∈ M. So l = 0, which proves the lemma.

(4.2.9) Theorem 4.4.2 allows us to study the structure of certain Λ-modules at height one primes which are of the form (f) for some irreducible distinguished polynomial f. For completeness, we also need a version to treat the height one prime (π). For this case, let

M = (\frac{Λ}{(π^{n_{1}})})^{α_{1}} ⊕ ⋯ ⊕ (\frac{Λ}{(π^{n_{r}})})^{α_{r}}  with  n_{1} > ⋯ > n_{r}.

Then for any k, X^{k} + π is an irreducible distinguished polynomial and we let M_{k} = M/(X^{k} + π). So M_{k} is an O_{k}-module, where O_{k} = Λ/(X^{k} + π). We get the following theorem.

⟨ , ⟩ : M_{k} × M_{k} ⟶ ℚ_{p}/ℤ_{p}

satisfying ⟨rx, y⟩ = ⟨x, ry⟩ for all r in O_{k} and all x and y in M_{k}. Then the α_{i} are all even.

The proof of this is exactly the same as the proof of Theorem 4.2.2.

(4.2.11) The modules that arise in the application to Tate-Šafarevič groups need not be of the standard form considered in 4.2.1, so we need the following generalization of Theorem 4.2.2. Rather than proving Theorem 4.2.2 for modules which are only pseudo-isomorphic to M (and non-degenerate pairings), we prove a version for M of standard form, but where the pairing is only non-degenerate up to finite groups.

(4.2.12) Theorem. Let M and M_{k} be as in 4.2.1. Suppose for each k sufficiently large we have an alternating pairing

⟨ , ⟩_{k} : M_{k} × M_{k} ⟶ ℚ_{p}/ℤ_{p}

satisfying ⟨rx, y⟩_{k} = ⟨x, ry⟩_{k} for all r in O_{k} and all x and y in M_{k}, and such that the left and right kernels are finite and bounded independently of k. Then the α_{i} are all even.

Proof. The proof is essentially the same as that of Theorem 4.2.2, but taking account of the finite bounded kernels. Let π^{c} kill the left and right kernels of ⟨ , ⟩_{k} for all k. We begin with the analogue of Lemma 4.2.4.

(4.2.13) Lemma. For all k sufficiently large, π^{c} kills the left and right kernels of ⟨ , ⟩_{k}|_{A}.

Proof. Suppose the lemma is false. Then for infinitely many k, there is a ∈ A such that ⟨a, x⟩_{k} = 0 for all x ∈ A but π^{c}a ≠ 0. Multiplying by a power of π if necessary, we may assume that π^{c}a ≠ 0 but π^{c+1}a = 0. Choosing k ≥ c + 1, we have π^{k}a = 0. So ⟨a, b⟩_{k} = 0 for all b ∈ B as in Lemma 4.2.4. So ⟨a, m⟩_{k} = 0 for all m ∈ M_{k}. So π^{c}a = 0, which is a contradiction.

(4.2.14) Lemma. For all k sufficiently large, π^{2c} kills the left and right kernels of ⟨ , ⟩_{k}|_{B}.

Proof. Suppose the lemma is false. Then for infinitely many k, there is b ∈ B with π^{2c}b ≠ 0, π^{2c+1}b = 0 and ⟨b, y⟩_{k} = 0 for all y ∈ B. By Lemma 4.2.13, the right kernel of ⟨ , ⟩_{k}|_{A} lies in the kernel of the map

A ⟶ ℚ_{p}/ℤ_{p}
a ⟼ ⟨b, π^{c}a⟩

so there is α ∈ A such that this map can be written as ⟨α, . ⟩_{k}, i.e. such that ⟨α, a⟩_{k} = ⟨b, π^{c}a⟩_{k} = ⟨π^{c}b, a⟩_{k} for all a ∈ A. Also, for all a ∈ A,

⟨π^{c+1}α, a⟩_{k} = ⟨α, π^{c+1}a⟩_{k} = ⟨π^{c}b, π^{c+1}a⟩_{k} = ⟨π^{2c+1}b, a⟩_{k} = ⟨0, a⟩ = 0.

So π^{2c+1}α = 0 by Lemma 4.2.13. Now take k ≥ 2c + 1, so π^{k}α = 0. So ⟨α, y⟩_{k} = 0 for all y ∈ B as in Lemma 4.2.4. So ⟨π^{c}b − α, m⟩_{k} = 0 for all m ∈ M_{k}. So π^{c}(π^{c}b − α) = 0. So π^{2c}b = π^{c}α = 0 which is a contradiction.

(4.2.15) We have now reduced Theorem 4.2.12 to the case M_{k} ⟶^{∼} (L_{n})^{α}. The arguments of 4.2.7 show that, if H is the orthogonal complement of π^{(n−1)k}M_{k}, then π^{k}M_{k} ⊆ H ⊆ π^{k−c}M_{k}, and so we deduce a pairing

π^{(n−1)k}M_{k} × \frac{M_{k}}{π^{k}M_{k}} ⟶ ℚ_{p}/ℤ_{p}.

and reduce to the case n = 1 as before. Finally, we need the analogue of Lemma 4.2.8.

(4.2.16) Lemma. With notation as in Lemma 4.2.8, suppose for all k sufficiently large, we have an alternating pairing

⟨ , ⟩ : M × M ⟶ ℚ_{p}/ℤ_{p}

satisfying ⟨rx, y⟩ = ⟨x, ry⟩ for all r in \overline{O}_{k} and all x and y in M, where \overline{O}_{k} and M are as in Lemma 4.2.8 (with α fixed) and such that the left and right kernels of ⟨ , ⟩ are killed by p^{c} for some c independent of k. Then α is even.

Proof. We rewrite the proof of Lemma 4.2.8, keeping track track of the small kernels involved. Briefly; since ⟨ , ⟩ is alternating, the left and right kernels are equal, call them K. Also, write K_{1} for the left kernel of

⟨ , ⟩ : X × N^{⊥} ⟶ ℚ_{p}/ℤ_{p}.

Then #K_{1} ≤ #K. Write M = X ⊕ N as in Lemma 4.2.8. Then M = X^{⊥} + N^{⊥} and X^{⊥} ∩ N^{⊥} = K. Define L = (N + K_{1}) ∩ X^{⊥}. Then one can show that L + N^{⊥} ⊆ N + K_{1} with index independent of k (explicitly, the index is ≤ #K ·p^{cα}). So we can choose a (N^{⊥})′ ⊃ N^{⊥} with index ≤ #K ·p^{cα} such that L+(N^{⊥})′ ⊃ N +K_{1} and so M = X + L + (N^{⊥})′. Now one can choose direct summands A ⊆ L and B ⊆ (N^{⊥})′ of rank α − 2 and 1 respectively, such that M = X ⊕A⊕B, and such that the index of A in L and B in (N^{⊥})′ are independent of k. Finally one checks that

⟨ , ⟩ : A × A ⟶ ℚ_{p}/ℤ_{p}.

has left and right kernels which are killed by p^{c′} for some new c′ (defined in terms of c and α, but independent of k). This completes the proof.

(4.2.17) One can use variants of Lemmas 4.2.13–16 to treat the case of the height one prime (π) of Λ, so that Theorem 4.2.12 holds in this case too (with M_{k} = M/(X^{k} + π) as in 4.2.9). In this case, suppose the left and right kernels of ⟨ , ⟩_{k} are finite and bounded independently of k; then there is some constant c such that X^{c} kills these kernels, independently of k. Now proceed using this X^{c} in place of the p^{c} in Lemmas 4.2.13–16. The proofs are essentially the same.

(4.3) Cassels-Tate pairings

(4.3.1) From now on, let p > 2. Returning to the situation of Section 4.1.1, if λ is any irreducible distinguished polynomial, then the specialization T_{λ} is a free rank two O_{λ}-module and there is an O_{λ}-bilinear skew-symmetric pairing

(4.3.1.1)  T_{λ} × T_{λ} ⟶ O_{λ}(1)

coming from specializing (4.1.1.2), giving an isomorphism T_{λ} ⟶^{∼} Hom_{O_{λ}} (T_{λ}, O_{λ}) (1).

(4.3.2) We will work with pairings taking values in ℤ_{p} rather than O_{λ}, and so we recall how to switch between the two. Recall that any O_{λ} is Gorenstein and that Hom_{ℤ_{p}} (O_{λ}, ℤ_{p}) is free of rank one as an O_{λ}-module. Now we have a canonical isomorphism

Hom_{O_{λ}} (T_{λ}, O_{λ}) ⊗_{O_{λ}} Hom_{ℤ_{p}} (O_{λ}, ℤ_{p}) ⟶^{∼} Hom_{ℤ_{p}} (T_{λ}, ℤ_{p}) .

θ ⊗ ϕ ↦ ϕ ∘ θ

If we fix a generator for Hom_{ℤ_{p}} (O_{λ}, ℤ_{p}), this gives an isomorphism

Hom_{O_{λ}} (T_{λ}, O_{λ}) ⟶^{∼} Hom_{ℤ_{p}} (T_{λ}, ℤ_{p})

and so the isomorphism T_{λ}→^{∼}Hom_{O_{λ}} (T_{λ}, O_{λ}) (1) translates to an isomorphism T_{λ}→^{∼}Hom_{ℤ_{p}} (T_{λ}, ℤ_{p}) (1), i.e. gives a pairing

(4.3.2.1)  ( , )_{λ} : T_{λ} × T_{λ} ⟶ ℤ_{p}(1)

satisfying (rx, y)_{λ} = (x, ry)_{λ} for all x, y ∈ T_{λ} and all r ∈ O_{λ}. The construction of (4.3.2.1) from (4.3.1.1) is non-canonical, depending on a choice of generator for Hom_{ℤ_{p}} (O_{λ}, ℤ_{p}). Once we fix such a generator, this also fixes isomorphisms A_{λ} ⟶^{∼} T_{λ} ⊗_{ℤ_{p}} ℚ_{p}/ℤ_{p} and so we also have pairings

(4.3.2.2)  ( , )_{λ} : A_{λ} × A_{λ} ⟶ ℚ_{p}/ℤ_{p}(1)

satisfying (rx, y)_{λ} = (x, ry)_{λ} for all x, y ∈ A_{λ}, r ∈ O_{λ}.

(4.3.3) The generalization by Flach [Fl] of the Cassels-Tate pairing gives the following result.

Theorem. For (λ) ∈ X_{good}, there is a non-degenerate alternating pairing

[ , ]_{λ} : � _{λ} × � _{λ} ⟶ ℚ_{p}/ℤ_{p}

satisfying [rx, y]_{λ} = [x, ry]_{λ} for all x, y ∈ � _{λ} and all r ∈ O_{λ}.

Proof. For (λ) ∈ X_{good}, H^{1}(ℚ_{p}, T_{λ}) has O_{λ}-rank two and the subspace H_{f}^{1}(ℚ_{p}, T_{λ}) has O_{λ}-rank one and is its own orthogonal complement under the local duality pairing

H^{1}(ℚ_{p}, T_{λ}) × H^{1}(ℚ_{p}, T_{λ}) ⟶ ℤ_{p}

coming from (4.3.2.1) (we know that this is true after tensoring with the field of fractions, but because we defined H_{f}^{1}(ℚ_{p}, T_{λ}) to be O_{λ}-saturated, it is true at the integral level too). So Flach’s construction gives a non-degenerate alternating pairing

[ , ]_{λ} : � _{λ} × � _{λ} ⟶ ℚ_{p}/ℤ_{p}.

One can easily check through the construction of the pairing using cocycles that the relation (rx, y)_{λ} = (x, ry)_{λ} on (4.3.2.2) implies the relation [rx, y]_{λ} = [x, ry]_{λ}.

(4.3.4) Dualizing this result, we deduce the existence of a non-degenerate alternating pairing

⟨ , ⟩_{λ} : ˆ�_{λ} × ˆ�_{λ} ⟶ ℚ_{p}/ℤ_{p},

depending upon our fixed choice of generator for the free module Hom_{ℤ_{p}} (O_{λ}, ℤ_{p}).

(4.3.5) We can now use the results of 4.2 to prove Proposition 4.0.1. We choose a pseudo-isomorphism between ˆ� and a module of standard form as in (4.1.22.1). Fix any irreducible distinguished polynomial f (or f = π) in the support of ˆ� and write this module of standard form as M ⊕ \overline{M} as in 4.1.22. So M can be written as

M = (\frac{Λ}{(f^{n_{1}})})^{α_{1}} ⊕ ⋯ ⊕ (\frac{Λ}{(f^{n_{r}})})^{α_{r}}  with  n_{1} > ⋯ > n_{r}.

We need to show that all the α_{i} in this expression are even. Write M_{k} = M/λ_{k}M where λ_{k} = f + π^{k} (or λ_{k} = X^{k} + π if f = π). Then we know from Lemma 4.1.23 that we have maps

M_{k} ⟶ ˆ�_{k}

with kernel and cokernel that are finite and bounded independently of k. If these kernels and cokernels are trivial we can apply the results of 4.2; if we take M as above, then the pairing of 4.3.4 on M_{k} ⟶^{∼} ˆ�_{k} satisfies the conditions of Theorem 4.2.2 (or Theorem 4.2.10 in the case f = π) and we deduce that the α_{i} are even. If the kernels and cokernels are non-trivial, we define a pairing on M_{k} as follows. Let ϕ_{k} : M_{k} ⟶ ˆ�_{k} be the maps of Lemma 4.1.23. Define a pairing

⟨ , ⟩′_{k} : M_{k} × M_{k} ⟶ ℚ_{p}/ℤ_{p}

by ⟨x, y⟩′_{k} = ⟨ϕ_{k}(x), ϕ_{k}(y)⟩_{k}, where ⟨ , ⟩_{k} = ⟨ , ⟩_{λ_{k}} is the pairing on ˆ�_{k} of 4.3.4. The left and right kernels of the pairings ⟨ , ⟩′_{k} are finite and bounded independently of k, as k → ∞, and we can apply Theorem 4.2.12 (and 4.2.17). This proves Proposition 4.0.1.

(4.3.6) Corollary. For any (λ) ∈ X_{good}, we have

rk_{O_{λ}} 𝕊̂_{λ} ≡ rk_{Λ}𝕊̂ (mod 2).

Proof. From Lemma 4.1.19, we get

rk_{O_{λ}} 𝕊̂_{λ} = rk_{Λ}𝕊̂ + rk_{O_{λ}} \frac{Tors_{Λ}(𝕊̂)}{λTors_{Λ}(𝕊̂)}

= rk_{Λ}𝕊̂ + 2 rk_{O_{λ}} \frac{X}{λX}

≡ rk_{Λ}𝕊̂ (mod 2).

(4.3.7) In the situation of 4.1.4, let P = (P_{k,ε}) ∈ Spec(Λ) be any arithmetic point. So P is the height one prime (λ) = (λ_{k,ε}) of O[[X]] under the identification Λ ⟶^{∼} O[[X]] of 4.1.4. If k is even and we are not in the exceptional case then the O_{λ}-ranks of the H^{i}(ℚ_{v}, M) for M = T, T^{+}, T^{−}, are equal to the “generic”ranks of 4.1.6 by 3.1.3–5, and so (λ) ∈ X_{good}. In this case, the local conditions are the same as those defined by the Bloch-Kato subgroups H_{f}^{1}(ℚ_{v}, T_{λ} ⊗_{ℤ_{p}} ℚ_{p}), again by the arguments of 3.1.3–5, and so 𝕊_{λ} is equal to the Bloch-Kato Selmer group H_{f}^{1}(ℚ, A_{λ}).

(4.3.8) Corollary. With notation as in 4.3.7, if k is even and we are not in the exceptional case, then

cork_{O_{λ}}H_{f}^{1}(ℚ, A_{λ}) ≡ rk_{Λ}𝕊̂ (mod 2).

(4.3.9) This Corollary proves Theorem A′ in the special case when R = Λ (for non-exceptional P). How often is R = Λ? In the notation of 3.2.2, this equality holds if g_{k_{0}} is not congruent to any other normalized eigenform of weight k_{0} on Γ_{1}(Np) (with respect to the embedding i_{p}). For example, let E be a modular elliptic curve over ℚ of conductor N and let f = ∑ a_{n}q^{n} be the corresponding modular form of weight 2 on Γ_{0}(N). Let p be a prime satisfying the following assumptions:

(1) p does not divide 6N.

(2) E has ordinary reduction at p, i.e. p ∤ a_{p}.

(3) E_{p}(\overline{ℚ}) is an irreducible 𝔽_{p}[G_{ℚ}]-module.

(4) p does not divide φ(N).

(5) p does not divide the degree of some modular parametrization X_{0}(N) ⟶ E of E.

(6) p ∤ (a_{p} ± 1). The first three conditions mean that the discussion in Sect. 3–4 applies to f. It follows from (4) that all modular forms corresponding to arithmetic points of R have trivial character modulo N. According to [Ri 2, Thm. 1.4] (cf. [Za, Thm. 3]), the conditions (1) and (5) imply that f is not congruent (modulo a prime above p) to any other normalized eigenform on Γ_{0}(N), and hence not even on Γ_{1}(N), by (4). It follows from [Mi, Thm. 4.6.17] and (6) that there is no congruence between f and a normalized eigenform on Γ_{1}(Np) (cf. 1.3.5). This implies that R/(γ − 1)R = ℤ_{p}, hence R = Λ = ℤ_{p}[[X]], for all primes satisfying (1)–(6). Note that, if E has no complex multiplication, the set of such p has density one.

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Department of Pure Mathematics and Mathematical Statistics University of Cambridge 16 Mill Lane Cambridge CB2 1SB UK

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