Jan Nekovář & Norbert Schappacher

On the asymptotic behaviour of Heegner points

Abstract. We prove that all but finitely many Heegner points on a given (modular) elliptic curve (or, more generally, on a given quotient of the modular Jacobian variety J_{0}(N)) are of infinite order in the Mordell-Weil group where they naturally live, i.e., over the corresponding ring class field.

1. Notations

1.1 Let N > 1. The quasi-projective curve Y_{0}(N) defined over Q classifies isogenies [E ⟶^{λ} E′] of elliptic curves with cyclic kernel kerλ ≅ Z/NZ. Over C, the isogeny

[C/Z+Zτ ⟶^{[×N]} C/Z+ZNτ] corresponds to the point Γ_{0}(N)·τ of the quotient Γ_{0}(N)\H = Y_{0}(N)(C) of the complex upper half-plane H. The dual isogeny [C/Z+ZNτ ⟶ C/Z+Zτ] induced by the identity on C corresponds to the point Γ_{0}(N) · w_{N}(τ), where w_{N}(τ) = \frac{−1}{Nτ} denotes the Fricke involution.

We write as usual X_{0}(N) for the smooth projective curve defined over Q which is the compactification of Y_{0}(N) and classifies cyclic N-isogenies between generalized elliptic curves. And we denote by J_{0}(N) the Jacobian of X_{0}(N). We embed X_{0}(N) in J_{0}(N) by sending ∞ to 0, where ∞ is the cusp corresponding to the Néron polygon with a single side.

Finally, we fix a nonzero quotient defined over Q, J_{0}(N) ⟶ A of the abelian variety J_{0}(N), and we let X_{0}(N) ⟶^{π_{A}} A be the nonconstant morphism defined over Q which arises from composing the fixed embedding of X_{0}(N) into J_{0}(N) with the projection of J_{0}(N) onto A. The following results will therefore apply in particular to the case of a (modular) elliptic curve A over Q.

1.2 Let K be an imaginary quadratic field such that all prime numbers dividing N split in K. Note right away that, for any given N, there are infinitely many K satisfying this so-called “Heegner-condition” (which was introduced by Birch). It implies that there exists an ideal 𝔫 of the ring of integers 𝔬_{K} such that one has 𝔬_{K}/𝔫 ≅ Z/NZ.

More explicitly, let D be the discriminant of K and let \sqrt{D} be the square root of D which belongs to H. Then 𝔬_{K} = Z + Zα and 𝔫 = NZ + Zα, with α = \frac{−B+\sqrt{D}}{2} such that α^{2} + Bα + AN = 0 and B^{2} − 4AN = D. Or again, 𝔬_{K} = Z + Z\frac{NA}{α}, 𝔫^{−1} = Z + Z\frac{A}{α}.

For every f ≥ 1 relatively prime to N, we write 𝔬_{f} = Z+f𝔬_{K} for the order of conductor f in K. Its discriminant is D_{f} = Df^{2}. And we put 𝔫_{f} = 𝔬_{f} ∩ 𝔫. Since (f, N) = 1, 𝔫_{f} is a proper 𝔬_{f}-ideal, that is to say, 𝔬_{f} = {x ∈ K | x𝔫_{f} ⊆ 𝔫_{f}}.

More explicitly, we have that 𝔬_{f} = Z + Zα_{f} and 𝔫_{f} = NZ + Zα_{f} for α_{f} = fα. Thus

α_{f} = ^{\frac{−B_{f} +\sqrt{D_{f}}}{2}} , α_{f}^{2} + B_{f}α_{f} + A_{f}N = 0 with B_{f} = fB, A_{f} = f^{2}A, B_{f}^{2} − 4A_{f}N = D_{f} = f^{2}D. Or again, 𝔬_{f} = Z + Z\frac{NA_{f}}{α_{f}} = Z + Z\frac{NfA}{α}, 𝔫_{f}^{−1} = Z + Z\frac{A_{f}}{α_{f}} = Z + Z\frac{fA}{α}.

Given our choice of 𝔫, the Heegner point y_{f} of conductor f on Y_{0}(N) is defined to be the point represented by the isogeny [C/𝔬_{f} ⟶ C/𝔫^{−1}] which is induced by the identity on C. Its image π_{A}(y_{f}) ∈ A(K_{f}) is called the Heegner point of conductor f on A. The point w_{N}(y_{f}) is represented by the dual isogeny

[C/𝔫^{−1} ⟶ C/𝔬_{f}] = [C/Z + Z\frac{fA}{α} ⟶ C/Z + Z\frac{NfA}{α}],

which is induced by multiplication by N. The point w_{N}(y_{f}) therefore corresponds to the point Γ_{0}(N) · τ_{f} of Γ_{0}(N)\H, where τ_{f} = −\frac{fA}{α} = ^{\frac{f(B+\sqrt{D})}{2N}}.

1.3 The field of definition K_{f} of y_{f} — and therefore also that of π_{A}(y_{f}) — is the field generated over K by the j-invariants of elliptic curves with complex multiplication by the order 𝔬_{f}. It is the ring class field of conductor f of K, i.e., the abelian extension of K which is unramified outside of f and in which a prime ideal of K not dividing f is totally split if and only if it is not only principal, but can be generated by an element which, modulo f, is congruent to a rational number.

If f and f′ are relatively prime, then K_{f} and K_{f′} are linearly disjoint over the Hilbert class field K_{1} of K. Also, K_{ff′} is the compositum of K_{f} and K_{f′}. The same holds true for the rings of integers.

1.4 We can now state our main finiteness result, in which N > 1 is a fixed positive integer, D varies over the discriminants of imaginary quadratic fields K satisfying the Heegner condition of 1.2, and f varies over positive integers prime to N.

1.5 Theorem. There are only a finite number of pairs (D, f) as above such that the point π_{A}(y_{f}) ∈ A(K_{f}) is a torsion point.

2. The first finiteness result Theorem 1.5 will result from a quantitative version of the following proposition—see 3.3 below. Proposition 2.1 has already been used in the literature—see [3].

2.1 Proposition. There exists f_{0} > 0, depending on the level N and the discriminant D, such that for every f > f_{0} relatively prime to N, the point π_{A}(y_{f}) ∈ A(K_{f}) is a point of infinite order on the abelian variety A.

The proof proceeds in three steps, 2.2 – 2.4. 2.2 Let K_{∞} = ⋃_{f≥1} K_{f}. We show that the subgroup of torsion points A(K_{∞})_{tors} is finite. Let ℓ be a prime number which is inert in K. Every prime ideal λ of K_{f} above ℓ has norm Nλ = ℓ^{2}. — In fact, writing f = ℓ^{a}f′ with f′ not divisible by ℓ one sees that the prime divisors of ℓ𝔬_{K_{1}} are totally ramified in K_{ℓ^{a}}/K_{1} and ℓ𝔬_{K} splits completely in K_{f′}.

Since ℓ does not divide N (the primes dividing N split in K), the abelian variety A, as any quotient of J_{0}(N), has good reduction at λ, and for every f ≥ 1 prime to N the torsion subgroup of order prime to ℓ of A(K_{f}) reduces injectively modulo λ. This gives

card(A(K_{f})_{tors}^{non−ℓ}) | card(Ã_{λ}(F_{ℓ^{2}})).

Taking a second prime ℓ′, distinct from ℓ, which remains inert in K we see that

card(A(K_{f})_{tors}) | card(Ã_{λ}(F_{ℓ^{2}})) · card(Ã_{λ′}(F_{ℓ′}^{2})[ℓ^{∞}]),

where the suffix [ℓ^{∞}] signifies taking the ℓ-primary part. This proves 2.2.

2.3 Over C, π_{A} lifts to a holomorphic mapping F on the completed upper half-plane.

H ∪ P^{1}(Q)  ⟶^{F}  C^{dim A}
↓  ↓
Γ_{0}(N)\H ∪ P^{1}(Q) ⟶^{π_{A}} A(C) ≅ C^{dim A}/Λ

We fix F by requiring that F (∞) = 0. Then, for every m ≥ 1, there exists M_{m} ∈ R such that for all τ ∈ H with Im(τ) > M_{m} we have:

0 < ||F (τ)|| < \frac{1}{m} inf_{0≠γ∈Λ} ||γ||.

2.4 We now conclude the proof of proposition 2.1 first under the additional hypothesis that the Fricke involution w_{N} induces an automorphism of A: According to 2.2, put m = card(A(K_{∞})_{tors}), and pick M_{m} for this choice of m as in 2.3. Let f_{0} = \frac{2N}{|D|^{1/2}} M_{m}. Then we find, in the notation introduced at the end of 1.2 above, that for every integer f greater than f_{0} and prime to N one has Im(τ_{f}) = \frac{f|D|^{1/2}}{2N} > M_{m}. 2.3 now ensures that the point π_{A}(w_{N}(y_{f})) ∈ A(K_{f}), which corresponds to F (τ_{f}), is not an m-torsion point on the Jacobian. In view of our choice of m it has to be of infinite order. Since the involution w_{N} induces an automorphism of A, the same holds for the Heegner point π_{A}(y_{f}) itself.

Finally, in order to prove the proposition for an arbitrary quotient A of J_{0}(N), not necessarily invariant under w_{N}, one only has to modify the preceding argument by applying 2.3 to w_{N}(A) rather than A. 2.5 Remarks. (1) For f as above, put G_{f} = Gal(K_{f}/K). Assume that A is of dimension 1, i.e., a (modular) elliptic curve. For a ring class character χ ∈ Ĝ_{f} of conductor dividing f, we define the L-function of A twisted by χ by the following Euler product, which converges for Re(s) > 3/2.

L(A/K, χ, s) = ∏_{𝔭⊂𝔬_{K}} det(1 − Frob_{𝔭} · χ(Frob_{𝔭})N𝔭^{−s} | V_{l}(A)_{I_{𝔭}})^{−1}

Here, Frob_{𝔭} denotes the arithmetic Frobenius, and we put χ(Frob_{𝔭}) = 0 for 𝔭 | f. It follows from the Heegner condition of 1.2 that the order of L(A/K, χ, s) at s = 1 is odd and therefore that L(E/K, χ, 1) = 0. Write ⟨ , ⟩_{f} for the sesquilinear extension to A(K_{f}) ⊗_{Z} C of the canonical Néron-Tate height pairing on A(K_{f}). Finally, put e_{χ} = \frac{1}{#G_{f}} ∑_{σ∈G_{f}} χ(σ)^{−1}σ. Then the following formula is conjectured to hold, with the real period ω_{A} and some nonzero rational number r = r(D, f).

(2.6)  L′(A/K, χ, 1) = r \frac{ω_{A}}{\sqrt{|D|}} ⟨e_{χ}y_{f}, e_{χ}y_{f}⟩_{f}

In the particular case where f = 1, this is the well-known theorem of B.H. Gross and D. Zagier [4]. The generalization 2.6 is not completely proved yet. Assuming it, our theorem shows that, for every sufficiently big f, there is at least one ring class character χ ∈ Ĝ_{f} such that L′(A/K, χ, 1) ≠ 0. On the other hand, guided by results of Rohrlich’s [11], one may wonder whether, given K, there are only a finite number of pairs (f, χ), with f ≥ 1 prime to N and χ ∈ Ĝ_{f} a character of conductor f, such that L′(E/K, χ, 1) = 0.

(2) The second step in the above proof transfers to our situation (and simplifies) an argument given in a more general context by S. Bloch and C. Schoen — see [12].

(3) The above proposition generalizes an analogous result proved in a particular case by P.F. Kurčanov [6]. The proof given by Kurčanov is certainly different from ours, but does rely on the similar principles.

3. Effectivity questions 3.1 It follows from the Pólya-Vinogradov theorem that we may always find distinct prime numbers ℓ, ℓ′ as in 2.2 such that ℓ, ℓ′ < |D|^{c} for an absolute constant c. This gives the following bound for m.

card(A(K_{∞})_{tors}) ≤ card(Ã_{λ}(F_{ℓ^{2}})) · card(Ã_{λ′}(F_{ℓ′}^{2})) ≤ ((ℓ + 1)(ℓ′ + 1))^{2 dim A} ≤ |D|^{4c dim A}

3.2 In 2.3, we may always take

M_{m} = c_{1} + \frac{log(m)}{2π},

for some constant c_{1} depending on the map π_{A}. Indeed, a local parameter at ∞ is given by e^{2iπτ}, and |e^{2iπτ} | = e^{−2πIm(τ)}.

3.3 Proof of theorem 1.5. Let us put together 2.4, 3.1 and 3.2. We see that there exists an absolute constant c_{0} and a constant c_{1} depending on A such that for all positive integers f prime to N and satisfying

f > \frac{2N}{|D|^{1/2}} \{c_{1} + c_{0} dim A log |D|\},

the Heegner point π_{A}(y_{f}) has infinite order in A. For |D| sufficiently big, this inequality holds for any f ≥ 1. This concludes the proof of theorem 1.5.

4. The anticyclotomic Z_{p}-extension and Mazur’s module of Heegner points We will now restrict to the case where A/Q is of dimension 1, i.e., A is a (modular) elliptic curve, assumed to be of conductor N. Let p be a prime number which stays prime in K, and such that a_{p} = p + 1 − #(Ã_{p}(F_{p})), the eigenvalue of the Hecke-operator T_{p} on A, is not divisible by p. In other words, assume that p is ordinary for A.

Let H_{∞} = ⋃ H_{n} be the anticyclotomic Z_{p}-extension of our fixed imaginary quadratic field K, i.e., H_{∞} is the unique Z_{p}-extension of K contained in K_{p^{∞}} = ⋃ K_{p^{n+1}}. We consider the Heegner points z_{n} = tr_{K_{p^{n+1}} /H_{n}}(π_{A}(y_{p^{n+1}})) ∈ A(H_{n}) and following Mazur [9], no. 19, we write E_{∞} for the projective limit (with respect to the trace maps) of the submodules E_{n} of (E(H_{n}) ⊗ Z_{p})/(torsion) which are generated by all the conjugates of z_{n}. For n ≥ 2, the points on different levels are linked by the following distribution relations, which are immediate consequences of [10], p. 430.

tr_{H_{n+1}/H_{n}}(z_{n+1}) = a_{p}z_{n} − z_{n−1}

E_{∞} is an Iwasawa module, i.e., a finitely generated module over Λ = Z_{p}[[Γ]] = Z_{p}[[T]], where Γ = Gal(H_{∞}/K). Moreover, as Mazur observed [9], no. 19, E_{∞} is a Λ-module (in fact, free) of rank 1 if and only if z_{n} is a point of infinite order for (any, and thus for all) n >> 0. Mazur conjectured that this is always the case. Note that this conjecture is a special instance of the question formulated at the end of remark 2.5(1).

Recall the definitions of the relevant Selmer groups. For any number field F and any m ≥ 2, the m-Selmer group of A over F is defined to be the torsion group

Sel_{m}(A/F) = ker(H^{1}(F, A_{m}) ⟶ ∏_{v} H^{1}(F_{v}, A)_{m}).

Via direct limits, we obtain Q_{p}/Z_{p}-modules

Sel_{p^{∞}}(A/F) = \underrightarrow{lim} Sel_{p^{n}}(A/F), Sel_{p^{∞}}(A/H_{∞}) = \underrightarrow{lim} Sel_{p^{∞}}(A/H_{n}).

4.1 Theorem. If z_{n} is a point of infinite order for n >> 0, then the Selmer group Sel_{p^{∞}}(A/K) contains a subgroup isomorphic to Q_{p}/Z_{p}.

The following immediate consequence of this is particularly interesting when the order of vanishing of L(A/Q, s) at s = 1 is at least 2.

4.2 Corollary. Assume that the p-part \underline{III}(A/K)(p^{∞}) of the Tate-Šafarevič group of A over K is finite and that z_{n} is a point of infinite order for n >> 0. Then dim A(K)⊗Q ≥ 1.

4.3 Proof of theorem 4.1 (argument suggested by K. Rubin). Put H_{∞} = \underrightarrow{lim} E_{n} ⊗ Q_{p}/Z_{p}. By the assumption of the theorem, E_{∞} is a rank one Λ-module, so its coinvariants (E_{∞})_{Γ} admit a quotient isomorphic to Z_{p}. Therefore the invariants H_{∞}^{Γ} ⊂ Sel_{p^{∞}}(A/H_{∞})^{Γ} ⊂ H^{1}(H_{∞}, A_{p^{∞}})^{Γ} contain a copy of Q_{p}/Z_{p}. However, in our case the p^{∞}-Selmer group along the anticyclotomic extension is “controlled” in the sense that the canonical map Sel_{p^{∞}}(A/K) ⟶ Sel_{p^{∞}}(A/H_{∞})^{Γ} has finite kernel and cokernel. This control can be wielded locally, the only interesting place being at p. More precisely, write the local descent sequences

0 ⟶ A((H_{n})_{p}) ⊗ Q_{p}/Z_{p} ⟶ H^{1}((H_{n})_{p}, A_{p^{∞}}) ⟶ (A((H_{n})_{p}) ⊗ Z_{p}) ⟶ 0
↑  ↑  ↑
0 ⟶ A((K)_{p}) ⊗ Q_{p}/Z_{p} ⟶ H^{1}((K)_{p}, A_{p^{∞}}) ⟶ (A((K)_{p}) ⊗ Z_{p}) ⟶ 0

where the last vertical arrow is given by the dual of the trace map on local points. It’s kernel is bounded independently of n since p is ordinary for A and thus the universal local traces have finite index in the local points A((K)_{p}). This control theorem is due to Mazur [8]; for our situation see Manin [7], Thm. 4.5 together with Cor. 4.11(a).

4.4 Remarks. (1) Bertolini [1] (see also [2]) has established an Iwasawa theoretic analogue of Kolyvagin’s method to prove in particular (under additional hypotheses on the prime p) that, if E_{∞} is indeed of rank 1, then it agrees with the dual of the Selmer group up to a torsion module for which he can exhibit an annihilating power series.

(2) 4.1 provides an example of how the behaviour of higher Heegner points (granting the non-triviality assumption) govern the arithmetic of E over K, and therefore over Q. Another striking instance of such a relationship was given by Kolyvagin in [5]. It also depends on an initial non-triviality conjecture, and is more like an ℓ-adic descent, for some fixed prime ℓ different from the primes entering into the conductors of the Heegner points. It would be interesting to be able to combine these two theories.

References

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

[2] M. Bertolini, Growth of Mordell-Weil groups in anticyclotomic towers; in: Arithmetic geometry (Cortona, 1994; F. Catanese, ed.), Sympos. Math., XXXVII, Cambridge Univ. Press, Cambridge, 1997, pp. 23–44

[3] M. Bertolini and H. Darmon, Non-triviality of families of Heegner points and ranks of Selmer groups over anticyclotomic towers. J. Ramanujan Math. Soc. 13 (1998), 15–24

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

[5] V.A. Kolyvagin, On the structure of Selmer groups, Mathematische Annalen 291 (1991), 253–259

[6] P.F. Kurčanov, Elliptic Curves of infinite rank over Γ-extensions, Math. USSR Sbornik 19 (1973), 320–324

[7] Yu. Manin, Krugovye polja i moduljarnye krivye, Uspechi Mat. Nauk, 26 (1971), 7– 71. English translation: Cyclotomic fields and modular curves, Russian Math. Surveys 26, no. 6 (1971), 7–78

[8] B. Mazur, Rational points on abelian varieties with values in towers of number fields, Inventiones Math. 18 (1972), 183–266

[9] B. Mazur, Modular Curves and Arithmetic, Proc. ICM Warszawa 1983, vol. I, pp. 185–211

[10] B. Perrin-Riou, Fonctions L p-adiques, théorie d’Iwasawa et points de Heegner, Bull. Soc. Math. France 115 (1987), 399–456 [11] D. Rohrlich, Nonvanishing of L-functions for GL(2), Inventiones Math. 97 (1989),

381–403

[12] C. Schoen, Complex multiplication cycles on elliptic modular threefolds, Duke Math. J. 53 (1986), 771–794

Jan Nekovář Norbert Schappacher D.P.M.M.S. U.F.R. de mathématique et d’informatique University of Cambridge Université Louis Pasteur 16 Mill Lane 7, rue René Descartes Cambridge CB2 1SB, UK 67084 Strasbourg Cedex, France

nekovar@dpmms.cam.ac.uk                           schappa@math.u-strasbg.fr