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.

We did not thoroughly study the case p = 13.

1. We recall the results of [2]

Denote by S_{2}(Γ_{0}(p)) the space of cusp forms of weight 2 for the congruence subgroup Γ_{0}(p). Denote by T the subring of End S_{2}(Γ_{0}(p)) generated by the Hecke operators. Let f ∈ S_{2}(Γ_{0}(p)) have q-expansion ∑_{n=1}^{∞} a_{n}q^{n}. When χ is a Dirichlet character, denote by L(f, χ, s) the entire function which extends the Dirichlet series ∑_{n=1}^{∞} a_{n}χ(n)/n^{s}.

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} is anomalous 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 anomalous j ∈ F_{p} and all non-quadratic Dirichlet characters χ : (Z/pZ)^{∗} ⟶ C, there exists t_{χ} ∈ T and δ ∈ Δ_{S} such that L(f, χ, 1) ≠ 0 for every newform f ∈ t_{χ}S_{2}(Γ_{0}(p)) 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} of 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 is anomalous.

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(f, χ, 1) ≠ 0 for every newform f ∈ t_{χ}S_{2}(Γ_{0}(p)), translates into L(t_{χ}J_{0}(p), χ, 1) ≠ 0, which in turn 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 classifies (finely since 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, which, therefore, can not be a square.

3. Verification of the hypothesis of Proposition 1

Let p be a prime number. In this section we explain how to use a computer to verify that the hypothesis of Proposition 1 are satisfied. We have carried out this verification for p = 11 and 13 < p < 1000.

We first list the anomalous j-invariants j ∈ F_{p}. Since p is fairly small in the range of our computations, we created this list by simply enumerating all of the elliptic curves over F_{p} and counting the number of points on each curve. For example, when p = 31 the anomolous j-invariants are j = 10, 14.

Let χ : (Z/pZ)^{∗} ⟶ C^{∗} denote a non-quadratic Dirichlet character and suppose j ∈ P^{1}(F_{p}) is anomalous.

We study the following three T-modules: T, Δ_{S}, and S_{2}(Γ_{0}(p); Z). After extension of scalars to Q, these are T ⊗ Q-modules that are free of rank 1, of which the irreducible sub-T ⊗ Q modules are the annihilators of the minimal prime ideals of T. We compute a list of the minimal prime ideals of T by computing appropriate kernels and characteristic polynomials of Hecke operators of small index on Δ_{S}, which we find using the graph method of Mestre and Oesterlé [3].

Having computed the minimal prime ideals I of T, we verify that one of the I simultaneously satisfies the following three conditions:

1) There exists x ∈ Δ_{S} such that Ix = 0 and ι_{j}(x) ≠ 0.

2) Each of the conjugate newforms f ∈ S_{2}(Γ_{0}(p)) with If = 0 satisfies L(f, χ, 1) ≠ 0.

3) The image of I in the T-module T/pT is a direct factor.

Let I be a minimal prime ideal of T. When I satisfies the above three conditions, the following steps were used to prove that this is the case.

We verified that I satisfies the first condition by finding a T-eigenvector v of Δ_{S} ⊗ Z̄ that is annihilated by I such that ι_{j}(v) ≠ 0. Because ι_{j} is a homomorphism, this implies the existence of an x as in condition 1.

We verified the second condition using the theory of modular symbols, using a technique that avoids the need to compute integrals by approximating sums of infinite series. Using the algorithm described in, e.g., [1], we compute the space H_{1}(X_{0}(p); Q). Then we find a single T-eigenvector φ in the linear dual H^{1}(X_{0}(p); \overline{Q}) such that Iφ = 0. Let

e_{χ} = ∑_{a∈(Z/pZ)^{∗}} χ(a){0, a/p}

be the χ-winding element. Then α · L(f, χ, 1) = φ(e_{χ}), where α is a nonzero element of C. Thus φ(e_{χ}) ≠ 0 if and only if L(f, χ, 1) ≠ 0. A significant subtlety that arises in carrying out this computation is that the character χ takes values in one number field, whereas the eigenvector φ is defined over a different number field. These two number fields are frequently not linearly disjoint, so it is necessary to compute their compositum in order to compute the value φ(e_{χ}), and this can be very costly in practice. Since we only have to verify that φ(e_{χ}) ≠ 0, we avoided this difficulty by instead approximating φ and e_{χ} by vectors with approximate complex entries and computing an approximation to the complex number φ(e_{χ}); this suffices for the present application because we can recognize that an approximate complex number is nonzero (although we can not prove that an approximation is zero).

The third condition is satisfied for all p < 10000 except p = 389, because one of us has verified that the discriminant of T is prime to p and so the ring T/pT is semisimple. This was accomplished by computing discrimininants of characteristic polynomials mod p of the Hecke operators T_{ℓ}, for ℓ ≤ 7 a prime. In the few cases when all four of these characteristic polynomials had discriminant equal to 0 mod p, we resorted to modular symbols to compute several more characteristic polynomials until we found one with nonzero discriminant mod p.

For all prime numbers p < 1000 different than 2, 3, 5, 7, 13 and 389, we thus verified the existence of a minimal prime ideal that satisfies the three conditions given above. More precisely, we carried out this computation one p at a time by considering each Galois conjugacy class of non-quadratic characters χ. We find a single newform f such that L(f, χ, 1) ≠ 0 and then try to verify condition 1 for all of the anamolous j-invariants in F_{p}; if this succeeds, we then attempt to verify condition 2 for all conjugates of f and χ.

In the case when p = 389, we find I 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 I to be one of these that is appropriate.

When our three conditions are satisfied for an ideal I of T, there exists t_{χ} ∈ T which is annihilated by I and is the inverse image of a projector of T/pT on the complement of I + 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 I and t_{χ} ∈ 1+pT+P). Every newform f ∈ t_{χ}S_{2}(Γ_{0}(p)) satisfies If = 0, and therefore, by our second condition, L(f, χ, 1) ≠ 0.

The pair (t_{χ}, δ) thus satisfies the conditions required by Proposition 1.

Bibliography

[1] J. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997.

[2] L. Merel, Sur la nature non cyclotomique des points d’ordre fini des courbes elliptiques, Prépublication de l’Institut de mathématiques de Jussieu, 222, 1999.

[3] J.-F. Mestre, La méthode des graphes. Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata), 217–242, 1986.