EULER SYSTEMS AND MODULAR ELLIPTIC CURVES

KARL RUBIN

Introduction

This paper consists of two parts. In the first we present a general theory of Euler systems. The main results (see §§3 and 4) show that an Euler system for a p-adic representation T gives a bound on the Selmer group associated to the dual module Hom(T, μ_{p^{∞}}). These theorems, which generalize work of Kolyvagin [Ko], have been obtained independently by Kato [Ka1], Perrin-Riou [PR2], and the author [Ru3]. We will not prove these theorems here, or even attempt to state them in the greatest possible generality.

In the second part of the paper we show how to apply the results of Part I and an Euler system recently constructed by Kato [Ka2] (see the article of Scholl [Scho] in this volume) to obtain Kato’s theorem in the direction of the Birch and Swinnerton-Dyer conjecture for modular elliptic curves (Theorem 8.1).

Part 1. Generalities

1. Selmer groups attached to p-adic representations

A p-adic representation of G_{Q} = Gal(Q̄/Q) is a free Z_{p}-module T of finite rank with a continuous, Z_{p}-linear action of G_{Q}. We will assume in addition throughout this paper that T is unramified outside of a finite set of primes.

Given a p-adic representation T, we also define

V = T ⊗_{Z_{p}} Q_{p},
W = V/T = T ⊗ Q_{p}/Z_{p}.

(Note that T determines V and W, and W determines T and V, but in general there may be non-isomorphic Z_{p}-lattices T giving rise to the same vector space V.)

The following are basic examples of p-adic representations to keep in mind.

Example . If ρ : G_{Q} → Z_{p}^{×} is a continuous character we can take T to be a free, rankone Z_{p}-module with G_{Q} acting via ρ (clearly every one-dimensional representation arises in this way). For example, when ρ is the cyclotomic character we get

T = Z_{p}(1) = lim_{\overleftarrow{n}} μ_{p^{n}}.

Example . If A is an abelian variety we can take the p-adic Tate module of A

T = T_{p}(A) = lim_{\overleftarrow{n}} A_{p^{n}}.

This is the situation we will concentrate on in this paper, when A is an elliptic curve.

partially supported by NSF grant DMS-9306287.

Example . If T is a p-adic representation, then we define the dual representation

T^{∗} = Hom(T, Z_{p}(1))

and we denote the corresponding vector space and divisible group by

V^{∗} = T^{∗} ⊗ Q_{p} = Hom(V, Q_{p}(1)), W^{∗} = V^{∗}/T^{∗} = Hom(T, μ_{p^{∞}}).

If E is an elliptic curve then the Weil pairing gives an isomorphism T_{p}(E)^{∗} ≅ T_{p}(E).

Let Q_{∞} ⊂ Q(μ_{p^{∞}}) denote the cyclotomic Z_{p}-extension of Q. For every n let Q_{n} ⊂ Q(μ_{p^{n+1}}) be the extension of Q of degree p^{n} in Q_{∞}, and let Q_{n,p} denote the completion of Q_{n} at the unique prime above p.

Fix a p-adic representation T as above. We wish to define a Selmer group S(Q_{n}, W) ⊂ H^{1}(Q_{n}, W) for every n. If v is a place of Q_{n} not dividing p, let I_{v} denote an inertia group of v in G_{Q_{n}}, let Q_{n,v}^{ur} denote the maximal unramified extension of Q_{n,v}, and define

H_{S}^{1}(Q_{n,v}, V) = H_{ur}^{1}(Q_{n,v}, V) = ker(H^{1}(Q_{n,v}, V) → H^{1}(Q_{n,v}^{ur}, V))
= H^{1}(Q_{n,v}^{ur}/Q_{n,v}, V^{I_{v}}).

For the unique prime of Q_{n} above p, we will ignore all questions about what is the “correct” definition, and we just fix some choice of subspace H_{S}^{1}(Q_{n,p}, V) ⊂ H^{1}(Q_{n,p}, V). (For example, one could choose H_{S}^{1}(Q_{n,p}, V) = H^{1}(Q_{n,p}, V) or H_{S}^{1}(Q_{n,p}, V) = 0.)

For every place v of Q_{n} we now define

H_{S}^{1}(Q_{n,v}, W) ⊂ H^{1}(Q_{n,v}, W) and H_{S}^{1}(Q_{n,v}, T) ⊂ H^{1}(Q_{n,v}, T)

to be the image and inverse image, respectively, of H_{S}^{1}(Q_{n,v}, V) under the maps on cohomology induced by the exact sequence

0 → T → V → W → 0.

Finally, we define

S(Q_{n}, W) = ker (H^{1}(Q_{n}, W) → ⨁_{v of Q_{n}} H^{1}(Q_{n,v}, W)/H_{S}^{1}(Q_{n,v}, W)) .

Of course, this definition depends on the choice we made for H_{S}^{1}(Q_{n,p}, V).

2. Euler systems

Fix a p-adic representation T of G_{Q} as in §1, and fix a positive integer N divisible by p and by all primes where T is ramified. Define

R = R(N) = {squarefree integers r : (r, N) = 1}

For every prime q which is unramified in T, let Fr_{q} denote a Frobenius of q in G_{Q} and define the characteristic polynomial

P_{q}(x) = det(1 − Fr_{q}x|T) ∈ Z_{p}[x].

Since q is unramified in T, P_{q} is independent of the choice of Frobenius element Fr_{q}.

Definition . An Euler system c for T is a collection of cohomology classes

c_{Q_{n}(μ_{r})} ∈ H^{1}(Q_{n}(μ_{r}), T)

for every r ∈ R and every n ≥ 0, such that if m ≥ n, q is prime, and rq ∈ R, then

Cor_{Q_{n}(μ_{rq})/Q_{n}(μ_{r})}c_{Q_{n}(μ_{rq})} = P_{q}(q^{−1}Fr_{q}^{−1})c_{Q_{n}(μ_{r})},
Cor_{Q_{m}(μ_{r})/Q_{n}(μ_{r})}c_{Q_{m}(μ_{r})} = c_{Q_{n}(μ_{r})}.

Note that this definition depends on N (since R does), but not in an important way so we will suppress it from the notation.

Remarks . Kolyvagin’s original method (see [Ko] or [Ru1]) required the Euler system to satisfy an additional “congruence” condition. The fact that our Euler system “extends in the Q_{∞} direction” (i.e., consists of classes defined over the fields Q_{n}(μ_{r}) for every n, and not just over Q(μ_{r})) eliminates the need for the congruence condition.

There is some freedom in the exact form of the distribution relation in the definition of an Euler system. It is easy to modify an Euler system satisfying one distribution relation to obtain a new Euler system satisfying a slightly different one.

3. Results over Q

We now come to the fundamental applications of an Euler system. For the proofs of Theorems 3.1, 3.2, and 4.1 see [Ka1], [PR2], or [Ru3]. In fact, once the setting is properly generalized the proofs are similar to the original method of Kolyvagin [Ko]; see also [Ru1] and [Ru2].

For this section and the next fix a p-adic representation T as in §1. Fix also a choice of subspaces H_{S}^{1}(Q_{n,p}, V) and H_{S}^{1}(Q_{n,p}, V^{∗}) for every n, so that we have Selmer groups as defined in §1. We assume only that these choices satisfy the following conditions:

• H_{S}^{1}(Q_{n,p}, V) and H_{S}^{1}(Q_{n,p}, V^{∗}) are orthogonal complements under the cup product pairing

H^{1}(Q_{n,p}, V) × H^{1}(Q_{n,p}, V^{∗}) → H^{2}(Q_{n,p}, Q_{p}(1)) = Q_{p},

• if m ≥ n then

Cor_{Q_{m,p}/Q_{n,p}}H_{S}^{1}(Q_{m,p}, V) ⊂ H_{S}^{1}(Q_{n,p}, V),
Res_{Q_{m,p}}H_{S}^{1}(Q_{n,p}, V) ⊂ H_{S}^{1}(Q_{m,p}, V)

and similarly for V^{∗}. We will write H_{/S}^{1}(Q_{n,p}, T) = H^{1}(Q_{n,p}, T)/H_{S}^{1}(Q_{n,p}, T) (and similarly with T replaced by V or W), and we write loc_{p}^{ram} for the localization map

loc_{p}^{ram} : H^{1}(Q_{n}, T) → H_{/S}^{1}(Q_{n,p}, T).

We will make use of two different sets of hypotheses on the Galois representation T. Hypotheses Hyp(Q_{∞}, T) are stronger than Hyp(Q_{∞}, V), and will allow us to prove a stronger conclusion.

Hypotheses Hyp(Q_{∞}, T). (i) There is a τ ∈ G_{Q_{∞}} such that • τ acts trivially on μ_{p^{∞}},

• T/(τ − 1)T is free of rank one over Z_{p}. (ii) T/pT is an irreducible F_{p}[G_{Q_{∞}}]-module.

Hypotheses Hyp(Q_{∞}, V). (i) There is a τ ∈ G_{Q_{∞}} such that • τ acts trivially on μ_{p^{∞}},

• dim_{Q_{p}}(V/(τ − 1)V) = 1. (ii) V is an irreducible Q_{p}[G_{Q_{∞}}]-module.

Theorem 3.1. Suppose c is an Euler system for T, and V satisfies Hyp(Q_{∞}, V). If c_{Q} ∉ H^{1}(Q, T)_{tors} and [H_{/S}^{1}(Q_{p}, T) : loc_{p}^{ram}(H^{1}(Q, T))] is finite, then S(Q, W^{∗}) is finite. In particular if loc_{p}^{ram}(c_{Q}) ≠ 0 and rank_{Z_{p}}H_{/S}^{1}(Q_{p}, T) = 1, then S(Q, W^{∗}) is finite.

Define Ω = Q(W)Q(μ_{p^{∞}}), where Q(W) denotes the minimal extension of Q such that G_{Q(W)} acts trivially on W.

Theorem 3.2. Suppose c is an Euler system for T, and T satisfies Hyp(Q_{∞}, T). If p > 2 and loc_{p}^{ram}(c_{Q}) ≠ 0 then

|S(Q, W^{∗})| ≤ |H^{1}(Ω/Q, W)||H^{1}(Ω/Q, W^{∗})|[H_{/S}^{1}(Q_{p}, T) : Z_{p}loc_{p}^{ram}(c_{Q})].

Remark . Hypotheses Hyp(Q_{∞}, T) are satisfied if the image of the Galois representation on T is “sufficiently large.” They often hold in practice; see for example the discussion in connection with elliptic curves below. If rank_{Z_{p}}T = 1 then Hyp(Q_{∞}, T) holds with τ = 1.

4. Results over Q_{∞}

Essentially by proving analogues of Theorem 3.2 for each field Q_{n}, we can pass to the limit and prove an Iwasawa-theoretic version of Theorem 3.2.

Let Λ denote the Iwasawa algebra

Λ = Z_{p}[[Gal(Q_{∞}/Q)]] = lim_{\overleftarrow{n}} Z_{p}[[Gal(Q_{n}/Q)]],

so Λ is (noncanonically) isomorphic to a power series ring in one variable over Z_{p}. If B is a finitely generated torsion Λ-module then there is a pseudo-isomorphism (a Λ-module homomorphism with finite kernel and cokernel)

B → ⨁_{i} Λ/f_{i}Λ

with nonzero f_{i} ∈ Λ, and we define the characteristic ideal of B

char(B) = ∏_{i} f_{i}Λ.

The characteristic ideal is well-defined, although the individual f_{i} are not. If B is not a finitely-generated torsion Λ-module we define char(B) = 0.

Define Λ-modules

X_{∞} = Hom(lim_{\overrightarrow{n}} S(Q_{n}, W^{∗}), Q_{p}/Z_{p}).

and

H_{∞,/S}^{1}(Q_{p}, T) = lim_{\overleftarrow{n}} H_{/S}^{1}(Q_{n,p}, T).

(Our requirements on the choices of H_{S}^{1}(Q_{n,p}, V) and H_{S}^{1}(Q_{n,p}, V^{∗}) ensure that if m ≥ n, the restriction and corestriction maps induce maps

S(Q_{n}, W^{∗}) → S(Q_{m}, W^{∗}) and H_{/S}^{1}(Q_{m,p}, T) → H_{/S}^{1}(Q_{n,p}, T)

so these limits are well-defined.) If c is an Euler system let [loc_{p}^{ram}(c_{Q_{n}})] denote the corresponding element of H_{∞,/S}^{1}(Q_{p}, T).

Theorem 4.1. Suppose that V satisfies Hyp(Q_{∞}, V), c is an Euler system for T, [loc_{p}^{ram}(c_{Q_{n}})] ∉ H_{∞,/S}^{1}(Q_{p}, T)_{tors}, and H_{∞,/S}^{1}(Q_{p}, T)/Λ[loc_{p}^{ram}(c_{Q_{n}})] is a torsion Λ-module. Define

L = char(H_{∞,/S}^{1}(Q_{p}, T)/Λ[loc_{p}^{ram}(c_{Q_{n}})]).

Then (i) X_{∞} is a torsion Λ-module, (ii) there is a nonnegative integer t such that char(X_{∞}) divides p^{t}L, (iii) if T satisfies Hyp(Q_{∞}, T) then char(X_{∞}) divides L.

Part 2. Elliptic curves

The ‘Heegner point Euler system’ for modular elliptic curves used by Kolyvagin in [Ko] does not fit into the framework of §2, because the cohomology classes are not defined over abelian extensions of Q. However, Kato [Ka2] has constructed an Euler system for the Tate module of a modular elliptic curve, using Beilinson elements in the K-theory of modular curves. We now describe how, given Kato’s Euler system and its essential properties, one can use the general results above to study the arithmetic of elliptic curves. The main result is Theorem 8.1 below.

5. Local cohomology groups

Suppose E is an elliptic curve defined over Q, and take T = T_{p}(E), the p-adic Tate module of E. Then V = V_{p}(E) = T_{p}(E) ⊗ Q_{p} and W = E_{p^{∞}}. The Weil pairing gives isomorphisms V ≅ V^{∗}, T ≅ T^{∗}, and W ≅ W^{∗}.

If B is an abelian group, we will abbreviate B ⊗ Z_{p} = lim_{⟵} B/p^{n}B, the p-adic completion of B, and

B ⊗ Q_{p} = (lim_{\overleftarrow{n}} B/p^{n}B) ⊗_{Z_{p}} Q_{p}, B ⊗ Q_{p}/Z_{p} = (lim_{\overleftarrow{n}} B/p^{n}B) ⊗_{Z_{p}} Q_{p}/Z_{p}.

For every n define

H_{S}^{1}(Q_{n,p}, V) = image(E(Q_{n,p}) ⊗ Q_{p} ↪ H^{1}(Q_{n,p}, V)),

image under the natural Kummer map. Since V = V^{∗}, this also fixes a choice of H_{S}^{1}(Q_{p}, V^{∗}), and this subgroup is its own orthogonal complement as required.

Let � (E_{/Q_{n}}) denote the Tate-Shafarevich group of E over Q_{n}.

Proposition 5.1. With H_{S}^{1}(Q_{n,p}, V) as defined above, S(Q_{n}, E_{p^{∞}}) is the classical p-power Selmer group of E over Q_{n}, so there is an exact sequence

0 → E(Q_{n}) ⊗ Q_{p}/Z_{p} → S(Q_{n}, E_{p^{∞}}) → � (E_{/Q_{n}})_{p^{∞}} → 0

Proof. If v ∤ p then the p-part of E(Q_{n,v}) is finite, and one can check easily that H_{S}^{1}(Q_{n,v}, V_{p}(E)) = 0. Therefore for every v, H_{S}^{1}(Q_{n,v}, V_{p}(E)) is the image of E(Q_{n,v}) ⊗ Q_{p} under the Kummer map. It follows that for every v, H_{S}^{1}(Q_{n,v}, E_{p^{∞}}) is the image of E(Q_{n,v}) ⊗ Q_{p}/Z_{p} under the corresponding Kummer map, and so the definition of S(Q_{n}, E_{p^{∞}}) coincides with the classical definition of the Selmer group of E.

For every n let tan(E_{/Q_{n,p}}) denote the tangent space of E_{/Q_{n,p}} at the origin and consider the Lie group exponential map

exp_{E} : tan(E_{/Q_{n,p}}) ⟶^{∼} E(Q_{n,p}) ⊗ Q_{p}.

Fix a minimal Weierstrass model of E and let ω_{E} denote the corresponding holomorphic differential. Then the cotangent space cotan(E) is Q_{n,p}ω_{E}, and we let ω_{E}^{∗} be the corresponding dual basis of tan(E). We have a commutative diagram

tan(E_{/Q_{n,p}})  ⟶^{exp_{E}}  E(Q_{n,p}) ⊗ Q_{p}
·ω_{E}^{∗} ↑  ↑
λ_{E}(𝔭_{n}) ⟶^{∼} Ê(𝔭_{n}) ⟶^{∼} E_{1}(Q_{n,p})

where Ê is the formal group of E, λ_{E} is its logarithm map, 𝔭_{n} is the maximal ideal of Q_{n,p}, E_{1}(Q_{n,p}) is the kernel of reduction in E(Q_{p}), and the bottom maps are the formal group exponential followed by the isomorphism of [T] Theorem 4.2. (Note that Ê(𝔭_{n})_{tors} = 0 because Q_{p}(Ê_{p})/Q_{p} is totally ramified of degree p − 1, so λ_{E} is injective.) Extending λ_{E} linearly we will view it as a homomorphism defined on all of E(Q_{n,p}) ⊗ Q_{p}.

Since V ≅ V^{∗},

Hom(E(Q_{n,p}), Q_{p}) ≅ Hom(H_{S}^{1}(Q_{n,p}, V), Q_{p}) ≅ H_{/S}^{1}(Q_{n,p}, V).

Thus there is a dual exponential map

exp_{E}^{∗} : H_{/S}^{1}(Q_{n,p}, V) ⟶^{∼} cotan(E_{/Q_{n,p}}) = Q_{n,p}ω_{E}.

We write exp_{ω_{E}}^{∗} : H_{/S}^{1}(Q_{n,p}, V) ⟶^{∼} Q_{n,p} for the composition ω_{E}^{∗} ∘ exp_{E}^{∗} Since H_{/S}^{1}(Q_{n,p}, T) injects into H_{/S}^{1}(Q_{n,p}, V), exp_{ω_{E}}^{∗} is injective on H_{/S}^{1}(Q_{n,p}, T). The local pairing allows us to identify

H_{/S}^{1}(Q_{n,p}, V) ⟶^{∼} Hom(E(Q_{n,p}), Q_{p})
↑  ↑
H_{/S}^{1}(Q_{n,p}, T) ⟶^{∼} Hom(E(Q_{n,p}), Z_{p}).

(1)

Explicitly, z ∈ H_{/S}^{1}(Q_{n,p}, V) corresponds to the map

x ↦ Tr_{Q_{n,p}/Q_{p}}λ_{E}(x) exp_{ω_{E}}^{∗}(z). (2)

Proposition 5.2. exp_{ω_{E}}^{∗}(H_{/S}^{1}(Q_{p}, T)) = [E(Q_{p}) : E_{1}(Q_{p}) + E(Q_{p})_{tors}]p^{−1}Z_{p}.

Proof. By (2),

exp_{ω_{E}}^{∗}(H_{/S}^{1}(Q_{p}, T)) = p^{a}Z_{p}

where

λ_{E}(E(Q_{p})) = p^{−a}Z_{p}.

We have λ_{E}(E_{1}(Q_{p})) = pZ_{p} and, since rank_{Z_{p}}E(Q_{p}) = 1,

[λ_{E}(E(Q_{p})) : λ_{E}(E_{1}(Q_{p})] = [E(Q_{p}) : E_{1}(Q_{p}) + E(Q_{p})_{tors}].

Thus the proposition follows.

6. The p-adic L-function

Let

L(E, s) = ∑_{n≥1} a_{n}n^{−s} = ∏_{q} ℓ_{q}(q^{−s})^{−1}

denote the Hasse-Weil L-function of E, where ℓ_{q}(q^{−s}) is the usual Euler factor at q. If N ∈ Z^{+} we will also write

L_{N}(E, s) = ∑_{(n,N)=1} a_{n}n^{−s} = ∏_{q∤N} ℓ_{q}(q^{−s})^{−1}

for the L-function with the Euler factors dividing N removed. If F is an abelian extension of Q of conductor f and γ ∈ Gal(F/Q), define the partial L-function

L_{N}(E, γ, F/Q, s) = ∑_{n↦γ} a_{n}n^{−s}

where the sum is over n prime to fN which map to γ under

(Z/fZ)^{×} ⟶^{∼} Gal(Q(μ_{f})/Q) ↠ Gal(F/Q).

If χ is a character of G_{Q} of conductor f_{χ}, and ker(χ) = Gal(Q̄/F_{χ}), let

L_{N}(E, χ, s) = ∑_{(n,f_{χ}N)=1} χ(n)a_{n}n^{−s} = ∏_{q∤f_{χ}N} ℓ_{q}(q^{−s}χ(q))^{−1}

= ∑_{γ∈Gal(F_{χ}/Q)} χ(γ)L_{N}(E, γ, F_{χ}/Q, s).

When N = 1 we write simply L(E, χ, s), and then we have

L_{N}(E, χ, s) = ∏_{q|N} ℓ_{q}(q^{−s}χ(q))L(E, χ, s). (3)

If E is modular then these functions all have analytic continuations to C.

Fix a generator [ζ_{p^{n}}]_{n} of lim_{⟵} μ_{p^{n}}. Write G_{n} = Gal(Q_{n}/Q) = Gal(Q_{n,p}/Q_{p}). If χ is a character of Gal(Q_{∞}/Q) of conductor p^{n} define the Gauss sum

τ(χ) = ∑_{γ∈Gal(Q(μ_{p^{n}})/Q)} χ(γ)ζ_{p^{n}}^{γ}.

Fix also an embedding of Q̄_{p} into C so that we can identify complex and p-adic characters of G_{Q}.

The following theorem is proved in [MSD] in the case of good ordinary reduction. See [MTT] for the (even more) general statement.

Theorem 6.1. Suppose E is modular and E has good ordinary reduction or multiplicative reduction at p. Let α ∈ Z_{p}^{×} and β ∈ pZ_{p} be the eigenvalues of Frobenius if E has good ordinary reduction at p, and let (α, β) = (1, p) (resp. (−1, −p)) if E has split (resp. nonsplit) multiplicative reduction. Then there is a nonzero integer c_{E} independent of p, and a p-adic L-function L_{E} ∈ c_{E}^{−1}Λ such that for every character χ of Gal(Q_{∞}/Q) of finite order,

χ(L_{E}) = \begin{cases}(1 − α^{−1})^{2}L(E, 1)/Ω_{E} & if χ = 1 and E has good reduction at p \\ (1 − α^{−1})L(E, 1)/Ω_{E} & if χ = 1 and E is multiplicative at p \\ α^{−n}τ(χ)L(E, χ^{−1}, 1)/Ω_{E} & if χ has conductor p^{n} > 1 \end{cases}

where Ω_{E} is the fundamental real period of E.

If N ∈ Z^{+}, define

L_{E,N} = ∏_{q|N,q≠p} ℓ_{q}(q^{−1}Fr_{q}^{−1})L_{E} ∈ Λ.

Using (3) and Theorem 6.1 one obtains analogous expressions for χ(L_{E,N}) in terms of L_{N}(E, χ^{−1}, 1)

7. Kato’s Euler system

The following theorem of Kato is crucial for everything that follows.

Theorem 7.1 (Kato [Ka1], see also [Scho]). Suppose that E is modular, and let N be the conductor of E. There are a positive integer r_{E} independent of p, positive integers D ≢ 1, D′ ≢ 1 (mod p), and an Euler system c̄ = c̄(D, D′) for T_{p}(E),

{c̄_{Q_{n}(μ_{r})} ∈ H^{1}(Q_{n}(μ_{r}), T_{p}(E)) : r squarefree, (r, NpDD′) = 1, n ≥ 0}

such that for every such n ≥ 0 and every character χ : Gal(Q_{n}/Q) → C^{×},

∑_{γ∈Gal(Q_{n}/Q)} χ(γ) exp_{ω_{E}}^{∗}(loc_{p}^{ram}(c̄_{Q_{n}}^{γ}))

= r_{E}DD′(D − χ^{−1}(D))(D′ − χ^{−1}(D′))L_{Np}(E, χ, 1)/Ω_{E}.

Proof. See the paper of Scholl [Scho] in this volume (especially Theorem 5.2.7) for the proof of this theorem when p > 2 and E has good reduction at p.

Using the Coleman map Col_{∞} : H_{∞,/S}^{1}(Q_{n,p}, T) → Λ described in the Appendix, we can relate Kato’s Euler system to the p-adic L-function.

Corollary 7.2. With hypotheses and notation as in Theorems 7.1 and 6.1, there is an Euler system c for T_{p}(E) such that

(i) exp_{ω_{E}}^{∗}(loc_{p}^{ram}(c_{Q})) = r_{E}L_{Np}(E, 1)/Ω_{E}, (ii) Col_{∞}([loc_{p}^{ram}(c_{Q_{n}})]) = r_{E}L_{E,N}.

Proof. Let c̄ be the Euler system of Theorem 7.1 for some D, D′ ≢ 1 (mod p). Let σ_{D} ∈ Gal(Q(μ_{p^{∞}})/Q) denote the automorphism ζ ↦ ζ^{D} for ζ ∈ μ_{p^{∞}}, and similarly for σ_{D′}.

Since D, D′ ≢ 1 (mod p), (D − σ_{D})(D′ − σ_{D′}) is invertible in Λ. Let ρ_{D,D′} ∈ Z_{p}[[G_{Q}]] be any element which restricts to (D−σ_{D})^{−1}(D′ −σ_{D′})^{−1} in Λ, and define

c_{Q_{n}(μ_{r})} = D^{−1}D′^{−1}ρ_{D,D′} c̄_{Q_{n}(μ_{r})}.

It is clear that {c_{Q_{n}(μ_{r})}} is still an Euler system, and for every n ≥ 0 and every character χ : Gal(Q_{n}/Q) → C^{×}

χ(ρ_{D,D′}) = χ((D − σ_{D})(D′ − σ_{D′}))^{−1} = (D − χ(D))^{−1}(D′ − χ(D′))^{−1},

so by Theorem 7.1

∑_{γ∈Gal(Q_{n}/Q)} χ(γ) exp_{ω_{E}}^{∗}(loc_{p}^{ram}(c_{Q_{n}}^{γ})) = r_{E}L_{Np}(E, χ, 1)/Ω_{E}.

When χ is the trivial character this is (i), and as χ varies (ii) follows from Proposition A.2 of the Appendix along with the definition (Theorem 6.1) of L_{E} and

8. Consequences of Kato’s Euler system

Following Kato, we will apply the results of §§3 and 4 to bound the Selmer group of E.

8.1. The main theorem.

Theorem 8.1 (Kato [Ka2]). Suppose E is modular and E does not have complex multiplication.

(i) If L(E, 1) ≠ 0 then E(Q) and � (E) are finite. (ii) If L is a finite abelian extension of Q, χ is a character of Gal(L/Q), and

L(E, χ, 1) ≠ 0 then E(L)^{χ} and � (E_{/L})^{χ} are finite.

Remarks . We will prove below a more precise version of Theorem 8.1(i).

Kato’s construction produces an Euler system for T_{p}(E) ⊗ χ for every character χ of G_{Q} of finite order, with properties analogous to those of Theorem 7.1. This more general construction is needed to prove Theorem 8.1(ii). For simplicity we will not treat this more general setting here, so we will only prove Theorem 8.1(i) below. But the method for (ii) is the same.

Theorem 8.1(i) was first proved by Kolyvagin in [Ko], using a system of Heegner points, along with work of Gross and Zagier [GZ], Bump, Friedberg, and Hoffstein [BFH], and Murty and Murty [MM]. The Euler system proof given here, due to Kato, is ‘self-contained’ in the sense that it replaces all of those other analytic results with the calculation of Theorem 7.1.

Corollary 8.2. Suppose E is modular and E does not have complex multiplication. Then E(Q_{∞}) is finitely generated.

Proof. A theorem of Rohrlich [Ro] shows that L(E, χ, 1) ≠ 0 for almost all characters χ of finite order of Gal(Q_{∞}/Q). Serre’s [Se] Théorème 3 shows that E(Q_{∞})_{tors} is finite, and the corollary follows without difficulty from Theorem 8.1(ii).

Remark . When E has complex multiplication, the representation T_{p}(E) does not satisfy Hypothesis Hyp(Q_{∞}, V)(i) (see Remark 8.5 below), so we cannot apply the results of §§3 and 4 with Kato’s Euler system. However, Theorem 8.1 and Corollary 8.2 are known in that case, as Theorem 8.1 for CM curves can be proved using the Euler system of elliptic units. See [CW], [Ru2] §11, and [RW].

8.2. Verification of the hypotheses. Fix a Z_{p}-basis of T and let

ρ_{E,p} : G_{Q} → Aut(T) ⟶^{∼} GL_{2}(Z_{p})

be the p-adic representation of G_{Q} attached to E with this basis.

Proposition 8.3. (i) If E has no complex multiplication, then T_{p}(E) satisfies

hypotheses Hyp(Q_{∞}, V) and H^{1}(Q(E_{p^{∞}})/Q, E_{p^{∞}}) is finite. (ii) If the p-adic representation ρ_{E,p} is surjective, then T_{p}(E) satisfies hypotheses

Hyp(Q_{∞}, T) and H^{1}(Q(E_{p^{∞}})/Q, E_{p^{∞}}) = 0.

Proof. The Weil pairing shows that

ρ_{E,p}(G_{Q(μ_{p^{∞}})}) = ρ_{E,p}(G_{Q}) ∩ SL_{2}(Z_{p}).

If E has no complex multiplication then a theorem of Serre ([Se] Théorème 3) says that the image of ρ_{E,p} is open in GL_{2}(Z_{p}). It follows that V_{p}(E) is an irreducible G_{Q_{∞}}-representation, and if ρ_{E,p} is surjective then E_{p} is an irreducible F_{p}[G_{Q_{∞}}]- representation.

It also follows that we can find τ ∈ G_{Q(μ_{p^{∞}})} such that

ρ_{E,p}(τ) = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} )

with x ≠ 0, and such a τ satisfies Hypothesis Hyp(Q_{∞}, V)(i). If ρ_{E,p} is surjective we can take x = 1, and then τ satisfies Hypothesis Hyp(Q_{∞}, T)(i).

We have

H^{1}(Q(E_{p^{∞}})/Q, E_{p^{∞}}) = H^{1}(ρ_{E,p}(G_{Q}), (Q_{p}/Z_{p})^{2})

and the rest of the proposition follows.

Remark 8.4. Serre’s theorem (see [Se] Corollaire 1 of Théorème 3) also shows that if E has no complex multiplication then ρ_{E,p} is surjective for all but finitely many p.

Remark 8.5. The conditions on τ in hypotheses Hyp(Q_{∞}, V)(i) force ρ_{E,p}(τ) to be nontrivial and unipotent. Thus if E has complex multiplication then there is no τ satisfying Hyp(Q_{∞}, V)(i).

8.3. Bounding S(Q, E_{p^{∞}}).

Theorem 8.6. Suppose E is modular, E does not have complex multiplication, and L(E, 1) ≠ 0.

(i) E(Q) and � (E)_{p^{∞}} are finite. (ii) Suppose in addition that p ∤ 2r_{E} and ρ_{E,p} is surjective. If E has good reduction

at p and p ∤ |Ẽ(F_{p})| (where Ẽ is the reduction of E modulo p), then

|� (E)_{p^{∞}}| divides \frac{L_{N}(E, 1)}{Ω_{E}}

where N is the conductor of E.

Proof. Recall that ℓ_{q}(q^{−s}) is the Euler factor of L(E, s) at q, and that by Proposition 5.1, S(Q, E_{p^{∞}}) is the usual p-power Selmer group of E.

Since ℓ_{q}(q^{−1}) is nonzero for every q, Corollary 7.2(i) shows that loc_{p}^{ram}(c_{Q}) ≠ 0. By Proposition 8.3(i) and Proposition 5.2 we can apply Theorem 3.1 to conclude that S(Q, E_{p^{∞}}) is finite, which gives (i).

If E has good reduction at p then pℓ_{p}(p^{−1}) = |Ẽ(F_{p})| and

[E(Q_{p}) : E_{1}(Q_{p}) + E(Q_{p})_{tors}] divides |Ẽ(F_{p})|.

Therefore if p ∤ r_{E}|Ẽ(F_{p})| then

exp_{ω_{E}}^{∗}(H_{/S}^{1}(Q_{p}, T_{p}(E))) = p^{−1}Z_{p}
exp_{ω_{E}}^{∗}(Z_{p}loc_{p}^{ram}(c_{Q})]) = p^{−1}(L_{N}(E, 1)/Ω_{E})Z_{p}

by Proposition 5.2 and Corollary 7.2(i). By Proposition 8.3(ii), if p ≠ 2 we can apply Theorem 3.2, and (ii) follows.

Remarks . In Corollary 8.9 below, using Iwasawa theory, we will prove that Theorem 8.6(ii) holds for almost all p, even when p divides |Ẽ(F_{p})|. This is needed to prove Theorem 8.1(i), since |Ẽ(F_{p})| could be divisible by p for infinitely many p. However, since |Ẽ(F_{p})| < 2p for all primes p > 5, we see that if E(Q)_{tors} ≠ 0 then |Ẽ(F_{p})| is prime to p for almost all p. Thus Theorem 8.1(i) for such a curve follows directly from Theorem 8.6.

The Euler system techniques we are using give an upper bound for the order of the Selmer group, but no lower bound.

8.4. Bounding S(Q_{∞}, E_{p^{∞}}). Define

S(Q_{∞}, E_{p^{∞}}) = lim_{⟶} S(Q_{n}, E_{p^{∞}}) ⊂ H^{1}(Q_{∞}, E_{p^{∞}}),

and recall that X_{∞} = Hom(S(Q_{∞}, E_{p^{∞}}), Q_{p}/Z_{p}). Let r_{E} be the positive integer of Theorem 7.1 and let N be the conductor of E.

Theorem 8.7. Suppose E is modular, E does not have complex multiplication, and E has good ordinary reduction or nonsplit multiplicative reduction at p. Then X_{∞} is a finitely-generated torsion Λ-module and there is an integer t such that

char(X_{∞}) divides p^{t}L_{E,N}Λ.

If ρ_{E,p} is surjective and p ∤ r_{E} ∏_{q|N,q≠p} ℓ_{q}(q^{−1}) then char(X_{∞}) divides L_{E}Λ.

If E has split multiplicative reduction at p, the same results hold with char(X_{∞}) replaced by J char(X_{∞}) where J is the augmentation ideal of Λ.

Proof. Rohrlich [Ro] proved that L_{E} ≠ 0. Thus the theorem is immediate from Propositions 8.3 and A.2, Corollary 7.2, and Theorem 4.1.

Corollary 8.8. Let E be as in Theorem 8.7. There is a nonzero integer M_{E} such that if p is a prime where E has good ordinary reduction and p ∤ M_{E}, then X_{∞} has no nonzero finite submodules.

Proof. This corollary is due to Greenberg [Gr1], [Gr2]; we sketch a proof here. Let Σ be a finite set of primes containing p, ∞, and all primes where E has bad reduction, and let Q_{Σ} be the maximal extension of Q unramified outside of Σ. Then there is an exact sequence

0 → S(Q_{∞}, E_{p^{∞}}) → H^{1}(Q_{Σ}/Q_{∞}, E_{p^{∞}}) → ⊕_{q∈Σ} ⊕_{v|q} H_{/S}^{1}(Q_{∞,v}, E_{p^{∞}}). (4)

Suppose q ∈ Σ, q ≠ p, and v | q. If p ∤ |E(Q_{q})_{tors}| then it is not hard to show that E(Q_{∞,v}) has no p-torsion, and so by [Gr1] Proposition 2, H^{1}(Q_{∞,v}, E_{p^{∞}}) = 0. Thus for sufficiently large p the Pontryagin dual of (4) is

lim_{\overleftarrow{n}} E(Q_{n,p}) ⊗ Z_{p} → Hom(H^{1}(Q_{Σ}/Q_{∞}, E_{p^{∞}}), Q_{p}/Z_{p}) → X_{∞} → 0.

Since Q_{∞}/Q is totally ramified at p,

lim_{\overleftarrow{n}} E(Q_{n,p}) ⊗ Z_{p} = lim_{\overleftarrow{n}} E_{1}(Q_{n,p}) = lim_{\overleftarrow{n}} Ê(𝔭_{n})

and this is free of rank one over Λ (see for example [PR1] Théorème 3.1 or [Schn] Lemma 6, §A.1). It now follows, using the fact that X_{∞} is a torsion Λ-module (Theorem 8.7) and [Gr1] Propositions 3, 4, and 5 that Hom(H^{1}(Q_{Σ}/Q_{∞}, E_{p^{∞}}), Q_{p}/Z_{p}) has no nonzero finite submodules, and by the Lemma on p. 123 of [Gr1] the same is true of X_{∞}.

Corollary 8.9. Suppose E is modular, E does not have complex multiplication, E has good reduction at p, p ∤ 2r_{E}M_{E} ∏_{q|N} ℓ_{q}(q^{−1}) (where r_{E} is as in Theorem 7.1 and M_{E} is as in Corollary 8.8), and ρ_{E,p} is surjective. Then

|� (E)_{p^{∞}}| divides \frac{L(E, 1)}{Ω_{E}}.

Proof. First, if E has supersingular reduction at p then |Ẽ(F_{p})| is prime to p, so the corollary follows from Theorem 8.6(ii).

Thus we may assume that E has good ordinary reduction at p. In this case the corollary is a well-known consequence of Theorem 8.7 and Corollary 8.8; see for example [PR1] §6 or [Schn] §2 for details. The idea is that if X_{∞} has no nonzero finite submodules and char(X_{∞}) divides L_{E}Λ, then

|S(Q_{∞}, E_{p^{∞}})^{Gal(Q_{∞}/Q)}| divides χ_{0}(L_{E,N}),

where χ_{0} denotes the trivial character, and with α as in Theorem 6.1

χ_{0}(L_{E,N}) = (1 − α^{−1})^{2} ∏_{q|N} ℓ_{q}(q^{−1})(L(E, 1)/Ω_{E}).

On the other hand, one can show that the restriction map

S(Q, E_{p^{∞}}) → S(Q_{∞}, E_{p^{∞}})^{Gal(Q_{∞}/Q)}

is injective with cokernel of order divisible by (1 − α^{−1})^{2}, and the corollary follows.

Proof of Theorem 8.1(i). Suppose E is modular, E does not have complex multiplication, and L(E, 1) ≠ 0. By Theorem 8.6, E(Q) is finite and � (E)_{p^{∞}} is finite for every p. By Corollary 8.9 (and using Serre’s theorem, see Remark 8.4) � (E)_{p^{∞}} = 0 for almost all p. This proves Theorem 8.1(i).

We can also now prove part of Theorem 8.1(ii) in the case where E has good ordinary or multiplicative reduction at p and L ⊂ Q_{∞}. For in that case, by Theorem 8.7, χ(char(Hom(S(Q_{∞}, E_{p^{∞}}), Q_{p}/Z_{p}))) is a nonzero multiple of L(E, χ, 1)/Ω_{E}. If L(E, χ, 1) ≠ 0 it follows that S(Q_{∞}, E_{p^{∞}})^{χ} is finite. The kernel of the restriction map S(L, E_{p^{∞}}) → S(Q_{∞}, E_{p^{∞}}) is contained in the finite group H^{1}(Q_{∞}/L, E_{p^{∞}}^{G_{Q∞}}), and so we conclude that both E(L)^{χ} and � (E_{/L})_{p^{∞}}^{χ} are finite.

Appendix. Explicit description of the Coleman map

In this appendix we give an explicit description of the Coleman map from H_{∞,/S}^{1}(Q_{n,p}, T) to Λ. This map allows us to relate Kato’s Euler system with the p-adic L-function L_{E}.

Suppose for this appendix that E has good ordinary reduction or multiplicative reduction at p. As in Theorem 6.1, let α ∈ Z_{p}^{×} and β ∈ pZ_{p} be the eigenvalues of Frobenius if E has good ordinary reduction, let α = 1, β = p if E has split multiplicative reduction and let α = −1, β = −p if E has nonsplit multiplicative reduction.

Recall we fixed before Theorem 6.1 a generator [ζ_{p^{n}}]_{n} of lim_{⟵} μ_{p^{n}}. For every n ≥ 0 define

x_{n} = α^{−n−1}Tr_{Q_{p}(μ_{p^{n+1}} )/Q_{n,p}} (∑_{k=0}^{n} \frac{ζ_{p^{n+1−k}} − 1}{β^{k}} + \frac{β}{β − 1}) ∈ Q_{n,p}.

Lemma A.1. (i) If n ≥ m then Tr_{Q_{n,p}/Q_{m,p}}x_{n} = x_{m}. (ii) If χ is a character of G_{n} then

χ( ∑_{γ∈G_{n}} x_{n}^{γ}γ) = \begin{cases}α^{−m}τ(χ) & if χ has conductor p^{m} > 1 \\ (1 − α^{−1})(1 − β^{−1})^{−1} & if χ = 1. \end{cases}

Proof. Exercise.

Proposition A.2. (i) For every n ≥ 0 there is a Z_{p}[G_{n}]-module map

Col_{n} : H_{/S}^{1}(Q_{n,p}, T) → Z_{p}[G_{n}]

such that for every z ∈ H_{/S}^{1}(Q_{n,p}, T) and every nontrivial character χ of G_{n} of conductor p^{m},

χ(Col_{n}(z)) = α^{−m}τ(χ) ∑_{γ∈G_{n}} χ^{−1}(γ) exp_{ω_{E}}^{∗}(z^{γ}).

If χ_{0} is the trivial character then

χ_{0}(Col_{n}(z)) = (1 − α^{−1})(1 − β^{−1})^{−1} ∑_{γ∈G_{n}} exp_{ω_{E}}^{∗}(z^{γ}).

(ii) The maps Col_{n} are compatible as n varies, and in the limit they induce a map

of Λ-modules

Col_{∞} : H_{∞,/S}^{1}(Q_{p}, T) → Λ.

(iii) The map Col_{∞} is injective. If E has split multiplicative reduction at p then the image of Col_{∞} is contained in the augmentation ideal of Λ.

Proof. The proof is based on work of Coleman [Co].

For the curves E which we are considering, Ê is a height-one Lubin-Tate formal group over Z_{p} for the uniformizing parameter β. It follows that, writing R for the ring of integers of the completion of the maximal unramified extension of Q_{p}, Ê is isomorphic over R to the multiplicative group G_{m}. Fix an isomorphism η : G_{m} ⟶^{∼} Ê, η ∈ R[[X]]. We define the p-adic period Ω_{p} of E

Ω_{p} = η′(0) ∈ R^{×}.

This period is unique up to Z_{p}^{×}, and is also characterized by the identity

λ_{E}(η(X)) = Ω_{p} log(1 + X). (5)

By [dS] §I.3.2 (4), if ϕ is the Frobenius automorphism of R/Z_{p} then

Ω_{p}^{ϕ} = α^{−1}Ω_{p}.

By [Co] Theorem 24 (applied to the multiplicative group) there is a power series g in the maximal ideal (p, X)R[[X]] of R[[X]] such that

log(1 + g(X)) = Ω_{p}^{−1} ((p − 1)\frac{β}{β − 1} + ∑_{k=0}^{∞} ∑_{δ∈μ_{p−1}⊂Z_{p}^{×}} \frac{(1 + X)^{δp^{k}} − 1}{β^{k}} ) .

In particular if we set X = ζ_{p^{n+1}} − 1 and use (5),

λ_{E}(η(g(ζ_{p^{n+1}} − 1))) = Ω_{p} log(1 + g(ζ_{p^{n+1}} − 1)) = α^{n+1}x_{n}.

Observe that Ê(𝔭_{n}R)_{tors} = 0 because Q_{p}(Ê_{p})/Q_{p} is totally ramified of degree p−1. Therefore λ_{E} is injective on Ê(𝔭_{n}R) and so

x_{n} ∈ λ_{E}(𝔭_{n}R)^{Gal(Q_{n,p}R/Q_{n,p})} = λ_{E}(𝔭_{n}) ⊂ λ_{E}(E(Q_{n,p})).

Define Col_{n} on H_{/S}^{1}(Q_{n,p}, T) by

Col_{n}(z) = ( ∑_{γ∈G_{n}} x_{n}^{γ}γ) ( ∑_{γ∈G_{n}} exp_{ω_{E}}^{∗}(z^{γ})γ^{−1})

= ∑_{γ∈G_{n}}(Tr_{Q_{n,p}/Q_{p}}x_{n}^{γ} exp_{ω_{E}}^{∗}(z))γ

Clearly this gives a Galois-equivariant map H_{/S}^{1}(Q_{n,p}, T) → Q_{p}[G_{n}]. Since x_{n}^{γ} ∈ λ_{E}(E(Q_{n,p})), (1) and (2) show that Col_{n}(z) ∈ Z_{p}[G_{n}]. The equalities of (i) follow from Lemma A.1(ii), and (ii) follows easily.

For (iii), suppose first that E has good ordinary reduction or nonsplit multiplicative reduction at p, so that α ≠ 1. Then the injectivity of Col_{n} (and of Col_{∞}) follows from (i), the nonvanishing of the Gauss sums, and the injectivity of exp_{ω_{E}}^{∗}.

If E has split multiplicative reduction at p, then α = 1. In this case it follows from (i) that ker(Col_{n}) = H_{/S}^{1}(Q_{n,p}, T)^{G_{n}}, which is free of rank one over Z_{p}, for every n. But one can show using (1) that H_{∞,/S}^{1}(Q_{p}, T) has no Λ-torsion, so Col_{∞} must be injective in this case as well. The assertion about the cokernel is clear from (i), since α = 1.

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Department of Mathematics, Ohio State University, Columbus, OH 43210 USA

   E-mail address: rubin@math.ohio-state.edu