On the parity of ranks of Selmer groups II

Jan Nekovář 11 November 2000

Let

E : y^{2} = x^{3} + Ax + B (A, B ∈ Q)

be an elliptic curve over Q of conductor N. Thanks to the work of Wiles and his followers [BCDT] we know that E is modular, i.e. there exists a non-constant map π : X_{0}(N) ⟶ E defined over Q and

L(E, s) = ∑_{n=1}^{∞} a_{n}n^{−s} = L(f, s)

for a normalized newform f ∈ S_{2}(Γ_{0}(N)).

For a large class of number fields F (which includes all solvable Galois extensions of Q), this is known to imply that the L-function L(E/F, s) has a holomorphic continuation to C and a functional equation relating the values at s and 2 − s. For such F, denote by

r_{an}(E/F) := ord_{s=1}L(E/F, s)

the analytic rank of E over F.

Over an arbitrary number field F, the m-descent on E gives rise (for every integer m ≥ 1) to the Selmer group S(E/F, m) sitting in the standard exact sequence

0 ⟶ E(F) ⊗ Z/mZ ⟶ S(E/F, m) ⟶ � (E/F)[m] ⟶ 0.

Fix a prime number p and put

S(E/F) = S_{p}(E/F) = \underrightarrow{lim}_{n} S(E/F, p^{n})
X(E/F) = X_{p}(E/F) = \underleftarrow{lim}_{n} S(E/F, p^{n})
s_{p}(E/F) = corank_{Z_{p}}S_{p}(E/F) = rk_{Z_{p}}X_{p}(E/F).

The conjecture of Birch and Swinnerton-Dyer predicts that

(BSD)  r_{an}(E/Q) =^{?} rk_{Z}E(Q).

As in [NePl], we are interested in a rather weak consequence of (BSD) (and the conjectural finiteness of the Tate-Šafarevič group), namely the

Parity conjecture for Selmer groups: r_{an}(E/Q) ≡^{?} s_{p}(E/Q) (mod 2). Our main result is the following

Theorem A. Let E be an elliptic curve over Q with good ordinary reduction at p. Then the parity conjecture

r_{an}(E/Q) ≡ s_{p}(E/Q) (mod 2)

holds.

See [NePl] for a discussion of earlier results in this direction. Our method of proof is similar to that in [NePl]; the only difference is that we use anticyclotomic deformations instead of Hida families. A recently proved conjecture of Mazur ([Maz2], [Co], [Va2]) on non-vanishing of Heegner points in anticyclotomic Z_{p}-extensions plays the role of Greenberg’s conjecture assumed in [NePl].

1. Heegner points

In this section we recall the basic setup of Heegner points on E (see [Gro] for a more detailed account).

1.1 Let K = Q(\sqrt{D}) be an imaginary quadratic field of discriminant D < 0. We assume that K satisfies the following “Heegner condition” of Birch [Bi]:

(Heeg)  Every prime q|N splits in K.

Under this assumption there exists an ideal N ⊂ O_{K} such that O_{K}/N ⟶^{∼} Z/NZ (of course, N is not unique; we choose one).

1.2 For every integer c ≥ 1 denote by O_{c} = Z + cO_{K} the unique order of O_{K} of conductor c. If (c, N) = 1, then

N_{c} := O_{c} ∩ N

is an invertible ideal in O_{c} satisfying O_{c}/N_{c} ⟶^{∼} Z/NZ (and hence also N_{c}^{−1}/O_{c} ⟶^{∼} Z/NZ, since N_{c} is invertible). The cyclic N-isogeny

[C/O_{c} ⟶ C/N_{c}^{−1}]

induced by the identity map on C defines a non-cuspidal point on the modular curve X_{0}(N), which is defined over H_{c}, the ring class field of conductor c over K. The image of this point under the fixed modular parametrization π : X_{0}(N) ⟶ E will be denoted by \overline{x}_{c} ∈ E(H_{c}) – this is a Heegner point of conductor c on E.

1.3 From now on, we consider only conductors of the form c = p^{n} for a fixed prime number p. The field H_{p^{∞}} = ⋃ H_{p^{n}} then contains the anticyclotomic Z_{p}-extension K_{∞} of K, which can be characterized by the following properties: K_{∞} = ⋃ K_{n}, G(K_{n}/K) ⟶^{∼} Z/p^{n}Z, K_{n}/Q is a Galois extension with G(K_{n}/Q) ⟶^{∼} D_{2p^{n}} (the dihedral group). There is n_{0} ≥ 0 such that

H_{p^{n+1}} ∩ K_{∞} = K_{n+n_{0}} (n ≥ 0).

We use the following standard notation:

Γ = G(K_{∞}/K),  Γ_{n} = G(K_{∞}/K_{n}),  Λ = Z_{p}[[Γ]].

Fix an isomorphism Γ ⟶^{∼} Z_{p} (i.e. fix a topological generator γ of Γ).

From now on, we assume that the following condition of “good reduction” is satisfied:

(G)  p |̸ N.

This assumption implies that the Heegner points

\overline{x}_{p^{n+1}} ∈ E(H_{p^{n+1}})

are defined. We put

x_{n+n_{0}} := Tr_{H_{p^{n+1}} /K_{n+n_{0}}} (\overline{x}_{p^{n+1}}) = Tr_{H_{p^{∞}} /K_{∞}} ∈ E(K_{n+n_{0}}) (n ≥ 0).

1.4 In his lecture at ICM 1983, Mazur formulated (among others) the following Conjecture [Ma2]. (∃ n ≥ n_{0}) x_{n} ∉ E(K_{n})_{tors}.

This conjecture was recently proved by Vatsal [Va2] under the assumptions that |D| is prime and p does not divide the class number of K, and by Cornut [Co] under the assumption (G). Both [Co] and [Va2] build upon an earlier work [Va1] of Vatsal. 1.5 The fundamental distribution relation for Heegner points ([PR], Lemma 2, p.432) states that

Tr_{K_{n+1}/K_{n}}(x_{n+1}) = a_{p}x_{n} − x_{n−1} (n ≥ n_{0} + 1).

The assumption (G) is equivalent to E having good reduction at p. From now on, we assume in addition that E has ordinary reduction at p, which is equivalent to

(Ord)  p |̸ a_{p}.

This assumption implies that the local Euler factor of E at p factorizes as

1 − a_{p}X + pX^{2} = (1 − αX)(1 − βX),

where α, β ∈ Z_{p} satisfy ord_{p}(α) = 0, ord_{p}(β) = 1.

Define, for n ≥ n_{0} + 1,

y_{n} = x_{n} ⊗ α^{1−n} − x_{n−1} ⊗ α^{−n} ∈ E(K_{n}) ⊗ Z_{p}.

Then

Tr_{K_{n+1}/K_{n}}(y_{n+1}) = y_{n} (n ≥ n_{0} + 1),

i.e. y = (y_{n}) is an element of the projective limit

\underleftarrow{lim}_{n>n_{0}} (E(K_{n}) ⊗ Z_{p}) = \underleftarrow{lim}_{n} (E(K_{n}) ⊗ Z_{p}) ⊆ \underleftarrow{lim}_{n} X(E/K_{n}) =: X_{∞}.

We shall also be interested in the inductive limit

S_{∞} = \underrightarrow{lim}_{n} S(E/K_{n}).

Both X_{∞} and S_{∞} are Λ-modules (of finite and co-finite type, respectively).

2. Iwasawa theory

In this section we recall basic results of Iwasawa theory of elliptic curves relating the Λ-modules X_{∞} and S_{∞} to Selmer groups over the fields K_{n}. The assumptions (Heeg), (G) and (Ord) are in force.

Lemma 2.1. (i) (∀n ≥ 0) the canonical map

S(E/K_{n}) ⟶ (S_{∞})^{Γ_{n}}

has finite kernel and cokernel. (ii) There is an isomorphism of Λ-modules (of finite type)

X_{∞} ⟶^{∼} Hom_{Λ}(Ŝ_{∞}, Λ)

(where M̂ denotes the Pontryagin dual of M), hence rk_{Λ}(X_{∞}) = corank_{Λ}(S_{∞}). (iii) (∀n ≥ 0) the canonical map

(X_{∞})_{Γ_{n}} ⟶ X(E/K_{n})

has finite kernel. (iv) X_{∞} ⟶^{∼} Λ^{r} for some r ≥ 0.

(v) E(K_{∞})_{tors} is finite.

Proof. (i) [Man], Thm. 4.5 or [Maz1], Prop. 6.4 (note that [Man], Lemma 4.6 eliminates the need for the second assumption in [Maz1], 6.1; however, the latter is satisfied in our situation, because of (Heeg)). (ii) [PR], Lemma 5, p. 417. (iii) This follows from (i) and (ii) (cf. [PR], Lemma 4, p. 415). (iv) The R.H.S. in (ii) is reflexive, hence free over Λ. (v) [NeSc], 2.2.

Lemma 2.2. y ≠ 0 in X_{∞}.

Proof. If y_{n} = 0 in E(K_{n}) ⊗ Z_{p} for all n > n_{0}, then

x_{n} ⊗ α = x_{n−1} ⊗ 1

in E(K_{n}) ⊗ Q_{p} for all n > n_{0}. As both x_{n} ⊗ 1 and x_{n−1} ⊗ 1 lie in E(K_{n}) ⊗ Q, but α ∉ Q, we must have x_{n} ⊗ 1 = 0 in E(K_{n}) ⊗ Q for all n > n_{0}, which contradicts (now proven) Mazur’s conjecture 1.4.

Lemma 2.3. X_{∞} ⟶^{∼} Λ.

Proof. (Note that the implication [Mazur’s conjecture ⟹ X_{∞} ⟶^{∼} Λ] was proved under more restrictive assumptions by Bertolini [Be]). By Lemma 2.2 there is an exact sequence of Λ-modules of finite type

0 ⟶ Λy ⟶ X_{∞} ⟶ X_{∞}/Λy ⟶ 0.

For every surjective character

χ : Γ ⟶ μ_{p^{n}}

we put c(χ) = n and denote by

e_{χ} = \frac{1}{p^{n}} ∑_{g∈Γ/Γ_{n}} χ(g)^{−1}g ∈ Q_{p}(μ_{p^{n}})[Γ/Γ_{n}]

the corresponding idempotent. Denote by A the (finite) set of integers n ≥ 1 such that

\frac{ω_{n}}{ω_{n−1}} ∈ Supp_{Λ}((X_{∞}/Λy)_{tors})

(where ω_{n} = γ^{p^{n}} − 1, as usual). If c(χ) ∉ A, then

e_{χ} (Q_{p}(μ_{p^{n}}) · y mod ω_{n}X_{∞}) ≠ 0

in (X_{∞})_{Γ_{n}} ⊗_{Z_{p}} Q_{p}(μ_{p^{n}}), hence also

e_{χ} (Q_{p}(μ_{p^{n}}) · y_{n}) ≠ 0

in X(E/K_{n}) ⊗_{Z_{p}} Q_{p}(μ_{p^{n}}), by Lemma 2.1(iii).

According to a mild generalization of the main result of [Be-Da],

e_{χ} (Q_{p}(μ_{p^{n}}) · y_{n}) ≠ 0 ⟹ e_{χ} (Q_{p}(μ_{p^{n}}) · y_{n}) = e_{χ} (X(E/K_{n}) ⊗_{Z_{p}} Q_{p}(μ_{p^{n}})) .

Putting everything together and again appealing to Lemma 2.1(iii), we see that

rk_{Z_{p}}((X_{∞})_{Γ_{n}}) ≤ p^{n} + O(1),

hence r ≤ 1 in Lemma 2.1(iv). However, r ≠ 0 by Lemma 2.2. 2.4 Recall that finite Λ-modules are also called pseudo-null. Denote by (_{Λ}Mod) the category of all Λmodules and by (_{Λ}Mod)/(ps − null) the category obtained from (_{Λ}Mod) by inverting all morphisms which have pseudo-null both kernel and cokernel. This is again an abelian category.

Lemma 2.5. In (_{Λ}Mod)/(ps − null) there is an exact sequence

0 ⟶ Y ⊕ Y ⊕ Z ⟶ Ŝ_{∞} ⟶ Λ ⟶ 0

and an isomorphism

Z ⟶^{∼} ⨁_{i=1}^{k}(Λ/p^{m_{i}}Λ).

Proof. Combining Lemma 2.3 with Lemma 2.1(ii) we get an exact sequence in (_{Λ}Mod)/(ps − null)

0 ⟶ (Ŝ_{∞})_{tors} ⟶ Ŝ_{∞} ⟶ Λ ⟶ 0.

The duality results of [Ne] imply that

(Ŝ_{∞})_{tors} ⟶^{∼} Y ⊕ Y ⊕ Z

in (_{Λ}Mod)/(ps − null), with Z as in the statement of the Lemma (in fact, the condition (Heeg) implies that Z itself is pseudo-null, at least if p > 2; however, we do not need this fact).

3. Main Results

Theorem B. Under the assumptions (Heeg) and p |̸ Na_{p} we have

s_{p}(E/K) ≡ 1 ≡ r_{an}(E/K) (mod 2).

Proof. First of all, (Heeg) implies that r_{an}(E/K) is odd. Lemma 2.5 together with Lemma 2.1(i) give an exact sequence

0 ⟶ (Y_{Γ} ⊕ Y_{Γ}) ⊗_{Z_{p}} Q_{p} ⟶ Hom_{Z_{p}}(X(E/K), Q_{p}) ⟶ Q_{p} ⟶ 0,

hence

s_{p}(E/K) = 1 + 2 rk_{Z_{p}}(Y_{Γ}) ≡ 1 (mod 2).

Theorem A. Let E be an elliptic curve over Q with good ordinary reduction at p. Then

r_{an}(E/Q) ≡ s_{p}(E/Q) (mod 2).

Proof. For every K = Q(\sqrt{D}) satisfying (Heeg) we have

\begin{aligned}s_{p}(E/K) & = s_{p}(E/Q) + s_{p}(E_{D}/Q) \\ r_{an}(E/K) & = r_{an}(E/Q) + r_{an}(E_{D}/Q), \end{aligned}

where

E_{D} : Dy^{2} = x^{3} + Ax + B

is the quadratic twist of E over K. We distinguish two cases: (I) r_{an}(E/Q) is odd. According to [Wa] there exists K satisfying (Heeg) such that

r_{an}(E_{D}/Q) = 0.

Results of Kolyvagin [Ko] then imply s_{p}(E_{D}/Q) = 0, hence

s_{p}(E/Q) = s_{p}(E/K) ≡ r_{an}(E/K) = r_{an}(E/Q) (mod 2)

by Theorem B. (II) r_{an}(E/Q) is even. Choose any K satisfying (Heeg) and p |̸ D. Applying the result of Case (I) to E_{D} (which has good ordinary reduction at p), we obtain

s_{p}(E_{D}/Q) ≡ r_{an}(E_{D}/Q) (mod 2),

hence

s_{p}(E/Q) ≡ 1 − s_{p}(E_{D}/Q) ≡ 1 − r_{an}(E_{D}/Q) ≡ r_{an}(E/Q) (mod 2),

again using Theorem B.

4. Higher dimensional quotients of J_{0}(N)

Results of Section 3 can be generalized as follows. 4.1 Let f = ∑_{n≥1} a_{n}q^{n} ∈ S_{2}(Γ_{0}(N)) be a normalized (a_{1} = 1) newform. Put F (f) = Q(a_{1}, a_{2}, ⋯); this is a totally real number field. Let A be a quotient abelian variety of J_{0}(N) corresponding to f; it has dimension [F (f) : Q], is defined over Q and is unique up to isogeny. One has an embedding ι : F (f) ↪ End(A) ⊗ Q; for each n ≥ 1, ι(a_{n}) is induced by the Hecke correspondence T (n) : J_{0}(N) ⟶ J_{0}(N) (with respect to the Albanese functoriality). 4.2 For each prime number p, V_{p}(A) = T_{p}(A) ⊗_{Z_{p}} Q_{p} is a free module of rank 2 over F (f) ⊗_{Q} Q_{p} = ⨁_{𝔭|p} F (f)_{𝔭}. Fix a prime 𝔭 above p in F (f); then the F (f)_{𝔭}-component V_{𝔭}(A) of V_{p}(A) defines a twodimensional Galois representation

G_{Q} = G(\overline{Q}/Q) ⟶ Aut_{F (f)_{𝔭}}(V_{𝔭}(A)) ⟶^{∼} GL_{2}(F (f)_{𝔭})

which is unramified outside Np and satisfies

det(1 − X · F r_{geom}(ℓ)|V_{𝔭}(A)) = 1 − a_{ℓ}ℓ^{−1}X + ℓ^{−1}X^{2}

for all prime numbers ℓ |̸ Np. 4.3 Let O_{𝔭} be the ring of integers of F (f)_{𝔭}. For every number field F, denote by S_{𝔭}(A/F) the 𝔭-primary part of the Selmer group S_{p}(A/F) = \underrightarrow{lim}_{n} S(A/F, p^{n}). This is an O_{𝔭}-module of cofinite type; denote by s_{𝔭}(A/F)

its O_{𝔭}-corank. Note that S_{𝔭}(A/F) depends only on the G(\overline{Q}/F)-module V_{𝔭}(A)/T_{𝔭}(A), where T_{𝔭}(A) is the O_{𝔭}-component of T_{p}(A), as it coincides with a Bloch-Kato Selmer group

S_{𝔭}(A/F) = H_{f}^{1}(F, V_{𝔭}(A)/T_{𝔭}(A)).

The role of r_{an}(E/F) is played by the order of vanishing

r_{an}(f, F) = ord_{s=1}L(f ⊗ F, s).

Under the assumptions (Heeg), (G) and

(Ord′)  ord_{𝔭}(a_{p}) = 0,

the arguments in Section 2 go through for the O_{𝔭}-modules S_{𝔭}(A/K_{n}) (Mazur’s control Theorem has to be replaced by a purely cohomological “control theorem”; see [Gre]). Similarly, all of the arguments of Section 3 work, if we replace the reference [Ko] by [KoLo]. The final results are the following.

Theorem B’. Under the assumptions (Heeg), p |̸ N and ord_{𝔭}(a_{p}) = 0 we have

s_{𝔭}(A/K) ≡ 1 ≡ r_{an}(f, K) (mod 2).

Theorem A’. Assume that p |̸ N and ord_{𝔭}(a_{p}) = 0. Then

ord_{s=1}L(f, s) ≡ s_{𝔭}(A/Q) (mod 2).

More general results can be deduced by applying the techniques of [NePl]. This will be discussed in a separate publication.

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Department of Pure Mathematics and Mathematical Statistics University of Cambridge Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB UK

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