Λ-Adic Euler Characteristics of Elliptic Curves

To John Coates on the occasion of his sixtieth birthday

Daniel Delbourgo

Received: July 11, 2005 Revised: February 17, 2006

Abstract. Let E_{/ℚ} be a modular elliptic curve, and p > 3 a good ordinary or semistable prime.

Under mild hypotheses, we prove an exact formula for the μ-invariant associated to the weight-deformation of the Tate module of E. For example, at ordinary primes in the range 3 < p < 100, the result implies the triviality of the μ-invariant of X_{0}(11).

2000 Mathematics Subject Classification: 11G40; also 11F33, 11R23, 11G05

0. Introduction

A central aim in arithmetic geometry is to relate global invariants of a variety, with the behaviour of its L-function. For elliptic curves defined over a number field, these are the numerical predictions made by Birch and Swinnerton-Dyer in the 1960’s. A decade or so later, John Coates pioneered the techniques of Iwasawa’s new theory, to tackle their conjecture prime by prime. Together with Andrew Wiles, he obtained the first concrete results for elliptic curves admitting complex multiplication.

Let p be a prime number, and F_{∞} a p-adic Lie extension of a number field F. From the standpoint of Galois representations, one views the Iwasawa theory of an elliptic curve E defined over F, as being the study of the p^{∞}-Selmer group

Sel_{F_{∞}}(E) ⊂ H^{1}(Gal(\overline{F} /F ), 𝔸_{F_{∞}}) .

Here 𝔸_{F_{∞}} = Hom_{cont}(Ta_{p}(E)[[Gal(F_{∞}/F)]], ℚ/ℤ) denotes the Pontrjagin dual to the Gal(F_{∞}/F)-deformation of the Tate module. The field F_{∞} is often taken to be the cyclotomic ℤ_{p}-extension of F, or sometimes the anti-cyclotomic extension. Hopefully a more complete picture becomes available over F_{∞} = F (E[p^{∞}]), the field obtained by adjoining all p-power division points on E. If E has no complex multiplication, then Gal(F_{∞}/F ) is an open subgroup of GL_{2}(ℤ_{p}) by a theorem of Serre, which means the underlying Iwasawa algebras are no longer commutative.

In this article we study a special kind of Selmer group, namely the one which is associated to a Hida deformation of Ta_{p}(E). This object is defined by imposing the local condition that every 1-cocycle lies within a compatible family of points, living on the pro-jacobian of X̂ = \underleftarrow{lim}_{r} X_{1}(Np^{r}). There is a natural action of the diamond operators on the universal nearly-ordinary representation, which extends to a continuous action of Λ = ℤ_{p}[[1 + pℤ_{p}]] on our big Selmer group. By the structure theory of Λ-modules, we can define an analogue of the μ-invariant for a weight deformation, μ^{wt} say. One can also deform both the Tate-Shafarevich group and the Tamagawa factors [E(F_{ν}) : E_{0}(F_{ν})], as sheaves over weight-space. Conjecturally the deformation of \underline{III} should be mirrored by the behaviour of the improved p-adic L-function in [GS, Prop 5.8], which interpolates the L-values of the Hida family at the point s = 1. The Λ-adic Tamagawa factors Tam_{Λ,l} are related to the arithmetic of F_{∞} = F (E[p^{∞}]), as follows.

For simplicity suppose that E is defined over F = ℚ, and is without complex multiplication. Let p ≥ 5 be a prime where E has good ordinary reduction, and assume there are no rational cyclic p-isogenies between E and any other elliptic curve. Both Howson and Venjakob have proposed a definition for a μ-invariant associated to the full GL_{2}-extension. Presumably, this invariant should represent the power of p occurring in the leading term of a hypothetical p-adic L-function, interpolating critical L-values of E at twists by Artin representations factoring through Gal(F_{∞}/ℚ).

Recall that for a discrete p-primary Gal(F_{∞}/ℚ)-module M, its Gal(F_{∞}/ℚ)- Euler characteristic is the product

χ(Gal(F_{∞}/ℚ), M) := ∏_{j=0}^{∞} (#H^{j}(F_{∞}/ℚ, M))^{(−1)^{j}} .

Under the twin assumptions that L(E, 1) ≠ 0 and Sel_{F_{∞}}(E) is cotorsion over the non-abelian Iwasawa algebra, Coates and Howson [CH, Th 1.1] proved that

χ(Gal(F_{∞}/ℚ), Sel_{F_{∞}}(E)) = ∏_{bad primes l} |L_{l}(E, 1)|_{p} × (#Ẽ(𝔽_{p})[p^{∞}])^{2}

× (the p-part of the BS,D formula) .

Let μ^{GL^{2}} denote the power of p occurring above. It’s straightforward to combine the main result of this paper (Theorem 1.4) with their Euler characteristic calculation, yielding the upper bound

μ^{wt} ≤ μ^{GL^{2}} + ∑_{bad primes l}{ord_{p}(L_{l}(E, 1)) − ord_{p}(Tam_{Λ,l})} .

In other words, the arithmetic of the weight-deformation is controlled in the p-adic Lie extension. This is certainly consistent with the commonly held belief, that the Greenberg-Stevens p-adic L divides the projection (to the Iwasawa algebra of the maximal torus) of some ‘non-abelian L-function’ living in ℤ_{p}[[Gal(ℚ(E[p^{∞}])/ℚ)]]. The non-commutative aspects currently remain shrouded in mystery, however.

Finally, we point out that many elliptic curves E possess Λ-adic Tamagawa factors, which differ from the p-primary component of the standard factor Tam(E). P. Smith has estimated this phenomenon occurs infrequently – a list of such curves up to conductor < 10, 000 has been tabulated in [Sm, App’x A].

Acknowledgement: We dedicate this paper to John Coates on his sixtieth birthday. The author thanks him heartily for much friendly advice, and greatly appreciates his constant support over the last decade.

1. Statement of the Results

Let E be an elliptic curve defined over the rationals. We lose nothing at all by supposing that E be a strong Weil curve of conductor N_{E}, and denote by ±ϕ the non-constant morphism of curves ϕ : X_{0}(N_{E}) ↠ E minimal amongst all X_{0}(N_{E})-parametrisations. In particular, there exists a normalised eigenform f_{E} ∈ S_{2}^{new}(Γ_{0}(N_{E})) satisfying ϕ^{∗}ω_{E} = c_{E}^{Man}f_{E}(q)dq/q, where ω_{E} denotes a Néron differential on E and c_{E}^{Man} is the Manin constant for ϕ.

Fix a prime number p ≥ 5, and let’s write N = p^{−ord_{p}N_{E}}N_{E} for the tame level. We shall assume E has either good ordinary or multiplicative reduction over ℚ_{p},

hence f_{2} := \{ \begin{matrix}f_{E}(q) − β_{p}f_{E}(q^{p}) & if p ∤ N_{E} \\ f_{E}(q) & if p||N_{E} \end{matrix} will be the p-stabilisation of f_{E}

at p.

Hypothesis(R_{E}). f_{2} is the unique p-stabilised newform in S_{2}^{ord}(Γ_{0}(Np)).

Throughout Λ = ℤ_{p}[[Γ]] denotes the completed group algebra of Γ = 1 + pℤ_{p}, and L = Frac(Λ) its field of fractions. There are non-canonical isomorphisms Λ ≅ ℤ_{p}[[X]] given by sending a topological generator u_{0} ∈ Γ to the element 1 + X. In fact the ℤ_{p}-linear extension of the map σ_{k} : u_{0} ↦ u_{0}^{k−2} transforms Λ into the Iwasawa functions A_{ℤ_{p}} = ℤ_{p}⟨⟨k⟩⟩, convergent everywhere on the closed unit disk.

Under the above hypothesis, there exists a unique Λ-adic eigenform f ∈ Λ[[q]] lifting the cusp form f_{2} at weight two; furthermore

f_{k} := ∑_{n=1}^{∞} σ_{k}(a_{n}(f))q^{n} ∈ S_{k}^{ord}(Γ_{1}(Np^{r}))

is a p-stabilised eigenform of weight k and character ω^{2−k}, for all integers k ≥ 2. Hida and Mazur-Wiles [H1,H2,MW] attached a continuous Galois representation

ρ_{∞} : G_{ℚ} ⟶ GL_{2}(Λ) = Aut_{Λ}(𝕋_{∞})

interpolating Deligne’s p-adic representations for every eigenform in the family. The rank two lattice 𝕋_{∞} is always free over Λ, unramified outside of Np, and the characteristic polynomial of ρ_{∞}(Frob_{l}) will be 1−a_{l}(f)x+l⟨l⟩x^{2} for primes l ∤ Np. If we restrict to a decomposition group above p,

ρ_{∞} ⊗_{Λ} A_{ℤ_{p}} |_{G_{ℚp}} ∼ ( \begin{matrix}χ_{cy} < χ_{cy} >^{k−2} ϕ_{k}^{−1} & ∗ \\ 0 & ϕ_{k} \end{matrix} ) where ϕ_{k} : G_{ℚ_{p}}/I_{p} → ℤ_{p}^{×}

is the unramified character sending Frob_{p} to the eigenvalue of U_{p} at weight k.

Question. Can one make a Tamagawa number conjecture for the Λ-adic form f, which specialises at arithmetic primes to each Bloch-Kato conjecture?

The answer turns out to be a cautious ‘Yes’, provided one is willing to work with p-primary components of the usual suspects. In this article, we shall explain the specialisation to weight two (i.e. elliptic curves) subject to a couple of simplifying assumptions. The general case will be treated in a forthcoming work, and includes the situation where the nearly-ordinary deformation ring R_{E} is a non-trivial finite, flat extension of Λ. Let’s begin by associating local points to ρ_{∞}...

For each pair of integers m, r ∈ ℕ, the multiplication by p^{m} endomorphism on the p-divisible group J_{r} = jac X_{1}(Np^{r}) induces a tautological exact sequence

0 → J_{r}[p^{m}] → J_{r} →^{×p^{m}} J_{r} → 0. Upon taking Galois invariants, we obtain a long exact sequence in G_{ℚ_{p}}-cohomology

0 → J_{r}(ℚ_{p})[p^{m}] → J_{r}(ℚ_{p}) →^{×p^{m}} J_{r}(ℚ_{p})

→^{∂_{r,m}} H^{1}(ℚ_{p}, J_{r}[p^{m}]) → H^{1}(ℚ_{p}, J_{r})[p^{m}] → 0.

The boundary map ∂_{r,m} injects J_{r}(ℚ_{p})/p^{m} into H^{1}(ℚ_{p}, J_{r}[p^{m}]), so applying the functors \underleftarrow{lim}_{m} and \underleftarrow{lim}_{r} yields a level-compatible Kummer map

\underleftarrow{lim}_{r,m} ∂_{r,m} : J_{∞}(ℚ_{p})ˆ⊗ℤ_{p} ↪ H^{1}(ℚ_{p}, Ta_{p}(J_{∞})) which is Hecke-equivariant; here J_{∞} denotes the limit \underleftarrow{lim}_{r} jac X_{1}(Np^{r}) induced from X_{1}(Np^{r+1}) ↠^{π_{p}} X_{1}(Np^{r}).

For a compact Λ-module M, we define its twisted dual A_{M} := Hom_{cont}(M, μ_{p^{∞}}). Recall that Hida [H1] cuts 𝕋_{∞} out of the massive Galois representation Ta_{p}(J_{∞}) using idempotents e_{ord} = lim_{n→∞}U_{p}^{n!} and e_{prim} living in the abstract Hecke algebra (the latter is the projector to the p-normalised primitive part, and in general exists only after extending scalars to L).

Definition 1.1. (a) We define X(ℚ_{p}) to be the pre-image of the local points

e_{prim}. ((e_{ord}. \underleftarrow{lim}_{r,m} ∂_{r,m}(J_{∞}(ℚ_{p})ˆ⊗ℤ_{p})) ⊗_{Λ} L)

under the canonical homomorphism H^{1}(ℚ_{p}, 𝕋_{∞}) ⟶^{−⊗1} H^{1}(ℚ_{p}, 𝕋_{∞}) ⊗_{Λ} L.

(b) We define the dual group X^{D}(ℚ_{p}) to be the orthogonal complement

{x ∈ H^{1}(ℚ_{p}, A_{𝕋_{∞}}) such that inv_{ℚ_{p}}(X(ℚ_{p}) ∪ x) = 0}

under Pontrjagin duality H^{1}(ℚ_{p}, 𝕋_{∞}) × H^{1}(ℚ_{p}, A_{𝕋_{∞}}) → H^{2}(ℚ_{p}, μ_{p^{∞}}) ≅ ℚ_{p}/ℤ_{p}.

The local condition X(ℚ_{p}) will be Λ-saturated inside its ambient cohomology group. These groups were studied by the author and Smith in [DS], and are intimately connected to the behaviour of big dual exponential maps for the family.

Let Σ denote a finite set containing p and primes dividing the conductor N_{E}. Write ℚ_{Σ} for the maximal algebraic extension of the rationals, unramified outside the set of bad places Σ ∪ {∞}. Our primary object of study is the big Selmer group

Sel_{ℚ}(ρ_{∞}) := Ker (H^{1}(ℚ_{Σ}/ℚ, A_{𝕋_{∞}}) ⟶^{⊕res_{l}} ⨁_{l≠p} H^{1}(ℚ_{l}, A_{𝕋_{∞}}) ⊕ \frac{H^{1}(ℚ_{p}, A_{𝕋_{∞}})}{X^{D}(ℚ_{p})})

which is a discrete module over the local ring Λ. For each arithmetic point in Spec(Λ)^{alg}, the Λ-adic object Sel_{ℚ}(ρ_{∞}) interpolates the Bloch-Kato Selmer groups associated to the p-stabilisations f_{k} of weight k ≥ 2. At k = 2 it should encode the Birch and Swinnerton-Dyer formulae, up to some easily computable fudge-factors.

Proposition 1.2. (a) The Pontrjagin dual

{ \atop Sel_{ℚ}(ρ_{∞})} = Hom_{cont}(Sel_{ℚ}(ρ_{∞}), ℚ/ℤ)

is a finitely-generated Λ-module;

(b) If L(E, 1) ≠ 0 then { \atop Sel_{ℚ}(ρ_{∞})} is Λ-torsion, i.e. Sel_{ℚ}(ρ_{∞}) is Λ-cotorsion.

In general, one can associate a characteristic element to Sel_{ℚ}(ρ_{∞}) via

\underline{III}_{ℚ}(ρ_{∞}) := char_{Λ}(Hom_{cont}(Sel_{ℚ}(ρ_{∞})_{/Λ-div} , ℚ/ℤ))

where _{/Λ-div} indicates we have quotiented by the maximal 𝔪_{Λ}-divisible sub-

module; equivalently \underline{III}_{ℚ}(ρ_{∞}) is a generator of the characteristic ideal of Tors_{Λ} \binom{}{Sel_{ℚ}(ρ_{∞})}. If the L-function doesn’t vanish at s = 1 then by 1.2(b), the

Pontrjagin dual { \atop Sel_{ℚ}(ρ_{∞})} is already pseudo-isomorphic to a compact Λ-module of the form

⨁_{i=1}^{t} ℤ/p^{μ_{i}}ℤ ⊕ ⨁_{j=1}^{s} Λ/F_{j}^{e_{j}}Λ

where the F_{j}’s are irreducible distinguished polynomials, and all of the μ_{i}, e_{j} ≥ 0. In this particular case \underline{III}_{ℚ}(ρ_{∞}) will equal p^{μ_{1}+···μ_{t}} × ∏_{j=1}^{s} F_{j}^{e_{j}} modulo Λ^{×},

and so annihilates the whole of { \atop Sel_{ℚ}(ρ_{∞})}.

Definition/Lemma 1.3. For each prime l ≠ p and integer weight k ≥ 2, we set

Tam_{l}(ρ_{∞}; k) := #Tors_{Λ}(H^{1}(I_{l}, 𝕋_{∞}))^{Frob_{l}=1}⊗_{Λ,σ_{k}} ℤ_{p} ∈ p^{ℕ∪{0}} .

Then at weight two,

∏_{l≠p} Tam_{l}(ρ_{∞}; 2) divides the p-part of ∏_{l≠p} [C^{min}(ℚ_{l}) : C_{0}^{min}(ℚ_{l})]

where C_{/ℚ}^{min} refers to the ℚ-isogenous elliptic curve of Stevens, for which ev-

ery optimal parametrisation X_{1}(Np) ↠ E admits a factorisation X_{1}(Np) → C^{min} → E.

These mysterious Λ-adic Tamagawa numbers control the specialisation of our big Tate-Shafarevich group \underline{III} at arithmetic points. In particular, for the weight k = 2 they occur in the leading term of \underline{III}_{ℚ}(ρ_{∞}) viewed as an element of Λ ≅ ℤ_{p}[[X]]. It was conjectured in [St] that C^{min} is the same elliptic curve for which the Manin constant associated to X_{1}(Np) ↠ C^{min} is ±1. Cremona pointed out the Tamagawa factors [C^{min}(ℚ_{l}) : C_{0}^{min}(ℚ_{l})] tend to be smaller than the [E(ℚ_{l}) : E_{0}(ℚ_{l})]’s.

To state the simplest version of our result, we shall assume the following:

Hypothesis(Frb). Either (i) p ∤ N_{E} and a_{p}(E) ≠ +1, or (ii) p||N_{E} and a_{p}(E) = −1

or (iii) p||N_{E} and a_{p}(E) = +1, p ∤ ord_{p}(q_{Tate}(C^{min})).

Note that in case (iii), the condition that p does not divide the valuation of the Tate period q_{Tate}(C^{min}) ensures the p-part of [C^{min}(ℚ_{p}) : C_{0}^{min}(ℚ_{p})] is trivial.

Theorem 1.4. Assume both (R_{E}) and (Frb) hold. If L(E, 1) ≠ 0, then

σ_{2}(\underline{III}_{ℚ}(ρ_{∞}))

≡ L_{p}^{wt}(E) × [E(ℚ_{p}) : E_{0}(ℚ_{p})] ∏_{l≠p} \frac{[E(ℚ_{l}) : E_{0}(ℚ_{l})]}{Tam_{l}(ρ_{∞}; 2)} × \frac{#\underline{III}_{ℚ}(E)}{#E(ℚ)^{2}}

modulo ℤ_{p}^{×}, where the L^{wt}-invariant at weight two is defined to be

L_{p}^{wt}(E) := \frac{∫_{E(ℝ)} ω_{E}}{∫_{C^{min}(ℝ)} ω_{C^{min}}} × \frac{#C^{min}(ℚ)}{#A_{𝕋_{∞}}(ℚ)_{Γ}} .

In particular, the Γ-coinvariants of A_{𝕋_{∞}}(ℚ) = H^{0}(ℚ_{Σ}/ℚ, A_{𝕋_{∞}}) are always finite, and the denominator #A_{𝕋_{∞}}(ℚ)_{Γ} divides into #C^{min}(ℚ)[p^{∞}].

This equation is a special case of a more general Tamagawa number formalism. Whilst none of the assumptions (R_{E}), (Frb) and L(E, 1) ≠ 0 are actually necessary, the full result requires a weight-regulator term, the relative covolume of X(ℚ_{p}) and various other additional factors – we won’t consider these complications here.

Example 1.5. Consider the modular curve E = X_{0}(11) given by the equation

E : y^{2} + y = x^{3} − x^{2} − 10x − 20 .

The Tamagawa number of E at the bad prime 11 equals 5, whereas elsewhere the curve has good reduction. Let’s break up the calculation into three parts:

(a) Avoiding the supersingular prime numbers 19 and 29, one checks for every good ordinary prime 7 ≤ p ≤ 97 that both of the hypotheses (R_{E}) and (Frb) hold true (to check the former, we verified that there are no congruences modulo p between f_{E} and any newform at level 11p). Now by Theorem 1.4,

σ_{2}(\underline{III}_{ℚ}(ρ_{∞})) ≡ \frac{L_{p}^{wt}(E) × 5 × #\underline{III}_{ℚ}(E)}{Tam_{11}(ρ_{∞}; 2) × 5^{2}} ≡ 1 modulo ℤ_{p}^{×}

since the L_{p}^{wt}-invariant is a p-adic unit, and the size of \underline{III}_{ℚ}(E) is equal to one.

(b) At the prime p = 11 the elliptic curve E has split multiplicative reduction. The optimal curve C^{min} is X_{1}(11) whose Tamagawa number is trivial, hence so is Tam_{11}(ρ_{∞}; 2). Our theorem implies σ_{2}(\underline{III}_{ℚ}(ρ_{∞})) must then be an 11-adic unit.

(c) When p = 5 the curve E fails to satisfy (Frb) as the Hecke eigenvalue a_{5}(E) = 1. Nevertheless the deformation ring R_{E} ≅ Λ, and E has good ordinary reduction. Applying similar arguments to the proof of 1.4, one can show that

|σ_{2}(\underline{III}_{ℚ}(ρ_{∞}))|_{5}^{−1} divides \frac{#{ \atop X_{1}(11)}(𝔽_{5})[5^{∞}] × #\underline{III}_{ℚ}(X_{1}(11))[5^{∞}]}{#A_{𝕋_{∞}}(ℚ)_{Γ} × #X_{1}(11)(ℚ)[5^{∞}]} .

The right-hand side equals one, since X_{1}(11)(ℚ) and the reduced curve { \atop X_{1}(11)}(𝔽_{5}) possess a non-trivial 5-torsion point. As the left-hand side is 5integral, clearly #A_{𝕋_{∞}}(ℚ)_{Γ} = 1 and it follows that σ_{2}(\underline{III}_{ℚ}(ρ_{∞})) is a 5-adic unit.

Corollary 1.6. For all prime numbers p such that 5 ≤ p ≤ 97 and a_{p}(X_{0}(11)) ≠ 0,

the μ^{wt}-invariant associated to the Hida deformation of Sel_{ℚ}(X_{0}(11))[p^{∞}] is zero.

In fact the μ^{wt}-invariant is probably zero at all primes p for which X_{0}(11) has good ordinary reduction, but we need a more general formula than 1.4 to prove this.

2. Outline of the Proof of Theorem 1.4

We begin with some general comments. The rank two module 𝕋_{∞} ⊗_{Λ,σ_{2}} ℤ_{p} is isomorphic to the dual of H_{ét}^{1}(\overline{E}, ℤ_{p}), in general only after tensoring by ℚ_{p}. Consider instead the arithmetic provariety X̂ = \underleftarrow{lim}_{r≥1} X_{1}(Np^{r}) endowed with its canonical ℚ-structure. The

specialisation (σ_{2})_{∗} : 𝕋_{∞} ↠ (𝕋_{∞})_{Γ} ↪ Ta_{p}(jac X_{1}(Np)) is clearly induced

from X̂ ⟶^{proj} X_{1}(Np). It follows from [St, Th 1.9] that 𝕋_{∞}⊗_{Λ,σ_{2}} ℤ_{p} ≅ Ta_{p}(C^{min}) on an integral level, where C^{min} denotes the same elliptic curve occurring as a subvariety of jac X_{1}(Np), alluded to earlier in 1.3. Taking twisted duals of 0 → 𝕋_{∞} →^{u_{0}−1} 𝕋_{∞} → Ta_{p}(C^{min}) → 0, we obtain a corresponding short exact sequence

0 → Hom_{cont}(Ta_{p}(C^{min}), μ_{p^{∞}}) → A_{𝕋_{∞}} →^{u_{0}−1} A_{𝕋_{∞}} → 0

of discrete Λ-modules. The Weil pairing on the optimal curve C^{min} implies that Hom_{cont}(Ta_{p}(C^{min}), μ_{p^{∞}}) ≅ C^{min}[p^{∞}]. We thus deduce that Ta_{p}(C^{min}) ≅ Ta_{p}(E) if and only if there exists a cyclic p^{n}-isogeny defined over ℚ, between the two elliptic curves E and C^{min} (note this can only happen when the prime p is very small). Let G denote either Gal(ℚ_{Σ}/ℚ), or a decomposition group Gal(\overline{ℚ_{l}}/ℚ_{l}) at some prime number l. For indices j = 0, 1, 2 there are induced exact sequences

0 → H^{j}(G, A_{𝕋_{∞}})⊗_{Λ,σ_{2}} ℤ_{p} → H^{j+1}(G, C^{min}[p^{∞}]) → H^{j+1}(G, A_{𝕋_{∞}})^{Γ} → 0

and in continuous cohomology,

0 → H^{j}(G, 𝕋_{∞}) ⊗_{Λ,σ_{2}} ℤ_{p} → H^{j}(G, Ta_{p}(C^{min})) → H^{j+1}(G, 𝕋_{∞})^{Γ} → 0 .

From now on, we’ll just drop the ‘ _{σ_{2}}’ from the tensor product notation altogether.

Remark: Our strategy is to compare Sel_{ℚ}(ρ_{∞}) with the p-primary Selmer group for C^{min} over the rationals. We can then use the Isogeny Theorem to exchange the optimal curve C^{min} with the strong Weil curve E.

For each prime l ≠ p, we claim there is a natural map

δ_{l} : \frac{H^{1}(ℚ_{l}, C^{min}[p^{∞}])}{H_{nr}^{1}(ℚ_{l}, C^{min}[p^{∞}])} ⟶ H^{1}(ℚ_{l}, A_{𝕋_{∞}})^{Γ} ;

here H_{nr}^{1}(ℚ_{l}, C^{min}[p^{∞}]) denotes the orthogonal complement to the p-saturation of H^{1} (Frob_{l}, Ta_{p}(C^{min})^{I_{l}}) inside H^{1}(ℚ_{l}, Ta_{p}(C^{min})). To see why this map exists, note that H^{1}(ℚ_{l}, 𝕋_{∞}) is Λ-torsion, hence H^{1}(ℚ_{l}, 𝕋_{∞})⊗_{Λ} ℤ_{p} is p^{∞}-torsion and must lie in any p-saturated subgroup of H^{1}(ℚ_{l}, Ta_{p}(C^{min})). Consequently the Γ-coinvariants

H^{1}(ℚ_{l}, 𝕋_{∞})_{Γ} ↪ the p-saturation of H^{1} (Frob_{l}, Ta_{p}(C^{min})^{I_{l}}) ,

and then dualising we obtain δ_{l}.

Let’s now consider what happens when l = p. In [DS, Th 2.1] we identified the family of local points X(ℚ_{p}) with the cohomology subgroup

H_{G}^{1}(ℚ_{p}, 𝕋_{∞}) := Ker (H_{cont}^{1}(ℚ_{p}, 𝕋_{∞}) ⟶^{(−⊗1)⊗1} H_{cont}^{1}(ℚ_{p}, 𝕋_{∞} ⊗ 𝔹_{dR}) ⊗_{Λ} L)

where 𝔹_{dR} denotes Iovita and Stevens’ period ring. In particular, we showed that

X(ℚ_{p})_{Γ} = H_{G}^{1}(ℚ_{p}, 𝕋_{∞}) ⊗_{Λ} ℤ_{p} ↪ H_{g}^{1}(ℚ_{p}, Ta_{p}(C^{min})) ≅ C^{min}(ℚ_{p})ˆ⊗ℤ_{p}

the latter isomorphism arising from [BK, Section 3]. Dualising the above yields

δ_{p} : \frac{H^{1}(ℚ_{p}, C^{min}[p^{∞}])}{C^{min}(ℚ_{p}) ⊗ ℚ_{p}/ℤ_{p}} ⟶ (\frac{H^{1}(ℚ_{p}, A_{𝕋_{∞}})}{X^{D}(ℚ_{p})})^{Γ}

because H_{g}^{1}(ℚ_{p}, Ta_{p}(C^{min}))^{⊥} ≅ C^{min}(ℚ_{p}) ⊗ ℚ_{p}/ℤ_{p} and X(ℚ_{p})^{⊥} = X^{D}(ℚ_{p}).

Documenta Mathematica · Extra Volume Coates (2006) 301–323

Lemma 2.1. For all prime numbers l ∈ Σ, the kernel of δ_{l} is a finite p-group.

We defer the proof until the next section, but for l ≠ p it’s straightforward. This discussion may be neatly summarised in the following commutative diagram, with left-exact rows:

0 → Sel_{ℚ}(C^{min})[p^{∞}] → H^{1}(ℚ_{Σ}/ℚ, C^{min}[p^{∞}]) ⟶^{λ_{0}} ⨁_{l∈Σ} \frac{H^{1}(ℚ_{l}, C^{min}[p^{∞}])}{H_{⋆}^{1}(ℚ_{l}, C^{min}[p^{∞}])}

α↓  _{β}↓  _{⊕δ_{l}} ↓

0 → Sel_{ℚ}(ρ_{∞})^{Γ} → H^{1}(ℚ_{Σ}/ℚ, A_{𝕋_{∞}})^{Γ} ⟶^{λ_{∞}} ⨁_{l∈Σ} (\frac{H^{1}(ℚ_{l}, A_{𝕋_{∞}})}{H_{⋆}^{1}(ℚ_{l}, A_{𝕋_{∞}})})^{Γ}.

Figure 1.

At primes l ≠ p the notation H_{⋆}^{1} represents H_{nr}^{1}. When l = p we have written H_{⋆}^{1}(ℚ_{l}, C^{min}[p^{∞}]) for the points C^{min}(ℚ_{p}) ⊗ ℚ_{p}/ℤ_{p}, and analogously H_{⋆}^{1}(ℚ_{l}, A_{𝕋_{∞}}) in place of our family of local points X^{D}(ℚ_{p}). Applying the Snake Lemma to the above, we obtain a long exact sequence

0 → Ker(α) → Ker(β) → Im(λ_{0}) ∩ (⨁_{l∈Σ} Ker(δ_{l})) → Coker(α) → 0

as the map β is surjective. The kernel of β equals H^{0}(ℚ_{Σ}/ℚ, A_{𝕋_{∞}})⊗_{Λ} ℤ_{p} i.e.,

the Γ-coinvariants H^{1}(Γ, H^{0}(ℚ_{Σ}/ℚ, A_{𝕋_{∞}})). As Γ is pro-cyclic and A_{𝕋_{∞}} is discrete,

#H^{1}(Γ, H^{0}(ℚ_{Σ}/ℚ, A_{𝕋_{∞}})) ≤ #H^{0}(ℚ_{Σ}/ℚ, H^{0}(Γ, A_{𝕋_{∞}}))
= #H^{0}(ℚ_{Σ}/ℚ, Hom_{cont}(𝕋_{∞} ⊗_{Λ} ℤ_{p} , μ_{p^{∞}}))
= #H^{0}(ℚ_{Σ}/ℚ, C^{min}[p^{∞}]) = #C^{min}(ℚ)[p^{∞}] .

In other words, the size of Ker(β) is bounded by #C^{min}(ℚ)[p^{∞}]. By a wellknown theorem of Mazur on torsion points, the latter quantity is at most 16.

Remarks: (i) Let’s recall that for any elliptic curve A over the rational numbers, its Tate-Shafarevich group can be defined by the exactness of

0 → A(ℚ) ⊗ ℚ/ℤ → H^{1}(ℚ, A) → \underline{III}_{ℚ}(A) → 0 .

(ii) Lemma 2.1 implies every term occurring in our Snake Lemma sequence is finite, and as a direct consequence Sel_{ℚ}(C^{min})[p^{∞}] ⟶^{α} Sel_{ℚ}(ρ_{∞})^{Γ} is a quasiisomorphism. The coinvariants ({ \atop Sel_{ℚ}(ρ_{∞})})_{Γ} must then be of finite type over ℤ_{p}, Nakayama’s lemma forces { \atop Sel_{ℚ}(ρ_{∞})} to be of finite type over Λ, and Proposition 1.2(a) follows.

(iii) Assume further that L(E, 1) ≠ 0. By work of Kolyvagin and later Kato [Ka], both E(ℚ) and \underline{III}_{ℚ}(E) are finite. Since C^{min} is ℚ-isogenous to E, clearly the Mordell-Weil and Tate-Shafarevich groups of the optimal curve must also be finite. Equivalently #Sel_{ℚ}(C^{min}) < ∞, whence

rank_{Λ}({ \atop Sel_{ℚ}(ρ_{∞})}) ≤

≤ corank_{ℤ_{p}}(Sel_{ℚ}(ρ_{∞})^{Γ}) = corank_{ℤ_{p}}(Sel_{ℚ}(C^{min})[p^{∞}]) = 0.

It follows that Sel_{ℚ}(ρ_{∞}) is Λ-cotorsion, and Proposition 1.2(b) is established. The special value of \underline{III}_{ℚ}(ρ_{∞}) at σ_{2} is determined (modulo p-adic units) by the Γ-Euler characteristic of Sel_{ℚ}(ρ_{∞}), namely

χ(Γ, Sel_{ℚ}(ρ_{∞})) := ∏_{j=0}^{∞} (#H^{j}(Γ, Sel_{ℚ}(ρ_{∞})))^{(−1)^{j}} = \frac{#H^{0}(Γ, Sel_{ℚ}(ρ_{∞}))}{#H^{1}(Γ, Sel_{ℚ}(ρ_{∞}))}

as Γ has cohomological dimension one. After a brisk diagram chase around Figure 1, we discover that

χ(Γ, Sel_{ℚ}(ρ_{∞})) = \frac{#Sel_{ℚ}(C^{min})[p^{∞}] × #(Im(λ_{0}) ∩ (⨁_{l∈Σ} Ker(δ_{l})) )}{#Ker(β) × #H^{1}(Γ, Sel_{ℚ}(ρ_{∞}))}

= \frac{#\underline{III}_{ℚ}(C^{min})[p^{∞}] × ∏_{l∈Σ} #Ker(δ_{l})}{#A_{𝕋_{∞}}(ℚ)_{Γ} × #H^{1}(Γ, Sel_{ℚ}(ρ_{∞})) × ∏_{l∈Σ} [Ker(δ_{l}) : Im(λ_{0}) ∩ Ker(δ_{l})]}.

Proposition 2.2.

(a) #Ker(δ_{l}) = |[C^{min}(ℚ_{l}) : C_{0}^{min}(ℚ_{l})]|_{p}^{−1} × |Tam_{l}(ρ_{∞}; 2)|_{p} if l ≠ p;

(b) #Ker(δ_{p}) = 1 and |[C^{min}(ℚ_{p}) : C_{0}^{min}(ℚ_{p})]|_{p} = 1 if Hypothesis(Frb) holds

for E.

Proposition 2.3. If L(E, 1) ≠ 0, then

#H^{1}(Γ, Sel_{ℚ}(ρ_{∞})) × ∏_{l∈Σ} [Ker(δ_{l}) : Im(λ_{0}) ∩ Ker(δ_{l})] = #C^{min}(ℚ)[p^{∞}] .

The former result is proved in the next section, and the latter assertion in §4. Substituting them back into our computation of the Γ-Euler characteristic,

χ(Γ, Sel_{ℚ}(ρ_{∞})) ≈ \frac{#\underline{III}_{ℚ}(C^{min}) × ∏_{l∈Σ} [C^{min}(ℚ_{l}) : C_{0}^{min}(ℚ_{l})]}{#A_{𝕋_{∞}}(ℚ)_{Γ} × #C^{min}(ℚ) × ∏_{l∈Σ−\{p\}} Tam_{l}(ρ_{∞}; 2)}

Documenta Mathematica · Extra Volume Coates (2006) 301–323

where the notation x ≈ y is employed whenever x = uy for some unit u ∈ ℤ_{p}^{×}.

Setting L_{p}^{wt,†}(E) := #C^{min}(ℚ)/#A_{𝕋_{∞}}(ℚ)_{Γ} , the above can be rewritten as

\frac{L_{p}^{wt,†}(E)}{∏_{l∈Σ−\{p\}} Tam_{l}(ρ_{∞}; 2)} × \frac{#\underline{III}_{ℚ}(C^{min}) × ∏_{l∈Σ} [C^{min}(ℚ_{l}) : C_{0}^{min}(ℚ_{l})]}{#C^{min}(ℚ)^{2}} .

Cassels’ Isogeny Theorem allows us to switch C^{min} with the isogenous curve E, although this scales the formula by the ratio of periods ∫_{E(ℝ)} ω_{E}/ ∫_{C^{min}(ℝ)} ω_{C^{min}}.

Observing that σ_{2}(\underline{III}_{ℚ}(ρ_{∞})) ≈ χ(Γ, Sel_{ℚ}(ρ_{∞})), Theorem 1.4 is finally proved.

3. Computing the Local Kernels

We now examine the kernels of the homorphisms δ_{l} for all prime numbers l ∈ Σ. Let’s start by considering l ≠ p. By its very definition, δ_{l} is the dual of

δ̂_{l} : H^{1}(ℚ_{l}, 𝕋_{∞}) ⊗_{Λ} ℤ_{p} ↪ H_{nr}^{1}(ℚ_{l}, Ta_{p}(C^{min}))

where H_{nr}^{1}(⋯) denotes the p-saturation of H^{1} (Frob_{l}, Ta_{p}(C^{min})^{I_{l}}) ≅

\frac{Ta_{p}(C^{min})^{I_{l}}}{(Frob_{l}−1)} . The key term we need to calculate is

#Ker(δ_{l}) = #Coker(δ̂_{l}) = [H_{nr}^{1}(ℚ_{l}, Ta_{p}(C^{min})) : H^{1}(ℚ_{l}, 𝕋_{∞}) ⊗_{Λ} ℤ_{p}] .

Firstly, the sequence 0 → 𝕋_{∞}^{I_{l}} ⊗_{Λ} ℤ_{p} → Ta_{p}(C^{min})^{I_{l}} → H^{1}(I_{l}, 𝕋_{∞})^{Γ} → 0 is exact, and 𝕋_{∞}^{I_{l}} ⊗_{Λ} ℤ_{p} coincides with (𝕋_{∞} ⊗_{Λ} ℤ_{p})^{I_{l}} = Ta_{p}(C^{min})^{I_{l}} since the Galois action and diamond operators commute on 𝕋_{∞}. As a corollary H^{1}(I_{l}, 𝕋_{∞})^{Γ} must be zero. The group Gal(ℚ_{l}^{unr}/ℚ_{l}) is topologically generated by Frobenius, hence

H^{1}(Frob_{l}, 𝕋_{∞}^{I_{l}})_{Γ} ≅ (\frac{𝕋_{∞}^{I_{l}}}{(Frob_{l} − 1).𝕋_{∞}^{I_{l}}} ) ⊗_{Λ} ℤ_{p}

= (\frac{(𝕋_{∞} ⊗_{Λ} ℤ_{p})^{I_{l}}}{(Frob_{l} − 1).(𝕋_{∞} ⊗_{Λ} ℤ_{p})^{I_{l}}} ) ≅ H^{1}(Frob_{l}, Ta_{p}(C^{min})^{I_{l}}).

Since the local cohomology H^{1}(ℚ_{l}, 𝕋_{∞}) is always Λ-torsion when the prime l ≠ p, inflation-restriction provides us with a short exact sequence

0 → H^{1}(Frob_{l}, 𝕋_{∞}^{I_{l}}) →^{infl} H^{1}(ℚ_{l}, 𝕋_{∞}) →^{rest} Tors_{Λ}(H^{1}(I_{l}, 𝕋_{∞})^{Frob^{l}}) → 0 .

The boundary map Tors_{Λ}(H^{1}(I_{l}, 𝕋_{∞})^{Frob^{l}})^{Γ}→ H^{1}(Frob_{l}, 𝕋_{∞}^{I_{l}})_{Γ} trivialises be-

cause H^{1}(I_{l}, 𝕋_{∞})^{Γ} = 0, so the Γ-coinvariants H^{1}(Frob_{l}, 𝕋_{∞}^{I_{l}})_{Γ} inject into

H^{1}(ℚ_{l}, 𝕋_{∞})_{Γ} under inflation. We deduce that there is a commutative diagram, with exact rows and columns:

0 0

↓  ↓
H^{1}(Frob_{l}, 𝕋_{∞}^{I_{l}})_{Γ}  ↪^{infl} H^{1}(ℚ_{l}, 𝕋_{∞})_{Γ}  ↠^{rest} Tors_{Λ}(H^{1}(I_{l}, 𝕋_{∞})^{Frob^{l}})_{Γ}
||  ↓  _{θ}↓

H^{1}(Frob_{l}, Ta_{p}(C^{min})^{I_{l}}) ↪^{infl} H_{nr}^{1}(ℚ_{l}, Ta_{p}(C^{min})) ↠^{rest} H^{1}(I_{l}, Ta_{p}(C^{min}))^{Frob^{l}}[p^{∞}]

↓  ↓
H^{2}(ℚ_{l}, 𝕋_{∞})^{Γ}  ≅  Coker(θ)
↓  ↓

0 0 Figure 2.

Remark: Using Figure 2 to compute indices, general nonsense informs us that

#Ker(δ_{l}) = [H_{nr}^{1}(ℚ_{l}, Ta_{p}(C^{min})) : H^{1}(ℚ_{l}, 𝕋_{∞})_{Γ}] = #Coker(θ)

= \frac{#H^{1}(I_{l}, Ta_{p}(C^{min}))^{Frob^{l}}[p^{∞}]}{#Tors_{Λ}(H^{1}(I_{l}, 𝕋_{∞})^{Frob^{l}})_{Γ}} ≈ \frac{[C^{min}(ℚ_{l}) : C_{0}^{min}(ℚ_{l})]}{Tam_{l}(ρ_{∞}; 2)} .

In one fell swoop this proves Proposition 2.2(a), Lemma 1.3 and half of Lemma 2.1. Let’s concentrate instead on l = p. The kernel of δ_{p} is dual to the cokernel of

δ̂_{p} : H_{G}^{1}(ℚ_{p}, 𝕋_{∞}) ⊗_{Λ} ℤ_{p} ↪ H_{g}^{1}(ℚ_{p}, Ta_{p}(C^{min})) .

Clearly the ℤ_{p}-rank of H_{G}^{1}(ℚ_{p}, 𝕋_{∞}) ⊗_{Λ} ℤ_{p} is bounded below by the Λ-rank of H_{G}^{1}(ℚ_{p}, 𝕋_{∞}) which equals one, thanks to a specialisation argument in [DS, Th 3]. On the other hand

rank_{ℤ_{p}}(H_{g}^{1}(ℚ_{p}, Ta_{p}(C^{min}))) = dim_{ℚ_{p}}(C^{min}(ℚ_{p}) ˆ⊗ ℚ_{p}) = 1

because the formal group of C_{/ℤ_{p}}^{min} has semistable height one. We conclude that

#Ker(δ_{p}) = #Coker(δ̂_{p}) = [H_{g}^{1}(ℚ_{p}, Ta_{p}(C^{min})) : H_{G}^{1}(ℚ_{p}, 𝕋_{∞})_{Γ}] must be finite, which completes the demonstration of Lemma 2.1.

Remarks: (i) For any de Rham G_{ℚ_{p}}-representation V, Bloch and Kato [BK] define a dual exponential map

exp_{V}^{∗} : H^{1}(ℚ_{p}, V ) ⟶ Fil^{0}D_{dR}(V) := (V ⊗_{ℚ_{p}} B_{dR}^{+})^{G_{ℚp}}

whose kernel is H_{g}^{1}(ℚ_{p}, V ). If V equals the p-adic representation Ta_{p}(C^{min})⊗_{ℤ_{p}} ℚ_{p}, then the cotangent space Fil^{0}D_{dR}(V) ≅ ℚ_{p} ⊗_{ℚ} H_{dR}^{1}(C^{min}/ℚ) is a ℚ_{p}-line, generated by a Néron differential ω_{C^{min}} on the optimal elliptic curve.

(ii) Applying exp_{V}^{∗} above and then cupping with the dual basis ω_{C^{min}}^{∗}, we obtain a homomorphism

exp_{ω}^{∗} : \frac{H^{1}(ℚ_{p}, Ta_{p}(C^{min}))}{H_{g}^{1}(ℚ_{p}, Ta_{p}(C^{min}))} ⟶ (Ta_{p}(C^{min}) ⊗_{ℤ_{p}} B_{dR}^{+})^{G_{ℚp}} ⟶^{− ∪ ω_{C^{min}}^{∗}} ℚ_{p}

which sends Kato’s zeta element [Ka, Th 13.1] to a non-zero multiple of \frac{L_{Np}(C^{min},1)}{Ω_{C^{min}}^{+}}. In particular L_{Np}(C^{min}, 1) = L_{Np}(E, 1) ≠ 0, so the image of

the composition exp_{ω}^{∗} must be a lattice p^{n_{1}}ℤ_{p} ⊂ ℚ_{p} say. Let’s abbreviate the quotient H^{1}/H_{g}^{1} by H_{/g}^{1}. Notice also that the ℤ_{p}-rank of H_{/g}^{1}(ℚ_{p}, Ta_{p}(C^{min})) equals one and the module is p^{∞}-torsion free, hence exp_{ω}^{∗} is injective.

In [De, Th 3.3] we showed the existence of a big dual exponential map

EXP_{𝕋_{∞}}^{∗} : H^{1}(ℚ_{p}, 𝕋_{∞}) ⟶ Λ[1/p] , Ker(EXP_{𝕋_{∞}}^{∗}) = H_{G}^{1}(ℚ_{p}, 𝕋_{∞})

interpolating the standard exp^{∗}’s at the arithmetic points (we skip over the details). At weight two, EXP_{𝕋_{∞}}^{∗} modulo u_{0} − 1 coincides with exp_{ω}^{∗} up to a non-zero scalar. The weight-deformation of Kato’s zeta-element lives in loc_{p}(H^{1}(ℚ, 𝕋_{∞})), and via

H^{1}(ℚ_{p}, 𝕋_{∞}) ⟶^{mod u_{0}−1} H^{1}(ℚ_{p}, 𝕋_{∞})_{Γ} ↠^{proj} \frac{H^{1}(ℚ_{p}, 𝕋_{∞})_{Γ}}{H^{1}(ℚ_{p}, 𝕋_{∞})_{Γ} ∩ H_{g}^{1}} ↪^{exp_{ω}^{∗}} ℚ_{p}

is sent to the L-value \frac{L_{Np}(C^{min},1)}{Ω_{C^{min}}^{+}} × (a Λ-adic period). In this case, the image

of H^{1}(ℚ_{p}, 𝕋_{∞})_{Γ} under exp_{ω}^{∗} will be a lattice p^{n_{2}}ℤ_{p} ⊂ ℚ_{p} for some n_{2} ≥ n_{1}. Key Claim: There is a commutative diagram, with exact rows

0 → H_{G}^{1}(ℚ_{p}, 𝕋_{∞})_{Γ}  ⟶^{ε} H^{1}(ℚ_{p}, 𝕋_{∞})_{Γ}  ⟶^{exp^{∗}(−) ∪ ω_{C^{min}}^{∗}} p^{n_{2}}ℤ_{p} → 0
↓  _{nat}↓  _{id}↓

0 → H_{g}^{1}(ℚ_{p}, Ta_{p}(C^{min})) ⟶ H^{1}(ℚ_{p}, Ta_{p}(C^{min})) ⟶^{exp^{∗}(−) ∪ ω_{C^{min}}^{∗}} p^{n_{1}}ℤ_{p} → 0.

To verify this assertion, we need to prove the injectivity of the top-left map ε. Recall that H_{G}^{1}(ℚ_{p}, 𝕋_{∞}) = X(ℚ_{p}) is Λ-saturated inside the local H^{1}, thus the quotient H_{/G}^{1}(ℚ_{p}, 𝕋_{∞}) is Λ-free. In particular, both H_{G}^{1} and H^{1} share the same Λ-torsion submodules, so at weight two H_{G}^{1}(ℚ_{p}, 𝕋_{∞})_{Γ} and H^{1}(ℚ_{p}, 𝕋_{∞})_{Γ} must have identical ℤ_{p}-torsion. It follows from the invariants/coinvariants sequence

\begin{matrix}0 → H_{G}^{1}(ℚ_{p}, 𝕋_{∞})^{Γ} → & H^{1}(ℚ_{p}, 𝕋_{∞})^{Γ} → H_{/G}^{1}(ℚ_{p}, 𝕋_{∞})^{Γ} \\ →^{∂} H_{G}^{1}(ℚ_{p}, 𝕋_{∞})_{Γ} →^{ε} & H^{1}(ℚ_{p}, 𝕋_{∞})_{Γ} → H_{/G}^{1}(ℚ_{p}, 𝕋_{∞})_{Γ} → 0 \end{matrix}

that ε fails to be injective, if and only if the image of ∂ has ℤ_{p}-rank at least one. However,

rank_{ℤ_{p}}Im(∂) =

= rank_{ℤ_{p}}(H_{G}^{1}( ⋯)_{Γ}) − rank_{ℤ_{p}}(H^{1}( ⋯)_{Γ}) + rank_{ℤ_{p}}(H_{/G}^{1}( ⋯)_{Γ})

≤ rank_{ℤ_{p}}(H_{G}^{1}( ⋯)_{Γ}) − rank_{ℤ_{p}}(H^{1}( ⋯)_{Γ}) + rank_{ℤ_{p}}(p^{n_{2}}ℤ_{p})

as the rank of H_{/G}^{1}( ⋯)_{Γ} is bounded by the rank of H^{1}( ⋯)_{Γ}/(H^{1}( ⋯)_{Γ} ∩

H_{g}^{1}). The right-hand side above is equal to zero, hence rank_{ℤ_{p}}Im(∂) is forced to be zero. The non-triviality of the boundary map ∂ can therefore never happen, and the injectivity of ε follows as well.

Remark: Using our Key Claim to calculate [H_{g}^{1}( ⋯) : H_{G}^{1}( ⋯)_{Γ}], we find that

#Ker(δ_{p}) = p^{−(n_{2}−n_{1})} × [H^{1}(ℚ_{p}, Ta_{p}(C^{min})) : H^{1}(ℚ_{p}, 𝕋_{∞})_{Γ}]
= p^{−(n_{2}−n_{1})} × #H^{2}(ℚ_{p}, 𝕋_{∞})^{Γ} = p^{−(n_{2}−n_{1})} × #H^{0}(ℚ_{p}, A_{𝕋_{∞}})_{Γ}

where the very last equality arises from the non-degeneracy of the local pairing H^{2}(ℚ_{p}, 𝕋_{∞}) × H^{0}(ℚ_{p}, A_{𝕋_{∞}}) → ℚ_{p}/ℤ_{p}.

By an argument familiar from §2,

#H^{0}(ℚ_{p}, A_{𝕋_{∞}})_{Γ}
≤ #H^{0}(ℚ_{p}, A_{𝕋_{∞}}^{Γ}) = #H^{0}(ℚ_{p}, Hom_{cont}(𝕋_{∞} ⊗_{Λ} ℤ_{p}, μ_{p^{∞}}))
= #H^{0}(ℚ_{p}, Hom_{cont}(Ta_{p}(C^{min}), μ_{p^{∞}})) = #C^{min}(ℚ_{p})[p^{∞}]

again due to the pro-cyclicity of Γ. Because n_{2} − n_{1} ≥ 0, we get an upper bound

#Ker(δ_{p}) ≤ p^{−(n_{2}−n_{1})}#C^{min}(ℚ_{p})[p^{∞}] ≤ #C^{min}(ℚ_{p})[p^{∞}] ; we proceed by showing that the right-hand side is trivial under Hypothesis(Frb). Case (i): p ∤ N_{E} and a_{p}(E) ≠ +1. Here E and the isogenous curve C^{min} have good ordinary reduction at the prime p; in particular, the formal group of C_{/ℤ_{p}}^{min} possesses no points of order p since p ≠ 2. It follows that C^{min}(ℚ_{p})[p^{∞}] injects into the subgroup of 𝔽_{p}-rational points on C̃^{min}, the reduced elliptic curve. Moreover

#C̃^{min}(𝔽_{p}) = p + 1 − a_{p}(E) ≢ 0 ( mod p ) as a_{p}(E) ≢ +1,

meaning C^{min}(ℚ_{p})[p^{∞}] ≅ C̃^{min}(𝔽_{p})[p^{∞}] is the trivial group.

Case (ii): p||N_{E} and a_{p}(E) = −1. Both E and C^{min} have non-split multiplicative reduction at p. The Tamagawa factor [C^{min}(ℚ_{p}) : C_{0}^{min}(ℚ_{p})] is either 1, 2, 3 or 4, all of which are coprime to p ≥ 5. We thus have an isomorphism C^{min}(ℚ_{p})[p^{∞}] ≅ C_{0}^{min}(ℚ_{p})[p^{∞}]. Again the formal group is p-torsion free, so C_{0}^{min}(ℚ_{p})[p^{∞}] coincides with the p^{∞}-torsion in the group of non-singular points C̃^{min}(𝔽_{p}) − {node}. But these non-singular points look like 𝔽_{p}^{×} which has no points of order p, so neither does C^{min}(ℚ_{p}).

Case (iii): p||N_{E} and a_{p}(E) = +1, p ∤ ord_{p}(q_{Tate}(C^{min})).

This last situation corresponds to our elliptic curves being split multiplicative at p. The group of connected components C^{min}(ℚ_{p})/C_{0}^{min}(ℚ_{p}) ≅

ℤ/ord_{p}(q_{Tate}(C^{min}))ℤ has order coprime to p, by assumption. Again

C^{min}(ℚ_{p})[p^{∞}] ≅ C_{0}^{min}(ℚ_{p})[p^{∞}], and an identical argument to case (ii) establishes that the p-part of C^{min}(ℚ_{p}) is trivial.

4. Global Euler-Poincaré Characteristics

It remains to give the proof of Proposition 2.3, i.e. to demonstrate why

#H^{1}(Γ, Sel_{ℚ}(ρ_{∞})) × ∏_{l∈Σ} [Ker(δ_{l}) : Im(λ_{0}) ∩ Ker(δ_{l})] = #C^{min}(ℚ)[p^{∞}]

whenever the analytic rank of E is zero. Let’s start by writing down the Poitou-Tate sequence for the optimal curve. It is an easy exercise to verify that H^{1}(ℚ_{l}, C^{min}[p^{∞}])/H_{⋆}^{1}(ℚ_{l}, C^{min}[p^{∞}]) is

isomorphic to H^{1}(ℚ_{l}, C^{min})[p^{∞}] where ‘⋆ = nr’ if l ≠ p, and ‘⋆ = g’ if l = p. The exactness of the sequence

0 → Sel_{ℚ}(C^{min})[p^{∞}] → H^{1}(ℚ_{Σ}/ℚ, C^{min}[p^{∞}]) →^{λ_{0}} ⨁_{l∈Σ} H^{1}(ℚ_{l}, C^{min})[p^{∞}]

→ Hom_{cont}(C^{min}(ℚ)ˆ⊗ℤ_{p} , ℚ/ℤ) → H^{2}(ℚ_{Σ}/ℚ, C^{min}[p^{∞}]) → ⋯

is then an old result of Cassels.

Lemma 4.1. If Sel_{ℚ}(C^{min})[p^{∞}] is finite, then H^{2}(ℚ_{Σ}/ℚ, C^{min}[p^{∞}]) = 0.

The proof is well-known to the experts. It’s a basic consequence of the cyclotomic Iwasawa theory of elliptic curves, e.g. see Coates’ textbook on the subject. If we mimic the same approach Λ-adically, the Poitou-Tate exact sequence reads as

0 → Sel_{ℚ}(ρ_{∞}) →

→ H^{1}(ℚ_{Σ}/ℚ, A_{𝕋_{∞}}) ⟶^{λ_{∞}^{†}} ⨁_{l∈Σ} \frac{H^{1}(ℚ_{l}, A_{𝕋_{∞}})}{H_{⋆}^{1}(ℚ_{l}, A_{𝕋_{∞}})} → { \atop 𝔖𝔢𝔩_{ℚ}(𝕋_{∞})} → ⋯

where the compact Selmer group is defined to be

𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) :=

:= Ker (H^{1}(ℚ_{Σ}/ℚ, 𝕋_{∞}) ⟶^{⊕res_{l}} ⨁_{l≠p} \frac{H^{1}(ℚ_{l}, 𝕋_{∞})}{H^{1}(ℚ_{l}, A_{𝕋_{∞}})^{⊥}} ⊕ \frac{H^{1}(ℚ_{p}, 𝕋_{∞})}{X(ℚ_{p})}) .

In fact H^{1}(ℚ_{l}, A_{𝕋_{∞}}) is orthogonal to all of H^{1}(ℚ_{l}, 𝕋_{∞}) under Pontrjagin duality, so the local conditions at l ≠ p are completely redundant.

Proposition 4.2. If L(E, 1) ≠ 0, then the compact version 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) is zero.

The proof is rather lengthy – we postpone it till the end of this section.

As a corollary, the restriction map λ_{∞}^{†} must be surjective at the Λ-adic level. Taking Γ-cohomology, we obtain a long exact sequence

0 ⟶ Sel_{ℚ}(ρ_{∞})^{Γ} ⟶ H^{1}(ℚ_{Σ}/ℚ, A_{𝕋_{∞}})^{Γ} ⟶^{λ_{∞}} ⨁_{l∈Σ} (\frac{H^{1}(ℚ_{l}, A_{𝕋_{∞}})}{H_{⋆}^{1}(ℚ_{l}, A_{𝕋_{∞}})})^{Γ}

⟶ H^{1}(Γ, Sel_{ℚ}(ρ_{∞})) ⟶ H^{1}(Γ, H^{1}(ℚ_{Σ}/ℚ, A_{𝕋_{∞}})) .

The right-most term is zero, since it is contained inside H^{2}(ℚ_{Σ}/ℚ, C^{min}[p^{∞}]) which vanishes by Lemma 4.1. We can then compare the cokernels of λ_{0} and λ_{∞} via the commutative diagram, with exact columns:

⋮  ⋮
↓  ↓
H^{1}(ℚ_{Σ}/ℚ, C^{min}[p^{∞}]) ⟶^{β} H^{1}(ℚ_{Σ}/ℚ, A_{𝕋_{∞}})^{Γ}
λ_{0} ↓  _{λ_{∞}} ↓

⨁_{l∈Σ} H^{1}(ℚ_{l}, C^{min})[p^{∞}] ⟶^{⊕δ_{l}}  ⨁_{l∈Σ} (\frac{H^{1}(ℚ_{l}, A_{𝕋_{∞}})}{H_{⋆}^{1}(ℚ_{l}, A_{𝕋_{∞}})})^{Γ}
↓  ↓
Hom_{cont}(C^{min}(ℚ)ˆ⊗ℤ_{p} , ℚ/ℤ) −−→ H^{1}(Γ, Sel_{ℚ}(ρ_{∞}))
↓  ↓
0  0  .

Figure 3.

Remark: Focussing momentarily on the homomorphisms δ_{l} and λ_{0}, one deduces

[Ker( ⊕ δ_{l}) : Ker( ⊕ δ_{l}) ∩ Im(λ_{0})] = \frac{[ ⨁_{l∈Σ} H^{1}(ℚ_{l}, C^{min})[p^{∞}] : Im(λ_{0})]}{[Im( ⊕ δ_{l}) : ⊕δ_{l}(Im(λ_{0}))]}

upon applying the Snake Lemma to the diagram

0 ⟶ Ker( ⊕ δ_{l}) ⟶ H^{1}(ℚ_{l}, C^{min})[p^{∞}] ⟶^{⊕δ_{l}} Im( ⊕ δ_{l}) ⟶ 0

⋃  ⋃  ⋃
0 ⟶ Ker( ⊕ δ_{l}) ∩ Im(λ_{0}) ⟶ Im(λ_{0})  ⟶^{⊕δ_{l}} ⊕δ_{l}(Im(λ_{0})) ⟶ 0 .

The numerator above equals #Hom_{cont}(C^{min}(ℚ)ˆ⊗ℤ_{p}, ℚ/ℤ), which has the

same size as the p-primary subgroup of C^{min}(ℚ). Casting a cold eye over Figure 3, one exploits the surjectivity of ⊕δ_{l} to conclude the denominator term is #Coker(λ_{∞}). Equivalently,

∏_{l∈Σ} [Ker(δ_{l}) : Im(λ_{0}) ∩ Ker(δ_{l})] = \frac{#C^{min}(ℚ)[p^{∞}]}{#Coker(λ_{∞})}

which finishes off the demonstration of 2.3.

Documenta Mathematica · Extra Volume Coates (2006) 301–323

The proof of Proposition 4.2: There are three stages. We first show that the compact Selmer group is Λtorsion. Using a version of Nekovár̃’s control theory along the critical line (s, k) ∈ {1} × ℤ_{p}, we next establish its finiteness. Lastly, we embed 𝔖𝔢𝔩 inside a tower of rational points, whose structure is narrow enough to imply the Selmer group is zero.

Examining the behaviour of our big dual exponential EXP_{𝕋_{∞}}^{∗} from [De, Th 3.3], there is a tautological sequence of Λ-homomorphisms

0 ⟶ 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) ⟶ H^{1}(ℚ_{Σ}/ℚ, 𝕋_{∞}) ⟶^{loc_{p}(−) mod X(ℚ_{p})} \frac{H^{1}(ℚ_{p}, 𝕋_{∞})}{X(ℚ_{p})}
EXP_{𝕋∞}^{∗} ↓
Λ[1/p]

which is exact along the row. A global Euler characteristic calculation shows that

rank_{Λ}(H^{1}(ℚ_{Σ}/ℚ, 𝕋_{∞})) = rank_{Λ}(H^{2}(ℚ_{Σ}/ℚ, 𝕋_{∞})) + 1

≤ rank_{ℤ_{p}}(H^{2}(ℚ_{Σ}/ℚ, Ta_{p}(C^{min}))) + 1

=^{by Kato} 0 + 1

– the final equality lies very deep, and follows from [Ka, Th 14.5(1)]. On the other hand, the weight-deformation of Kato’s zeta-element will generate rank one Λ-submodules inside both of H^{1}(ℚ_{Σ}/ℚ, 𝕋_{∞}) and

H^{1}(ℚ_{p}, 𝕋_{∞})/X(ℚ_{p}). To verify this claim, observe that EXP_{𝕋_{∞}}^{∗} modulo u_{0} −1

sends the zeta-element to a multiple of \frac{L_{Np}(C^{min},1)}{Ω_{C^{min}}^{+}}, which is non-zero. This

means the image of EXP_{𝕋_{∞}}^{∗} ∘ loc_{p} is not contained in the augmentation ideal, and so is abstractly isomorphic to Λ.

Remark: In summary, we have just shown that the global H^{1} has Λ-rank one. Because the quotient H^{1}(ℚ_{p}, 𝕋_{∞})/X(ℚ_{p}) is Λ-torsion free and also has rank one, we may identify 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) with the Λ-torsion submodule of H^{1}(ℚ_{Σ}/ℚ, 𝕋_{∞}).

Question. Does 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) contain any pseudo-summands of the form Λ/F_{j}^{e_{j}}Λ

for some irreducible distinguished polynomial F_{j} and for e_{j} ∈ ℕ?

To provide an answer, we will need to specialise at arithmetic points of Spec(Λ)^{alg}. For any de Rham Gal(ℚ_{Σ}/ℚ)-lattice T, the Selmer group H_{g,Specℤ}^{1} is defined by

H_{g,Specℤ}^{1}(ℚ, T) := Ker (H^{1}(ℚ_{Σ}/ℚ, T) ⟶^{⊕res_{l}} ⨁_{l≠p} H^{1}(I_{l}, T) ⊕ \frac{H^{1}(ℚ_{p}, T)}{H_{g}^{1}(ℚ_{p}, T)}) .

Control Theorem. [Sm, Th 5.1] For all bar finitely many integral weights k ≥ 2, the induced specialisation

𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) ⊗_{Λ,σ_{k}} ℤ_{p} ⟶ H_{g,Specℤ}^{1}(ℚ , 𝕋_{∞} ⊗_{Λ,σ_{k}} ℤ_{p})

has finite kernel and cokernel, bounded independently of the choice of σ_{k} : Λ ↠ ℤ_{p}.

Kato’s Theorem. [Ka, Th 14.2] For all integral weights k ≥ 3, the Bloch-Kato compact Selmer group H_{f,Specℤ}^{1}(ℚ , 𝕋_{∞} ⊗_{Λ,σ_{k}} ℤ_{p}) is finite.

Actually Kato proves this result for discrete Selmer groups, but they are equivalent statements. Note that 𝕋_{∞} ⊗_{Λ,σ_{k}} ℤ_{p} is a lattice inside V_{f_{k}}^{∗}, the contragredient of Deligne’s G_{ℚ}-representation attached to the eigenform f_{k} ∈ S_{k}^{ord}(Γ_{0}(Np^{r}), ω^{2−k}). The non-vanishing of the L-value L(f_{k}, 1) forces these Selmer groups to be finite.

Corollary 4.3. For almost all k ≥ 2, the order of 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) ⊗_{Λ,σ_{k}} ℤ_{p} is bounded.

Proof: We first observe that H_{g,Specℤ}^{1}(ℚ, V_{f_{k}}^{∗}) coincides with H_{f,Specℤ}^{1}(ℚ, V_{f_{k}}^{∗}) unless the local condition H_{g}^{1}(ℚ_{p}, V_{f_{k}}^{∗}) is strictly larger than H_{f}^{1}(ℚ_{p}, V_{f_{k}}^{∗}). However,

dim_{ℚ_{p}}(H_{g/f}^{1}(ℚ_{p}, V_{f_{k}}^{∗})) = dim_{ℚ_{p}}(D_{cris}(V_{f_{k}}(1))/(φ − 1)) by [BK, Cor 3.8.4]

and an argument involving slopes of the Frobenius φ shows this dimension is zero. By Kato’s theorem H_{f,Specℤ}^{1} is finite, so it lies in H^{1}(ℚ_{Σ}/ℚ, 𝕋_{∞} ⊗_{Λ,σ_{k}} ℤ_{p})[p^{∞}];

the latter torsion is identified with H^{0}(ℚ_{Σ}/ℚ, (𝕋_{∞} ⊗_{Λ,σ_{k}} ℤ_{p}) ⊗ ℚ/ℤ) via a standard technique in continuous cohomology. It follows from the Control Theorem, that

𝔖𝔢𝔩_{ℚ}(𝕋_{∞})⊗_{Λ,σ_{k}} ℤ_{p} →^{nat} H_{f,Specℤ}^{1}(ℚ, 𝕋_{∞}⊗_{Λ,σ_{k}} ℤ_{p}) ↪ ((𝕋_{∞}⊗_{Λ,σ_{k}} ℤ_{p})⊗ℚ/ℤ)^{G_{ℚ}}

has kernel killed by a universal power p^{ν_{1}} say, independent of the weight k. Let us choose a prime l ∤ Np. By definition 1 − a_{l}(f_{k}).Frob_{l} + l < l >^{k−2} .Frob_{l}^{2}

is zero on V_{f_{k}}^{∗}, and 1−a_{l}(f_{k})+l < l >^{k−2} must kill off ((𝕋_{∞}⊗_{Λ,σ_{k}} ℤ_{p})⊗ℚ/ℤ)^{G_{ℚ}} because Frobenius acts trivially on the G_{ℚ}-invariants. We claim that there are infinitely many choices of l for which 1 − a_{l}(f_{k}) + l < l >^{k−2}≠ 0. If not,

1 − a_{l}(f_{k})l^{−s} + l < l >^{k−2} l^{−2s} = (1 − l^{−s})(1 − ω^{2−k}(l)l^{k−1−s}) for all l ∉ S

where S is some finite set containing Σ. Proceeding further down this cul-de-sac, we obtain an equality of incomplete L-functions L_{S}(f_{k}, s) = ζ_{S}(s)L_{S}(ω^{2−k}, s + 1 − k) which is patently ridiculous, as f_{k} is not an Eisenstein series! If k ≡ k′ mod (p − 1)p^{c}, then

1 − a_{l}(f_{k}) + l < l >^{k−2} ≡ 1 − a_{l}(f_{k′}) + l < l >^{k′−2} modulo p^{c+1} .

For each class τ modulo p−1, we can cover weight-space by a finite collection of open disks D_{1}^{τ}, …, D_{n(τ)}^{τ} upon which ord_{p}(1 − a_{l}(f_{k}) + l < l >^{k−2} ) is constant for every k ∈ D_{j}^{τ}, k ≡ τ( mod p − 1). Setting ν_{2} equal to the non-negative integer

max_{τ mod p−1} { max_{1≤j≤n(τ)} {ord_{p}(1 − a_{l}(f_{k}) + l < l >^{k−2} ) with k ∈ D_{j}^{τ}, k ≡ τ}} ,

clearly p^{ν_{2}} annihilates all the ((𝕋_{∞} ⊗_{Λ,σ_{k}} ℤ_{p}) ⊗ ℚ/ℤ)^{G_{ℚ}}’s. We deduce that p^{ν_{1}+ν_{2}} kills off 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) ⊗_{Λ,σ_{k}} ℤ_{p} for almost all k ≥ 3, and the corollary is proved.

Remark: The answer to the question posed above is therefore negative, i.e. there can exist no pseudo-summands of the shape Λ/F_{j}^{e_{j}}Λ lying inside of 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) (otherwise the specialisations 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) ⊗_{Λ,σ_{k}} ℤ_{p} would have unbounded order for varying weights k ≥ 2, which violates Corollary 4.3). The compact Selmer group is of finite-type over the local ring Λ, and it follows from the structure theory that 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) must be a finite abelian p-group, of order dividing p^{ν_{1}+ν_{2}}.

Let us recall the definition of the degeneration maps between modular curves. For integers d ≥ 1 and m, n ≥ 5 with dm|n, the finite map π_{d} : X_{1}(n) → X_{1}(m) operates on the affine curves Y_{1}(−) by the rule

π_{d} (A, μ_{n} ↪^{θ} A[n]) = (A′, μ_{m} ↪^{θ′} A′[m])

where A′ = A/θ(μ_{d}), and the injection θ′ : μ_{m} ↪ μ_{n/d} ←^{∼^{d}} μ_{n}/μ_{d} ↪^{θ mod μ_{d}}

A/θ(μ_{d}).

Hida [H1] identified the Γ^{p^{r−1}}-coinvariants of 𝕋_{∞}, with the Tate module of a p-divisible subgroup of jac X_{1}(n) at level n = Np^{r}. The natural composition

H^{1}(ℚ_{Σ}/ℚ, 𝕋_{∞}) ≅ \underleftarrow{lim}_{r≥1} H^{1}(ℚ_{Σ}/ℚ, (𝕋_{∞})_{Γ^{p^{r−1}}} ) ↪ \underleftarrow{lim}_{π_{p ∗}} H^{1}(ℚ_{Σ}/ℚ, Ta_{p}(J_{r})^{ord})

injects 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) into the projective limit \underleftarrow{lim}_{π_{p ∗}}(H^{1}(ℚ_{Σ}/ℚ, Ta_{p}(J_{r})^{ord})[p^{∞}]).

Again it’s continuous cohomology, so the ℤ_{p}-torsion in H^{1}(ℚ_{Σ}/ℚ, Ta_{p}(J_{r})^{ord})

is then isomorphic to H^{0}(ℚ_{Σ}/ℚ, Ta_{p}(J_{r})^{ord} ⊗ ℚ/ℤ) = J_{r}^{ord}(ℚ)[p^{∞}] as finite groups.

Lemma. (Nekovár̃) [NP, 1.6.6] (i) π_{1 ∗}(e_{ord}.Ta_{p}(J_{r+1})) ⊂ p (e_{ord}.Ta_{p}(J_{r}));

Let \frac{1}{p}π_{1 ∗} : Ta_{p}(J_{r+1})^{ord} → Ta_{p}(J_{r})^{ord} denote the map satisfying p (\frac{1}{p}π_{1 ∗}) = π_{1 ∗}.

(ii) (\frac{1}{p}π_{1 ∗}) ∘ π_{1}^{∗} = multiplication by p on e_{ord}.Ta_{p}(J_{r});

(iii) π_{1}^{∗} ∘ (\frac{1}{p}π_{1 ∗}) = ∑_{γ∈Γ_{r}/Γ_{r+1}}⟨γ⟩ on e_{ord}.Ta_{p}(J_{r+1}) where Γ_{r} = Γ^{p^{r−1}};

(iv) π_{p ∗} = U_{p} ∘ (\frac{1}{p}π_{1 ∗}) on e_{ord}.Ta_{p}(J_{r+1}).

We shall use these facts directly, to show the triviality of the compact Selmer group. Because it is finite of order dividing p^{ν_{1}+ν_{2}}, for large enough r ≫ 1 we can realise 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) as a subgroup 𝕊_{r} of jac X_{1}(Np^{r})^{ord}(ℚ)[p^{ν_{1}+ν_{2}}]. The sequence of 𝕊_{r}’s is compatible with respect to the degeneration maps π_{p ∗} and π_{1}^{∗} : jac X_{1}(Np^{r})(\overline{ℚ})[p^{∞}] ⟶ jac X_{1}(Np^{r+1})(\overline{ℚ})[p^{∞}], so for any e ≥ 0

𝕊_{r} = (π_{p ∗})^{e}(𝕊_{r+e}) ≅ (π_{p ∗})^{e} ∘ (π_{1}^{∗})^{e}(𝕊_{r}) .

By part (iv) of this lemma (π_{p ∗})^{e} coincides with (U_{p} ∘ (\frac{1}{p}π_{1 ∗}) )^{e}, and the covariant action of the U_{p}-operator is invertible on the ordinary locus. Consequently

𝕊_{r} ≅ a_{p}(f)^{e} × (\frac{1}{p}π_{1 ∗})^{e} ∘ (π_{1}^{∗})^{e}(𝕊_{r}) =^{by (ii)} a_{p}(f)^{e} × p^{e} (𝕊_{r})

and picking e ≥ ν_{1} +ν_{2}, we see that 𝔖𝔢𝔩_{ℚ}(𝕋_{∞}) ≅ 𝕊_{r} ⊂ J_{r}[p^{ν_{1}+ν_{2}}] must be zero. The proof of Proposition 4.2 is thankfully over.

References

[BK] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, in the Grothendieck Festchrift I, Progress in Math. 86, Birkhäuser (1990), 333-400.

[CH] J. Coates and S. Howson, Euler characteristics and elliptic curves II, Journal Math. Soc. Japan 53 (2001), 175-235.

[De] D. Delbourgo, Super Euler systems and ordinary deformations of modular symbols, preprint (2004).

[DS] D. Delbourgo and P. Smith, Kummer theory for big Galois representations, to appear in Math. Proc. of the Camb. Phil. Soc.

[GS] R. Greenberg and G. Stevens, p-adic L-functions and p-adic periods of modular forms, Invent. Math. 111 (1993), 401-447.

[H1] H. Hida, Galois representations into GL_{2}(ℤ_{p}[[X]]) attached to ordinary cusp forms, Invent. Math. 85 (1986), 545-613.

[H2] H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4) 19 (1986), 231-273.

[Ka] K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, preprint (2002).

[MW] B. Mazur and A. Wiles, On p-adic analytic families of Galois representations, Compositio Math. 59 (1986), 231-264.

[NP] J. Nekovár̃ and A. Plater, On the parity ranks of Selmer groups, Asian Journal Math. (2) 4 (2000), 437-498.

[Sm] P. Smith, PhD Thesis, University of Nottingham (2006).

[St] G. Stevens, Stickelberger elements and modular parametrizations of elliptic curves, Invent. Math. 98 (1989), 75-106.

Daniel Delbourgo

Department of Mathematics University Park Nottingham England NG7 2RD dd@maths.nott.ac.uk 324

Documenta Mathematica · Extra Volume Coates (2006)