Kolyvagin’s descent and Mordell-Weil groups over ring class fields^{1}

Massimo Bertolini^{2} Henri Darmon^{3}

May 23, 2000

1 Introduction

Let E/Q be a modular elliptic curve with the modular parametrization:

ϕ : X_{0}(N) → E,

where X_{0}(N) is the complete curve over Q which classifies pairs of elliptic curves related by a cyclic N-isogeny. The curve E is equipped with the collection of Heegner points defined over ring class fields of suitable imaginary quadratic fields.

More precisely, let K be an imaginary quadratic field in which all rational primes dividing N are split and let O be the order of K of conductor c prime to N. There exists a proper O-ideal N such that the natural projection of complex tori

C/O → C/N^{−1}  (1)

is a cyclic N-isogeny. The moduli interpretation of X_{0}(N) identifies the diagram (1) with a point of X_{0}(N). By the theory of complex multiplication, this point is defined over H, the ring class field of K of conductor c. Let α ∈ E(H) be its image under ϕ.

The group G = Gal(H/K) acts naturally on the Z-module E(H), and E(H) ⊗ C can be decomposed as a direct sum of eigenspaces under this action:

1The authors wish to thank B.H. Gross and K.C. Rubin for having proposed this problem and provided helpful suggestions in the write-up.

2Partially supported by a grant from the Consiglio Nazionale delle Ricerche of Italy.

3Partially supported by a Doctoral Fellowship from the Natural Sciences and Engineering Research Council of Canada (NSERC).

E(H) ⊗ C = ⊕_{χ∈Ĝ}E(H)^{χ},

where Ĝ = Hom(G, C^{∗}) is the group of complex characters of G. Let

e_{χ} = \frac{1}{#G} ∑_{σ∈G} χ^{−1}(σ)σ

be the idempotent in the group ring giving the projection onto the χ-eigenspace.

Gross and Zagier [3] proved the following limit formula when c = 1 (so that H is the Hilbert class field of K):

L′(E/K, χ, 1) = aĥ(e_{χ}α),

where L(E/K, χ, s) is the L-series of E/K twisted by the character χ, a is a nonzero invariant depending on E and K, and ĥ is the canonical height extended by linearity to E(H) ⊗ C. In view of the conjecture of Birch and Swinnerton-Dyer, Gross formulated the following:

Conjecture 1.1 If e_{χ}α ≠ 0, then dim_{C} E(H)^{χ} = 1.

In his paper on Euler systems [4], Kolyvagin proves the above conjecture when χ is the trivial character. We will apply Kolyvagin’s descent techniques to prove the general case when E has no complex multiplications.

2 Preliminaries

Our strategy will be to do a p-descent for a suitable prime p. We choose p so that

1. p|̸6cNDisc(K),

2. Q(E_{p})/Q is a GL_{2}(F_{p})-extension,

3. p ≡ 1 (mod #G).

These conditions can be imposed simultaneously, provided that E has no complex multiplications, by combining the “open image” theorem of Serre [8] with the result of Dirichlet on primes in arithmetic progressions. The following lemma is a simple consequence of conditions 1 and 2:

Lemma 2.1 If L is an extension of Q which is unramified at all primes dividing Np, then Gal(L(E_{p})/L) ≃ GL_{2}(F_{p}).

Proof: The extension Q(E_{p})/Q is ramified only at places dividing Np, and hence Q(E_{p}) and L are linearly disjoint over Q (the intersection of these two fields is an unramified extension of Q, which is Q by Minkowski’s theorem). Hence Gal(L(E_{p})/L) = Gal(Q(E_{p})/Q) = GL_{2}(F_{p}).

By condition 3, any F_{p}[G]-module M splits as a direct sum of primary representations:

M = ⊕_{χ∈Ĝ}M^{χ},

where χ ranges over the F_{p}-valued characters of G. (By choosing a reduction map

Z[μ_{#G}] → F_{p},

we identify complex and F_{p}-valued characters of G.) Notice that if e_{χ}α is nonzero in E(H) ⊗ C, then it is also non zero in E(H) ⊗ F_{p} for almost all primes p. Furthermore, lemma 2.1 implies that E_{p}(H) = 0 and hence

dim_{C} E(H)^{χ} = dim_{F_{p}}(E(H) ⊗ F_{p})^{χ}.

Conjecture 1.1 is thus reduced to the following “mod p” analogue:

Theorem 2.2 If e_{χ}α ≠ 0 in E(H) ⊗ F_{p} = E(H)/pE(H), then

dim_{F_{p}}(E(H)/pE(H))^{χ} = 1.

Let us introduce some conventions and results that will be used throughout. We fix an algebraic closure Q̄ of Q which contains all of the field extensions which will be introduced later on. Let τ ∈ Gal(Q̄/Q) be a fixed complex conjugation corresponding to a choice of an embedding Q̄ ↪ C, and denote by [τ] its conjugacy class. It will be convenient to identify τ with its images in finite quotients of Gal(Q̄/Q). If M is a module on which τ acts, the superscripts + and − are used to designate projection onto the eigenspaces for the action of τ (we assume that 2 is invertible in End_{Z}(M), so that M decomposes as a direct sum of such eigenspaces):

M^{±} = {m ∈ M|τm = ±m}.

Also, if x ∈ M and X ⊂ M, we let

x^{±} = \frac{1}{2}(x ± τx),

X^{±} = {x^{±}|x ∈ X}.

For conciseness, we shall occasionally use the notation M/p to denote the F_{p}- vector space M ⊗ F_{p} = M/pM.

Let H[n] ⊂ Q̄ denote the ring class field of K associated to the order O_{n} of conductor cn, where (n, N) = 1, let G_{n} = Gal(H[n]/H), and let α(n) be the Heegner point corresponding to the N-isogeny

C/O_{n} → C/(O_{n} ∩ N)^{−1}.

By class field theory, G_{n} is canonically isomorphic to (O/nO)^{∗}/(Z/nZ)^{∗}, and complex conjugation acts on the group Gal(H[n]/K) by

τxτ^{−1} = x^{−1}.  (2)

Let ϵ = ±1 denote the negative of the sign in the functional equation for L(E/Q, s). The following describes the action of τ on the Heegner points in E(H[n])/p:

Lemma 2.3 There exists σ_{0} ∈ Gal(H[n]/K) such that τα(n) = ϵσ_{0}α(n) in E(H[n])/p. Hence, τe_{χ}α(n) = ϵχ̄(σ_{0})eχ̄α(n).

Proof: In [2], it is observed that

τα(n) = ϵσ_{0}α(n) + torsion,

for some σ_{0} ∈ Gal(H[n]/K). By lemma 2.1, the group E(H[n]) has no p-torsion, and the first statement follows. The second is a consequence of the identity:

τe_{χ} = eχ̄τ,

which results from equation (2).

Kolyvagin’s idea is to construct elements in the dual of the Selmer group,

Sel_{p}^{dual}(E/H) = Hom(Sel_{p}(E/H), F_{p}),

via local Tate duality, and to control the size of this module by certain global cohomology classes related to Heegner points. Section 3 is devoted to the construction and study of these “Heegner cohomology classes”.

3 The Heegner cohomology classes

Definition 3.1 A rational prime l is said to be special if l|̸Npc and

Frob_{l}(K(E_{p})/Q) = [τ].

Observe that, if l is special, then

a_{l} ≡ l + 1 ≡ 0 (mod p),  (3)

where a_{l} denotes the trace of Frobenius acting on the Tate module T_{p}(E). This follows from comparing the minimal polynomials of Frob_{l} and τ acting on E_{p}.

Let n denote a squarefree product of special primes, and let l be a special prime not dividing n. The prime l is inert in K by definition; let λ = (l) be the unique prime of K above it. The prime λ splits completely in H[n]/K (its image in G_{n} = (O/nO)^{∗}/(Z/nZ)^{∗} by the Artin map is trivial), and any prime λ′ of H[n] above λ is totally ramified in H[nl]. Fix a choice of λ′, and denote by λ′′ the unique prime of H[nl] above it. The following proposition summarizes the properties of Heegner points that will be needed in the constructions:

Proposition 3.2 The system of Heegner points α(n) ∈ H[n] satisfies:

1. Tr_{H[nl]/H[n]}α(nl) = a_{l}α(n);

2. α(nl) ≡ Frob_{λ′} α(n) (mod λ′′).

These two properties are axiomatized by Kolyvagin in his definition of “Euler sytems” with congruence [4, §1]. For a proof, see for example [2, prop. 3.7]

Since H_{n} is the compositum of the extensions H_{l} which are linearly disjoint over H, we have a canonical isomorphism G_{n} = ∏_{l|n} G_{l}, allowing us to view G_{l} as a subgroup of G_{n}. By class field theory, each G_{l} is isomorphic to (O_{k}/λ)^{∗}/(Z/l)^{∗}. Choose for each l a generator σ_{l} of G_{l}, and let

Tr_{l} = ∑_{i=0}^{l} σ_{l}^{i} ∈ F_{p}[G_{l}]
D_{l} = ∑_{i=1}^{l} iσ_{l}^{i} ∈ F_{p}[G_{l}]
D_{n} = ∏_{l|n} D_{l} ∈ F_{p}[G_{n}].

Lemma 3.3 D_{n}α(n) ∈ (E(H[n])/p)^{G_{n}}.

Proof: For all primes l dividing n,

(σ_{l} − 1)D_{l} = 1 + l − Tr_{l} = −Tr_{l},

where the last equality follows from eq. (3). Combining prop. 3.2 with eq. (3), one has:

−Tr_{l}α(n) = −a_{l}α(n/l) = 0.

Hence σ_{l}D_{n}α(n) = D_{n}α(n); since the σ_{l} generate G_{n}, the lemma follows. By lemma 2.1, E_{p}(H[n]) = 0, and the following sequence is exact:

0 → E(H[n]) →^{p} E(H[n]) → E(H[n])/p → 0.

Taking G_{n}-invariants yields the exact sequence of F_{p}[G]-modules:

0 → E(H)/pE(H) → (E(H[n])/p)^{G_{n}} → H^{1}(G_{n}, E(H[n]))_{p} → 0.

Let ν(n) be the image of D_{n}α(n) in H^{1}(G_{n}, E(H[n]))_{p}. By abuse of notation, we identify ν(n) with its image in H^{1}(H, E)_{p} under inflation.

Let w be a prime of K lying above the rational prime v. There is a natural localization map

res_{w} : H → ⊕_{w′ |w}H_{w′} .

By abuse of notation, we identify the map res_{w} with its image by any functor (from the category of étale algebras to the category of F_{p}-vector spaces). The context will make it clear what the source and target of res_{w} are. Denote by F_{w′} the residue field of H_{w′}.

Lemma 3.4 The behaviour of the class ν(n) under the localization maps is given by:

1. If v does not divide n, then res_{w}ν(n) = 0.

2. For l special, there is a canonical G-equivariant isomorphism

T : ⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p} ≃ ⊕_{λ′ |λ}Hom(μ_{p}(F_{λ′}), E_{p}(F_{λ′}))

such that, when l divides n, the homomorphism T (res_{λ}ν(n)) maps each μ_{p}(F_{λ′}) onto the subgroup of E_{p}(F_{λ′}) generated by

(\frac{(l + 1)Frob_{l} − a_{l}}{p})D_{n/l}α(n/l),

where we identify the point D_{n/l}α(n/l) with its reduction mod λ′.

When v is a place of good reduction for E, part 1 follows from standard cohomological arguments, using the fact that ν(n) is inflated from a class in H^{1}(H[n]/H, E) and that the w′ are unramified in H[n]/H. The general case is proved in [2, prop. 6.2]. The isomorphism in part 2 is explicitely constructed in [2, prop. 6.2], using local class field theory.

Part 2 of lemma 3.4 will be applied via the following corollary:

Corollary 3.5 There is a G-equivariant isomorphism

⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p} → ⊕_{λ′ |λ}E(F_{λ′})/p

which sends res_{λ}ν(n) to res_{λ}D_{n/l}α(n/l).

(Observe that res_{λ}D_{n/l}α(n/l) is well-defined in ⊕_{λ′ |λ}E(F_{λ′})/p.) Using the map T of lemma 3.4, a choice of generators for μ_{p}(F_{λ′}) defines a (non-canonical) isomorphism

i : ⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p} →^{∼} ⊕_{λ′ |λ}E_{p}(F_{λ′}).

By choosing the generators appropriately, we may ensure that i sends res_{λ}ν(n) to

(\frac{(l + 1)Frob_{l} − a_{l}}{p})res_{λ}D_{n/l}α(n/l).

The operator W = (\frac{(l+1)Frob_{l}−a_{l}}{p}) induces an isomorphism E(F_{λ′})/p → E_{p}(F_{λ′}). (This can be shown by decomposing E(F_{λ′}) into eigencomponents for the action of the involution Frob_{l}. The trivial and non-trivial components are of order l + 1 − a_{l} and l + 1 + a_{l} respectively, and hence W is an isomorphism on the eigencomponents.) The composition W^{−1}i gives the desired map.

4 Local Tate duality: generating Sel_{p}^{dual}(E/H)

Local Tate duality [6] gives a perfect pairing

< >_{λ′}: E(H_{λ′})/p × H^{1}(H_{λ′} , E)_{p} → Z/pZ

which identifies ⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p} with (⊕_{λ′ |λ}E(H_{λ′})/p)^{dual}. The p–Selmer group Sel_{p}(E/H) consists of the cohomology classes s ∈ H^{1}(H, E_{p}) whose restrictions res_{v}(s) ∈ H^{1}(H_{v}, E_{p}) belong to E(H_{v})/p for all places v of H, where we view E(H_{v})/p as a subspace of H^{1}(H_{v}, E_{p}) by using the local p-descent exact sequence

0 → E(H_{v})/p → H^{1}(H_{v}, E_{p}) → H^{1}(H_{v}, E)_{p} → 0.

Transposing the map:

res_{λ} : Sel_{p}(E/H) → (⊕_{λ′ |λ}E(H_{λ′})/p),

and using the identification given by the local Tate pairing, we get a homomorphism

Ψ_{l} : ⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p} → Sel_{p}^{dual}(E/H).

Let X_{l} denote the image of Ψ_{l} in Sel_{p}^{dual}(E/H). We choose an auxiliary special prime l_{1} and define the following Galois extensions of Q:

F = H[l_{1}](E_{p}),
M_{0} = F (α/p)^{Gal},
M_{1} = F (D_{l_{1}}α(l_{1})/p)^{Gal},
M = M_{0}M_{1},

where the superscript “Gal” indicates taking normal closure over Q. (The reasons for these definitions will become clear in the next section.) By lemma 2.1,

Gal(F/Q) = Gal(H[l_{1}]/Q) × Gal(Q(E_{p})/Q) = Gal(H[l_{1}]/Q) × GL_{2}(F_{p}).

The Galois groups V_{0}, V_{1}, and V of M_{0}, M_{1}, and M over F are F_{p}-vector spaces equipped with a natural action of Gal(F/Q).

M
↗ | ↖
M_{0} |^{V} M_{1}
V_{0} ↖ | ↗_{V_{1}}
F
|
K

Given a subset U of V, define

L(U) = {l rational prime |Frob_{l}(M/Q) = [τu], for u ∈ U}.

Note that every l ∈ L(U) is special.

Proposition 4.1 If U^{+} generates V^{+}, then the X_{l}, with l ranging over L(U), generate Sel_{p}^{dual}(E/H).

Proof: Let s be in Sel_{p}(E/H). To prove the proposition, it suffices (by the nondegeneracy of the local Tate pairing) to show that res_{λ}(s) = 0 for all l ∈ L(U) implies s = 0. Assume without loss of generality that s is in an eigenspace for the action of τ. Let us identify s with its image by restriction in:

H^{1}(F, E_{p})^{Gal(F/H)} ⊂ Hom_{Gal(H(E_{p})/H)}(Gal(M̄/F), E_{p}),

where M̄ denotes the maximal abelian extension of F whose Galois group is of exponent p. The restriction is injective because it can be written as a composition

H^{1}(H, E_{p}) → H^{1}(H(E_{p}), E_{p})^{Gal(H(E_{p})/H)} → H^{1}(F, E_{p})^{Gal(H(E_{p})/H)}.

Both arrows are injections: the kernel of the first is

H^{1}(H(E_{p})/H, E_{p}) = H^{1}(GL_{2}(F_{p}), F_{p}^{2}) = 0,

and the kernel of the second is

Hom_{Gal(H(E_{p})/H)}(Gal(F/H(E_{p})), E_{p}) = 0.

Choose a minimal Galois extension M̃ of Q containing M with the property that s factors through Gal(M̃/F). Let x ∈ Gal(M̃/F) be such that x|_{M} ∈ U. By the Chebotarev density theorem, we may find l ∈ L(U) such that Frob_{l}(M̃/Q) = [τx]. The hypothesis res_{λ}(s) = 0 means that:

s(Frob_{λ′} (M̃/F)) = 0,

for all primes λ′ of M̃ above l. On the other hand, for some λ′ above l,

Frob_{λ′} (M̃/F) = (τx)^{2} = x^{τ} x = (x^{+})^{2},

and hence s(x^{+}) = 0. Since U^{+} generates V^{+}, the homomorphism s vanishes on Gal(M̃/F)^{+}. Hence the image of s is contained in an eigenspace of E_{p} for the action of τ. In particular, it is a proper Gal(H[l_{1}](E_{p})/H[l_{1}])-submodule of E_{p}. Hence it is trivial, since

Gal(H[l_{1}](E_{p})/H[l_{1}]) = GL_{2}(F_{p})

by lemma 2.1. Therefore s = 0.

5 Global Tate duality: relations in Sel_{p}^{dual}(E/H)

The tool for finding relations in Sel_{p}^{dual}(E/H) is:

Proposition 5.1 If s ∈ Sel_{p}(E/H) and γ ∈ H^{1}(H, E)_{p}, then

∑_{w} < res_{w}s, res_{w}γ >_{w}= 0,

where the sum is taken over all places of H.

This proposition is an immediate consequence of the global reciprocity law for elements in the Brauer group of H, taking into account the definition of the local Tate duality [6].

We suppose that the auxiliary special prime l_{1} of the previous section satisfies the following property:

res_{λ_{1}}eχ̄α ≠ 0  (4)

(and hence res_{λ_{1}}e_{χ}α ≠ 0, by lemma 2.3.) Such an l_{1} exists by the Chebotarev density theorem applied to the extension H(E_{p})(eχ̄α/p)/Q, using the hypothesis that eχ̄α ≠ 0 in E(H)/p. By corollary 3.5, condition (4) implies that

res_{λ_{1}}eχ̄ν(l_{1}) ≠ 0.  (5)

We need to examine the extensions M of F defined in the previous section. Let M χ̄ ; 0 (resp. M χ̄ ; 1, M χ̄) denote the extensions F (eχ̄α/p) (resp. F (eχ̄D_{l_{1}}α(l_{1})/p), F (eχ̄α/p, eχ̄D_{l_{1}}α(l_{1})/p)).

Lemma 5.2 The extensions M χ̄ ; 0 and M χ̄ ; 1 are linearly disjoint over F.

Proof: Indeed, linearly independent points in E(H[l_{1}])/p give rise to linearly disjoint extensions over F. This is because the map

E(H[l_{1}])/p → Hom_{Gal(F/H[l_{1}])}(V, E_{p})

is injective, and linearly independent elements of Hom_{Gal(F/H[l_{1}])}(V, E_{p}) cut out linearly disjoint extensions over F (use the fact that Gal(F/H[l_{1}]) ≃ GL_{2}(F_{p}), by lemma 2.1). Hence, if M χ̄ ; 0 and M χ̄ ; 1 were not linearly disjoint over F, we would have:

eχ̄D_{l_{1}}α(l_{1}) = ueχ̄α in E(H[l_{1}])/p, u ∈ F_{p}^{∗}.

The exact sequence

\begin{aligned}0 → (E(H)/p)χ̄ → & (E(H[l_{l}])/p)^{G_{l_{1}} ,χ̄} & → & H^{1}(G_{l_{1}}, E)χ̄ ; p → 0 \\ & eχ̄D_{l_{1}}α(l_{1}) & ↦ & eχ̄ν(l_{1}) \end{aligned}

and equation (5) show that this cannot happen.

We now describe the action of complex conjugation on V χ̄, by using 2.3 and the relation τD_{l} = −D_{l}τ. There are two cases: Case 1: χ = χ̄. Complex conjugation τ acts on V^{χ} = V_{0}^{χ} × V_{1}^{χ} = E_{p} × E_{p} by

τ(x, y)τ = (ϵχ(σ_{0})τx, −ϵχ(σ_{0})τy)

Case 2: χ ≠ χ̄. Then τ does not stabilize V^{χ} or V χ̄, but interchanges these two components. The action of τ on

V_{0}^{χ} × V χ̄ ; 0 × V_{1}^{χ} × V χ̄ ; 1 ≃ E_{p}^{4}

is given by:

τ(x, y, z, w)τ = (ϵχ̄(σ_{0})τy, ϵχ(σ_{0})τx, −ϵχ̄(σ_{0})τw, −ϵχ(σ_{0})τz)

We define a subset U of V as follows:

Case 1 : U = {(x, y)|ϵχ̄(σ_{0})τx + x and − ϵχ̄(σ_{0})τy + y generate E_{p}} (6)
Case 2 : U = {(x, y, z, w)|ϵχ(σ_{0})τx + y and
− ϵχ̄(σ_{0})τz + w generate E_{p}}  (7)

Note that, in both cases, U satisfies the property of prop. 4.1. Let l be a prime in L(U):

Lemma 5.3 The local cohomology classes res_{λ}eχ̄ν(l) and res_{λ}eχ̄ν(ll_{1}) generate (⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p})χ̄.

Proof: Since ⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p} ≃ (⊕_{λ′ |λ}E(H_{λ′})/p)^{dual} is isomorphic to two copies of the regular representation as an F_{p}[G]-module, we have

dim_{F_{p}}(⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p})χ̄ = 2.

The isomorphism of corollary 3.5 sends res_{λ}eχ̄ν(l) and res_{λ}eχ̄ν(ll_{1}) to eχ̄α and eχ̄D_{l_{1}}α(l_{1}). The frobenius condition on l in the definitions (6,7) of U show that these two points are linearly independant in ⊕_{λ′ |λ}E(F_{λ′})/p. Hence res_{λ}eχ̄ν(l) and res_{λ}eχ̄ν(ll_{1}) are linearly independant, and span (⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p})χ̄. We recall that X_{l} is the image of (⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p})^{dual} in Sel_{p}(E/H)^{dual}.

Proposition 5.4 The module Xχ̄ ; l is of dimension 1 over F_{p}.

Proof: By prop. 5.1 and part 1 of lemma 3.4, the kernel of the map

(⊕_{λ′ |λ}H^{1}(H_{λ′} , E)_{p})χ̄ → Xχ̄ ; l

contains the non-trival element res_{λ}eχ̄ν(l). On the other hand, Xχ̄ ; l ≠ 0: it does not vanish identically on eχ̄α, since res_{λ}eχ̄α ≠ 0, and the local Tate pairing is non-degenerate. Hence Xχ̄ ; l is one-dimensional.

Proposition 5.5 All of the Xχ̄ ; l are equal, for l ∈ L(U).

Proof: We show that Xχ̄ ; l = Xχ̄ ; l_{1} for all l ∈ L(U). By prop. 5.1 applied to the Heegner class eχ̄ν(ll_{1}), we have:

res_{λ}eχ̄ν(ll_{1}) + res_{λ_{1}}eχ̄ν(ll_{1}) = 0 in Sel_{p}^{dual}(E/H)χ̄.

By lemma 5.3, res_{λ}eχ̄ν(ll_{1}) generates Xχ̄ ; l. Hence res_{λ_{1}}eχ̄ν(ll_{1}) is non-zero and generates Xχ̄ ; l_{1}, and Xχ̄ ; l = Xχ̄ ; l_{1}.

6 Conclusion of the proof

Let δ denote the coboundary map E(H)/p → H^{1}(H, E_{p}). We can now show:

Theorem 6.1 The following are true:

1. Sel_{p}(E/H)^{χ} = F_{p}δ(e_{χ}α);

2. E(H)/pE(H)^{χ} = F_{p}(e_{χ}α).

Proof: By prop. 4.1, the Xχ̄ ; l generate Sel_{p}^{dual}(E/H)χ̄ when l ranges over L(U). On the other hand, each Xχ̄ ; l is one-dimensional, and all the Xχ̄ ; l are equal. Hence

dim_{F_{p}} Sel_{p}^{dual}(E/H)χ̄ = dim_{F_{p}} Sel_{p}(E/H)^{χ} = 1,

and Sel_{p}(E/H)^{χ} is generated by the non-zero element δ(e_{χ}α). It follows that (E(H)/p)^{χ} is one-dimensional, generated by the Heegner point e_{χ}α.

Remarks:

1. In [1], Gross formulates his conjecture for abelian varieties which are quotients of the jacobian of the modular curve X_{0}(N). The argument given above extends to this more general situation. For more details, see [5].

2. For applications of the formalism of Euler systems to different arithmetic situations, see [7].

References

[1] B.H. Gross, Heegner points on X_{0}(N), Modular Forms (R.A. Rankin, Ed.) Chichester, Ellis Horwood, 1984, 87-106.

[2] B.H. Gross, Kolyvagin’s work on modular elliptic curves, Proc. Durham symposium on L-functions and arithmetic, 1989, to appear.

[3] B.H. Gross and D.B. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84 (1986), 225-320.

[4] V.A. Kolyvagin, Euler Systems, (1988). To appear in a Birkhäuser volume in honor of Grothendieck.

[5] V.A. Kolyvagin and D.Y. Logachev, Finiteness of the Shafarevich-Tate group and group of rational points for some modular abelian varieties. Algebra and analysis (USSR), No. 5. (1989).

[6] J.S. Milne, Arithmetic duality theorems. Perspectives in mathematics. Academic Press, 1986.

[7] K. Rubin, Appendix to S. Lang, Cyclotomic Fields I and II, combined second edition, GTM 121, SPringer-Verlag, New York 1990.

[8] J-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15 (1972), 259-331.

Massimo Bertolini, Columbia U. Math. Dept., New York, NY 10027. Henri Darmon, Harvard U. Math. Dept., 1 Oxford St., Cambridge MA 02138.