Non-triviality of families of Heegner points and ranks of Selmer groups over anticyclotomic towers

Massimo Bertolini^{∗} Henri Darmon^{∗∗}

1 The result

Let E/ℚ be a modular elliptic curve of conductor N, and let K be an imaginary quadratic field of discriminant prime to N. Assume that E is semistable at all the prime divisors of N which are inert in K, and that the Hasse-Weil L-function L(E/K, s) vanishes to even order at s = 1. Since the sign of the functional equation of L(E/K, s) is −ϵ(N), where ϵ is the Dirichlet character attached to K (see [GZ], p. 71), it follows that the number of primes dividing N and inert in K is odd. Fix such a prime, say p, throughout the paper.

Let K_{∞} be the anticyclotomic ℤ_{p}-extension of K, and let Γ ≃ ℤ_{p} be its Galois group over K. Write Λ for the Iwasawa algebra ℤ_{p}[[Γ]]. The field K_{∞} is a Galois extension of ℚ, and the generator τ of Gal(K/ℚ) acts on Γ by the rule τγτ = γ^{−1} for all γ ∈ Γ. This property characterizes K_{∞} among the ℤ_{p}-extensions of K. Denote by Sel_{p^{∞}}(E/K_{∞}) the p-primary Selmer group of E over K_{∞}. It is a cofinitely generated Λ-module (i.e., its Pontryagin dual is a finitely generated Λ-module). It sits in the descent exact sequence

0 → E(K_{∞}) ⊗ ℚ_{p}/ℤ_{p} → Sel_{p^{∞}}(E/K_{∞}) → � (E/K_{∞})_{p^{∞}} → 0,

where � (E/K_{∞}) denotes the Shafarevich-Tate group of E over K_{∞}.

This note combines the results of [BD2] with techniques of Iwasawa theory to prove the following theorem.

Theorem 1.1 If L(E/K, 1) is non-zero, then the Λ-corank of Sel_{p^{∞}}(E/K_{∞}) is equal to 1. More precisely, E(K_{∞})⊗ℚ_{p}/ℤ_{p} has Λ-corank equal to 1, and � (E/K_{∞})_{p^{∞}} is a cotorsion Λ-module.

Remark 1.2

1. If χ : Γ → ℂ^{×} is a complex character of finite order which is ramified at p, the sign of the functional equation of L(E/K, χ, s) is −ϵ(N/p) = −1. One expects that L′(E/K, χ, s) is non-zero for almost all characters χ as above. Assuming this, theorem 1.1 is predicted by the Birch and Swinnerton-Dyer conjecture applied to the finite layers of the extension K_{∞}.

2. As explained in section 3, the proof of theorem 1.1 is achieved by showing along the way a non-triviality result for the family of Heegner points defined over K_{∞}. See theorem 3.2 for the precise statement. The proof of theorem 3.2 rests on one of the main results of [BD2].

∗Partially supported by grants from GNSAGA (CNR), MURST, and the European Community.

∗∗Partially supported by grants from FCAR, NSERC, and by an Alfred P. Sloan research award.

2 An upper bound for the corank of Sel_{p^{∞}}(E/K_{∞})

This section is devoted to the proof of the following:

Proposition 2.1 If L(E/K, 1) is non-zero, then the Λ-corank of Sel_{p^{∞}}(E/K_{∞}) is ≤ 1.

Proposition 2.1 is a consequence of the next two propositions.

Proposition 2.2 If L(E/K, 1) is non-zero, then corank_{ℤ_{p}}Sel_{p^{∞}}(E/K) = 0.

Proof. If L(E/K, 1) is non-zero, a theorem of Kolyvagin (see [K], Theorem A) shows that E(K) and the Shafarevich-Tate group � (E/K) of E over K are finite. In particular, the ℤ_{p}-corank of Sel_{p^{∞}}(E/K) is zero.

The next proposition does not depend on the assumption that L(E/K, 1) is nonzero.

Proposition 2.3 corank_{ℤ_{p}}Sel_{p^{∞}}(E/K_{∞})^{Γ} ≤ corank_{ℤ_{p}}Sel_{p^{∞}}(E/K) + 1.

Proof of Proposition 2.1 The structure theory of discrete Λ-modules shows that

corank_{Λ}Sel_{p^{∞}}(E/K_{∞}) ≤ corank_{ℤ_{p}}Sel_{p^{∞}}(E/K_{∞})^{Γ}.

(See [M], ch. 1, or also [L], ch. 5, sec. 3, for details.) But the propositions 2.2 and 2.3 imply that the ℤ_{p}-corank of Sel_{p^{∞}}(E/K_{∞})^{Γ} is ≤ 1.

It remains to prove proposition 2.3. Write K_{n} for the subfield of K_{∞} having degree p^{n} over K, and G_{n} for the Galois group Gal(K_{n}/K). Let K_{n_{0}}, with n_{0} ≥ 0 be the maximal unramified extension of K contained in K_{∞}. Thus, K_{n_{0}} = K_{∞} ∩ H, where H is the Hilbert class field of K. Note that p is inert in K, totally split in the extension K_{n_{0}}/K and, for n > n_{0}, all the primes of K_{n_{0}} above p are totally ramified in K_{n}/K_{n_{0}}. Denote by q ∈ pℤ_{p} Tate’s p-adic period of E, and by Φ_{n} the group of connected components of E over K_{n} ⊗ ℚ_{p}. By Tate’s theory of p-adic uniformization, the group Φ_{n} is a G_{n_{0}}-module, isomorphic to (ℤ/c_{p}p^{n−n_{0}}ℤ)[G_{n_{0}}], where c_{p} := ord_{p}(q).

Lemma 2.4 The torsion subgroup E(K_{∞})_{tors} of E(K_{∞}) is finite.

Proof. Let q_{1} and q_{2} be primes of good reduction for E which are inert in K. Then q_{1} and q_{2} are totally split in K_{∞}/K, and E(K_{∞})_{tors} injects in the finite group E(𝔽_{q_{1}^{2}}) ⊕ E(𝔽_{q_{2}^{2}}).

Proof of Proposition 2.3 The proof is an application of the inflation-restriction sequence. First, note the exact sequence

H^{1}(Γ, E_{p^{∞}}(K_{∞})) → H^{1}(K, E_{p^{∞}}) → H^{1}(K_{∞}, E_{p^{∞}})^{Γ} → H^{2}(Γ, E_{p^{∞}}(K_{∞})).

The first term is finite, by lemma 2.4, and the fourth one is zero, since Γ has cohomological dimension equal to 1. If ℓ is a rational prime, there are also the local exact sequences

H^{1}(Γ, E(K_{∞,ℓ}))_{p^{∞}} → H^{1}(K_{ℓ}, E)_{p^{∞}} → H^{1}(K_{∞,ℓ}, E)_{p^{∞}}^{Γ},

where K_{ℓ} denotes K ⊗ ℚ_{ℓ} and K_{∞,ℓ} denotes ⋃_{n}(K_{n} ⊗ ℚ_{ℓ}). If ℓ ≠ p, the cohomology group H^{1}(Γ, E(K_{∞,ℓ})) is finite, since K_{∞} is unramified outside p. Moreover, if ℓ ∤ N, H^{1}(Γ, E(K_{∞,ℓ})) is zero, since E has good reduction at ℓ. (See [Mi], ch. 1.) The theory of p-adic uniformization can be used to prove that the group H^{1}(Γ, E(K_{∞,p}))_{p^{∞}} has ℤ_{p}-corank ≤ 1. One starts from the exact sequence of Γ-modules

0 → Q_{E} → K_{∞,p}^{×} → E(K_{∞,p}) → 0,

where Q_{E} denotes the lattice of p-adic periods of E. The action of Γ on Q_{E} factors through G_{n_{0}}, and Q_{E} is isomorphic to ℤ[G_{n_{0}}]. Taking cohomology of the above sequence shows that H^{1}(Γ, E(K_{∞,p})) injects in H^{2}(Γ, Q_{E}). Combining the exact sequence in cohomology induced by

0 → Q_{E} → Q_{E} ⊗ ℚ → Q_{E} ⊗ ℚ/ℤ → 0

with an inflation-restriction argument identifies H^{2}(Γ, Q_{E}) with the group of homomorphisms Hom(Gal(K_{∞}/K_{n_{0}}), (Q_{E} ⊗ ℚ/ℤ)^{G_{n_{0}}} ), which is isomorphic to ℚ_{p}/ℤ_{p}. Proposition 2.3 now follows from the snake lemma applied to the commutative diagram

0 ⟶ Sel_{p^{∞}}(E/K) ⟶ H^{1}(K, E_{p^{∞}}) ⟶ ∏_{ℓ} H^{1}(K_{ℓ}, E)_{p^{∞}}
↓  ↓  ↓
0 ⟶ Sel_{p^{∞}}(E/K_{∞})^{Γ} ⟶ H^{1}(K_{∞}, E_{p^{∞}})^{Γ} ⟶ ∏_{ℓ} H^{1}(K_{∞,ℓ}, E)_{p^{∞}}^{Γ},

where the vertical maps are restriction maps.

Remark 2.5

1. The ℤ_{p}-corank of the group H^{1}(Γ, E(K_{∞,p}))_{p^{∞}} considered in the proof of proposition 2.3 is in fact equal to 1. To see this, note the exact sequence

0 → H^{1}(Γ, E(K_{∞,p})) → H^{2}(Γ, Q_{E}) → H^{2}(Γ, K_{∞,p}^{×}) → H^{2}(Γ, E(K_{∞,p})) → 0.

The “Brauer group” H^{2}(Γ, K_{∞,p}^{×}) has ℤ_{p}-corank equal to 1, and it was already observed that H^{2}(Γ, Q_{E}) has ℤ_{p}-corank equal to 1. Let Ê(K_{p}) be the p-adic completion of E(K_{p}), and let ÛE(K_{p}) be the submodule of universal norms along the local extension K_{∞,p}/K_{p}. The ℤ_{p}-corank of H^{2}(Γ, E(K_{∞,p})) is equal to the ℤ_{p}- rank of Ê(K_{p})/ÛE(K_{p}). The theory of p-adic uniformization, combined with class field theory and the fact that the Tate period of E/K_{p} is a universal norm from K_{∞,p}^{×}, shows that Ê(K_{p})/ÛE(K_{p}) has ℤ_{p}-rank equal to 1. The claim follows.

2. Recall that proposition 2.2 is a special case of a theorem of Kolyvagin [K]. The condition L(E/K, 1) ≠ 0 is equivalent to L(E/ℚ, 1) ≠ 0 and L(E′/ℚ, 1) ≠ 0, where E′ denotes the quadratic twist of E by K. The opening step in Kolyvagin’s proof consists in choosing auxiliary imaginary quadratic fields F and F′ such that the first derivatives L′(E/F, 1) and L′(E/F′, 1) are both non-zero, and such that the primes dividing the conductors of E and E′ are split in F and F′, respectively. The proof is then achieved by proving the finiteness of the Selmer groups of E/ℚ and E′/ℚ one at a time, and deducing the finiteness of the Selmer group of E/K. A simpler approach to the proof of proposition 2.2 rests on the methods of [BD2], which allow to bound directly the p-primary Selmer group of E/K.

3 A lower bound for the corank of Sel_{p^{∞}}(E/K_{∞})

Theorem 1.1 is a consequence of the next proposition, combined with proposition 2.1.

Proposition 3.1 If L(E/K, 1) is non-zero, then the Λ-corank of E(K_{∞})⊗ℚ_{p}/ℤ_{p} is ≥ 1. In particular, the Λ-corank of Sel_{p^{∞}}(E/K_{∞}) is ≥ 1.

Some preliminary results are needed. Recall the integer n_{0} defined in the previous section. The field K_{∞} is contained in the union of all the ring class fields of K of p-power conductor. More precisely, for n > n_{0} let H_{n} be the ring class field of conductor p^{n+1−n_{0}}. Thus, H_{n} is an extension of the Hilbert class field H of degree e_{n} := p^{n−n_{0}}(p + 1)/u, where u is one half the order of the group of units of K. The field H_{n} is the smallest ring class field containing K_{n}.

For n > n_{0}, a Heegner point construction (which is described in [BD1], sec. 2.5) defines a collection of points β_{n} ∈ E(H_{n}), satisfying the compatibility relations

Trace_{H_{n+1}/H_{n}}β_{n+1} = β_{n}, Trace_{H_{n}/H}β_{n} = 0.

Set α_{n} := Trace_{H_{n}/K_{n}}β_{n} ∈ E(K_{n}). Thus,

Trace_{K_{n+1}/K_{n}}α_{n+1} = α_{n}, Trace_{K_{n}/K_{n_{0}}} α_{n} = 0.

Theorem 3.2 If L(E/K, 1) is non-zero, then there is an integer n_{1} > n_{0} such that α_{n} has infinite order for all n ≥ n_{1}.

Proof. Let Ψ_{n} denote the group of connected components of E over H_{n} ⊗ ℚ_{p}. The group Ψ_{n} is a Gal(H/K)-module, isomorphic to (ℤ/c_{p}e_{n})[Gal(H/K)]. Write β̄_{n}, resp. ᾱ_{n} for the natural image of β_{n} in Ψ_{n}, resp. of α_{n} in Φ_{n}. Moreover, set

β̄_{n}^{1} := Trace_{H/K} β̄_{n}, ᾱ_{n}^{1} := Trace_{K_{n_{0}} /K} ᾱ_{n}.

The operator Trace_{H_{n}/K_{n}} induces a surjective map t_{H_{n}/K_{n}} : Ψ_{n} → Φ_{n}. Note that

t_{H_{n}/K_{n}} β̄_{n}^{1} = ᾱ_{n}^{1}.

Theorem A of [BD2] relates the elements β̄_{n}^{1} to the special value L(E/K, 1). In particular, since L(E/K, 1) is non-zero, it implies that the order of β̄_{n}^{1} tends to infinity with n. The same property holds for the order of ᾱ_{n}^{1}, since the kernel of t_{H_{n}/K_{n}} is bounded independently of n. This shows that either the points α_{n} have infinite order for n sufficiently large, or the α_{n} are a collection of torsion points of unbounded order. But the second possibility is ruled out by lemma 2.4.

Corollary 3.3 The Mordell-Weil group E(K_{∞}) has infinite rank over ℤ.

Proof. Suppose instead that E(K_{∞}) has finite rank. Since E(K_{∞})_{tors} is finite by lemma 2.4, it follows that E(K_{∞}) is finitely generated. Thus, there is a positive integer n_{2} such that E(K_{∞}) = E(K_{n_{2}}), and such that the Heegner point α_{n} has infinite order for all n ≥ n_{2}. By the compatibility of the Heegner points under traces, one obtains that α_{n_{2}} = p^{n−n_{2}}α_{n} for all n ≥ n_{2}. But the point α_{n_{2}} has infinite order, and therefore it cannot be infinitely divisible in E(K_{n_{2}}).

Proof of Proposition 3.1 By corollary 3.3, E(K_{∞}) ⊗ ℚ_{p}/ℤ_{p} has infinite ℤ_{p}-corank. On the other hand, a cotorsion Λ-module has finite ℤ_{p}-corank, by the structure theory of discrete Λmodules ([M], ch. 1). This completes the proof of proposition 3.1, and of theorem 1.1.

The next result gives information on the growth of the Mordell-Weil groups E(K_{n}).

Proposition 3.4 If L(E/K, 1) is non-zero, then there is a sequence of integers ι_{n} having absolute value bounded independently of n such that

rank_{ℤ}E(K_{n}) = p^{n} + ι_{n}.

Proposition 3.4 follows from theorem 1.1 and theorem 3.2. More precisely, theorem 1.1 implies that rank_{ℤ}E(K_{n}) ≤ p^{n} + ι_{n}, for a bounded sequence of integers ι_{n}. By theorem 3.2, the Heegner points α_{n} yield a norm-compatible sequence of points of infinite order. The opposite inequality follows from the structure of the modules of universal norms over the layers of K_{∞}. See [B], ch. 2 and 3. The details of the proof are omitted.

Remark 3.5

1. With the other assumptions on E, K and p as in the rest of the paper, now assume that L(E/K, s) vanishes to even order at s = 1 and that L(E/K, 1) = 0. The first part of remark 1.2 suggests that in this setting Sel_{p^{∞}}(E/K_{∞}) still has Λ-corank equal to 1. Moreover, the Heegner point construction carries over, and for n > n_{0} provides a norm-compatible collection of points α_{n} ∈ E(K_{n}). A natural generalization (not yet proved) of the Gross-Zagier formula [GZ] to the derivatives L′(E/K, χ, 1) for ramified characters χ of Γ leads one to expect that again the point α_{n} has infinite order for n sufficiently large. (However, this cannot be shown by mapping α_{n} to the group of connected components Φ_{n} as in the proof of theorem 3.2, since L(E/K, 1) is zero.) In the remainder of this remark, assume that α_{n} has infinite order for n large enough. The proofs of proposition 3.1 and of corollary 3.3 show that the Λ-corank of Sel_{p^{∞}}(E/K_{∞}) is ≥ 1. In order to show the opposite inequality, the techniques of Iwasawa theory exploited in section 2 do not apply, since the ℤ_{p}-corank of Sel_{p^{∞}}(E/K) is not zero in the current setting. However, the results of [B], which generalize the methods of [K] to the study of the Selmer groups Sel_{p^{∞}}(E/K_{n}), can be used to show that also in this case the Λ-corank of Sel_{p^{∞}}(E/K_{∞}) is equal to 1.

2. The following slight generalization of theorem 3.2 can be proved with similar techniques. Assume that L(E/K, ψ, 1) is non-zero for a character ψ of Gal(K_{n_{0}}/K). Then, there is a positive integer n_{1} > n_{0} such that α_{n} has infinite order for all n ≥ n_{1}.

3. Theorem 3.2 is interesting in its own right, as it gives a way of constructing families of Heegner points over anticyclotomic ℤ_{p}-extensions having infinite order. The standard technique for producing families of this kind applies to imaginary quadratic fields K such that −ϵ(N) is −1, and to ℤ_{p}-extensions K_{∞} such that p is a good ordinary prime for E, or a multiplicative prime which is not inert in K. In this setting, one can construct a family of Heegner points α_{n} ∈ E(K_{n}) for all n ≥ 0. (See [BD1], sec. 2.5 for more details.) Assume that α_{0} has infinite order in E(K). (If all primes dividing N are split in K, this is equivalent to the condition that L′(E/K, 1) is non-zero, by the Gross-Zagier formula.) Then, the compatibility of the Heegner points under traces implies that α_{n} has infinite order for all n.

References

[B] M. Bertolini, Selmer groups and Heegner points in anticyclotomic ℤ_{p}-extensions, Compositio Math. 99, n. 2 (1995) 153-182.

[BD1] M. Bertolini, H. Darmon, Heegner points on Mumford-Tate curves, Inv. Math. 126, Fasc. 3 (1996) 413-456.

[BD2] M. Bertolini, H. Darmon, A rigid-analytic Gross-Zagier formula and arithmetic applications, (With an Appendix by B. Edixhoven), Ann. Math., to appear.

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

[K] V.A. Kolyvagin, Euler Systems, in: The Grothendieck Festschrift, P. Cartier, et al., eds., vol. II, Progr. in Math. 87, Birkhäuser (1990) 435-483.

[L] S. Lang, Cyclotomic fields I and II, combined 2nd ed., Springer-Verlag (1990).

[M] Yu. Manin, Cyclotomic Fields and Modular Curves, Engl. Transl.: Russian Math. Surveys 26 (1971) 7-78.

[Mi] J.S. Milne, Arithmetic duality theorems, Perspective in Math., Academic Press