The field generated by the points of small prime order on an elliptic curve

Loïc Merel and William A. Stein

Introduction

Let p be a prime number. Let Q̄ be an algebraic closure of Q. Denote by Q(μ_{p}) the cyclotomic subfield of Q̄ generated by the pth roots of unity. Let E be an elliptic curve over Q(μ_{p}), such that the points of order p of E(Q̄) are all Q(μ_{p})-rational.

Theorem . — One has p > 1000, p < 6 or p = 13.

We note that the case p = 7 was treated by Emmanuel Halberstadt. The part of the theorem that concerns the case p ≡ 3 (mod 4) is given in [2]. We propose to give the details that permit our treating the more difficult case in which p ≡ 1 (mod 4). We treat this last case with the aid of Proposition 2 below, which is not present in loc. cit.

It may be possible to exclude the case when p = 13 by studying the modular curve X_{1}(13) and its Jacobian J_{1}(13).

1. We recall the results of [2]

Denote by J_{0}(p) the Jacobian of the modular curve X_{0}(p) (whose points are isomorphism classes of generalized elliptic curves equipped with a cyclic subgroup of order p). Consider the subring T of End J_{0}(p) generated by the Hecke operators. We refer the reader to [1] for an in-depth study of these objects.

Let S be the set of isomorphism classes of supersingular elliptic curves in characteristic p. Denote by Δ_{S} the group formed by the divisors of degree 0 with support on S. It is equipped with a structure of T-module (deduced, for example, from the action of the Hecke correspondences on the fiber at p of the regular minimal model of X_{0}(p) over Z).

Let j ∈ F̄_{p} − J_{S}, where J_{S} denotes the set of supersingular modular invariants. We denote by ι_{j} the homomorphism of groups Δ_{S} ⟶ F̄_{p} that associates to ∑_{E} n_{E}[E] the quantity ∑_{E} n_{E}/(j − j(E)), where j(E) denotes the modular invariant of E.

One says that an element j ∈ F_{p} presents an anomaly if there exists an elliptic curve over F_{p} with modular invariant j that possesses an F_{p}-rational point of order p (then necessarily j ∉ J_{S}).

Proposition 1. — Suppose that p is congruent to 1 modulo 4. Suppose that for all j ∈ F_{p} that present an anomaly and all non-quadratic Dirichlet characters χ : (Z/pZ)^{∗} ⟶ C, there exists t_{χ} ∈ T and δ ∈ Δ_{S} such that L(t_{χ}J_{0}(p), χ, 1) ≠ 0 and ι_{j}(t_{χ}δ) ≠ 0.

Then for all subgroups C of order p of E(Q̄), there exists an elliptic curve E_{C} over Q(\sqrt{p}) equipped with a Q(\sqrt{p})-rational subgroup D_{C} or order p, and the pairs (E, C) and (E_{C}, D_{C}) are Q̄-isomorphic. Proof. — We indicate how this is deduced from [2]. The hypothesis ι_{j}(t_{χ}δ) ≠ 0 forces t_{χ} ∉ pT and, a fortiori, t_{χ} ≠ 0 ; in addition, the non-vanishing hypothesis on the L-series forces the hypothesis H_{p}(χ) of loc. cit, introduction.

According to Corollary 3 of Proposition 6 of loc. cit, E has potentially good reduction at the prime ideal P of Z[μ_{p}] that lies above p once we know that hypothesis H_{p}(χ) is satisfied for all non-quadratic Dirichlet characters χ of conductor p (this is the case by hypothesis).

Denote by j the modular invariant of the fiber at P of the Néron model of E. According to the corollary of Proposition 15 of loc. cit., j presents an anomaly.

Let C be a subgroup of E(Q̄) of order p. By assumption E is an elliptic curve over Q(μ_{p}) whose points of order p are all Q(μ_{p})-rational, so the pair (E, C) defines a Q(μ_{p})-rational point P of the modular curve X_{0}(p).

Consider the morphism ϕ_{χ} = ϕ_{t_{χ}} (see loc. cit. section 1.3). When ι_{j}(t_{χ}δ) ≠ 0, this is a formal immersion at the point P_{/F_{p}}, according to loc. cit., Proposition 4. The hypothesis that L(t_{χ}J_{0}(p), χ, 1) ≠ 0 implies that the χ-isotypical component of t_{χ}J_{0}(p)(Q(μ_{p})) is finite (this is Kato’s theorem, see the discussion in loc. cit. section 1.5). We can then apply Corollary 1 of Proposition 6 of loc. cit. This proves that P is Q(\sqrt{p})-rational ; this translates into the conclusion of Proposition 1.

2. A lemma about elliptic curves

Proposition 2. — Let p be a prime number that is congruent to 1 modulo 4. Let E be an elliptic curve over Q̄. There exists a cyclic subgroup C of order p of E(Q̄)[p], such that for all elliptic curves E′ over Q(\sqrt{p}) equipped with a Q(\sqrt{p})-rational subgroup C′, the pairs (E, C) and (E′, C′) are not Q̄-isomorphic. Proof. — We procede by contradiction. Let E_{0} be an elliptic curve over Q(\sqrt{p}) that is Q̄ isomorphic to E (it exists by hypothesis). We first show that the subgroup Gal(Q̄/Q(\sqrt{p})) acts by scalars on the F_{p}-vector space E_{0}(Q̄)[p].

Denote by X(p) the algebraic curve over Q that parametrizes classes (fine because p > 2) of generalized elliptic curves equipped with an embedding π : (Z/pZ)^{2} ⟶ E[p]. Consider the morphism (of algebraic varieties over Q) ϕ : X(p) ⟶ X_{0}(p)^{P^{1}(F_{p})} that to (E, π) associates ∏_{t∈P^{1}(F_{p})}(E, π(t)). Denote by X_{Δ}(p) the image of ϕ. The covering (of algebraic curves over Q) ϕ′ : X(p) ⟶ X_{Δ}(p) is Galois with Galois group isomorphic to F_{p}^{∗} (the action being deduced from the scalar action of F_{p}^{∗} on E[p]).

Let π_{0} be an embedding (Z/pZ)^{2} ⟶ E_{0}[p]. Denote by P the Q̄-rational point of X(p) deduced from (E_{0}, π_{0}). Its image by ϕ is Q(\sqrt{p})-rational by hypothesis. We have then a character α : Gal(Q̄/Q(\sqrt{p})) ⟶ F_{p}^{∗} such that σ(P) = α(σ).P (σ ∈ Gal(Q̄/Q(\sqrt{p}))). In other words, Gal(Q̄/Q(\sqrt{p})) acts by scalars on E_{0}(Q̄)[p] via the character α.

Because of the Weil pairing, α^{2} coincides with the cyclotomic character modulo p, and it factors through Gal(Q(μ_{p})/Q(\sqrt{p})). But, when p ≡ 1 (mod 4), the group Gal(Q(μ_{p})/Q(\sqrt{p})) is of even order, and the characters modulo p form a group generated by the reduction modulo p of the cyclotomic character ; it thus can not be a square.

3. Verification of the hypothesis of Proposition 1

Let p be a prime number. We now indicate how to computationally verify that the hypothesis of Proposition 1 are satisfied. We denote by S_{2}(Γ_{0}(p)) the group formed by the cuspidal modular forms of weight 2 for the congruence subgroup Γ_{0}(p) which have integer Fourier coefficients. By a newform we mean a newform in the Q̄-vector space generated by S_{2}(Γ_{0}(p)).

We first list the modular j-invariants that present an anomaly. “On ne s’intéresse qu’à ces derniers.” ( ? ?) Let χ be a Dirichlet character (Z/pZ)^{∗} ⟶ C^{∗} and suppose j ∈ P^{1}(F_{p}) presents an anomaly.

We study the following three T-modules : T, Δ_{S}, and S_{2}(Γ_{0}(p)). After extension of scalars to Q, these are T ⊗ Q-modules that are free of rank 1, of which the irreducible sub-T modules are the annihilators of the minimal prime ideals of T. We establish a list of the minimal prime ideals of T. From an algorithmic point of view, these are generated by the minimal polynomials of small degree of Hecke operators of small index ; we find these polynomials by utilizing the graph method of Mestre and Oesterlé [3].

For each ideal P de T, we can determine if the following three conditions are satisfied :

i) There exists x ∈ Δ_{S} such that Px = 0 and ι_{j}(x) ≠ 0. ii) All newforms f such that Pf = 0 satisfy L(f, χ, 1) ≠ 0. iii) The image of P in the T-module T/pT is a direct factor.

We study the case when P is a minimal prime. The first condition is studied by the graph method of Mestre and Oesterlé. The second condition is verified using the theory of modular symbols (without recourse to the calculation of integrals). The third condition is verified for p < 1000 and p ≠ 389, because one of us has verified that the discriminant of T is prime to p and so the ring T/pT is semi-simple. For all prime numbers p different than 2, 3, 5, 7, 13 and 389, we verified the existence of a minimal prime ideal that satisfies the three conditions given above.

In the case when p = 389, we find P in the following way. There exists two minimal prime ideals P_{1} and P_{2} that satisfy the first two conditions. Because the discriminant of T has p-adic valuation 1, the image of at least one of the ideals P_{1}, P_{2} and P_{1} ∩P_{2} is a direct factor of T/pT. We choose P to be one of these that is appropriate.

When our three conditions are satisfied for an ideal P of T, there exists t_{χ} ∈ T which is annihilated by P and is the inverse image of a projector of T/pT on the complement of P + pT. Putting δ = x, one has ι_{j}(t_{χ}δ) = ι_{j}(δ) ≠ 0 (because ι_{j} takes its values in characteristic p, δ is annihilated by P and t_{χ} ∈ 1+pT+P). The assertion L(t_{χ}J_{0}(p), χ, 1) ≠ 0 follows easily from the second condition, since L(t_{χ}J_{0}(p), χ, s) is the product of L(f, χ, s), where f runs over the newforms that are not annihilated by t_{χ}.

The pair (t_{χ}, δ) thus satisfy the required conditions.

Bibliographie

[1] B. Mazur, Modular curves and the Eisenstein ideal, Pub. math. de l’IHES 47, 33–186, 1977.

[2] L. Merel, Sur la nature non cyclotomique des points d’ordre fini des courbes elliptiques, Prépublication , 1999.

[3] J.-F. Mestre et J. Oesterlé, Courbes elliptiques de conducteur premier, Manuscrit non publié.