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

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


Loïc Merel   and   William A. Stein





Contents:
Introduction
1. We recall the results of [2]
2. A lemma about elliptic curves
3. Verification of the hypothesis of Proposition 1 (includes source code and data)
Bibliography
Printable dvi version of this document




Introduction


Let p be a prime number. Let $\bar{\bf Q}$ be an algebraic closure of ${\bf Q}$. Denote by ${\bf Q}(\mu_p)$ the cyclotomic subfield of $\bar{\bf Q}$generated by the pth roots of unity. Let E be an elliptic curve over ${\bf Q}(\mu_p)$, such that the points of order p of $E(\bar{\bf Q})$ are all ${\bf Q}(\mu_p)$-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 $p\equiv 3\pmod 4$ is given in [2]. We propose to give the details that permit our treating the more difficult case in which $p\equiv 1 \pmod 4$. 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 X1(13) and its Jacobian J1(13).


1. We recall the results of [2]


Denote by J0(p) the Jacobian of the modular curve X0(p) (whose points are isomorphism classes of generalized elliptic curves equipped with a cyclic subgroup of order p). Consider the subring ${\bf T}$ of ${\rm End}\,J_0(p)$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 $\Delta_S$ the group formed by the divisors of degree 0 with support on S. It is equipped with a structure of ${\bf T}$-module (deduced, for example, from the action of the Hecke correspondences on the fiber at p of the regular minimal model of X0(p) over ${\bf Z}$).

Let $j\in\bar{\bf F}_p-J_S$, where JS denotes the set of supersingular modular invariants. We denote by $\iota_j$ the homomorphism of groups $\Delta_S\longrightarrow\bar{\bf F}_p$ that associates to $\sum_E n_E[E]$ the quantity $\sum_E n_E/(j-j(E))$, where j(E) denotes the modular invariant of E.

One says that an element $j\in{\bf F}_p$ presents an anomaly if there exists an elliptic curve over ${\bf F}_p$ with modular invariant j that possesses an ${\bf F}_p$-rational point of order p (then necessarily $j\notin{}J_S$).


PROPOSITION 1. -- Suppose that p is congruent to 1 modulo 4. Suppose that for all $j\in{\bf F}_p$ that present an anomaly and all non-quadratic Dirichlet characters $\chi$ : $({\bf Z}/p{\bf Z})^*\longrightarrow{\bf C}$, there exists $t_\chi\in {\bf T}$ and $\delta\in\Delta_S$ such that $L(t_\chi J_0(p),\chi,1)\ne0$ and $\iota_j(t_\chi\delta)\ne0$.

Then for all subgroups C of order p of $E(\bar{\bf Q})$, there exists an elliptic curve EC over ${\bf Q}(\sqrt p)$ equipped with a ${\bf Q}(\sqrt p)$-rational subgroup DC or order p, and the pairs (E,C) and (EC,DC) are $\bar{\bf Q}$-isomorphic.

Proof. -- We indicate how this is deduced from [2]. The hypothesis $\iota_j(t_\chi\delta)\ne0$ forces $t_\chi\notin p{\bf T}$and, a fortiori, $t_\chi\ne0$; in addition, the non-vanishing hypothesis on the L-series forces the hypothesis $H_p(\chi)$ of loc. cit, introduction.

According to Corollary 3 of Proposition 6 of loc. citE has potentially good reduction at the prime ideal ${\cal P}$ of ${\bf Z}[\mu_p]$ that lies above p once we know that hypothesis $H_p(\chi)$ is satisfied for all non-quadratic Dirichlet characters $\chi$ of conductor p (this is the case by hypothesis).

Denote by j the modular invariant of the fiber at ${\cal P}$ 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 $E(\bar{\bf Q})$ of order p. By assumption E is an elliptic curve over  ${\bf Q}(\mu_p)$ whose points of order p are all ${\bf Q}(\mu_p)$-rational, so the pair (E,C) defines a ${\bf Q}(\mu_p)$-rational point P of the modular curve X0(p).

Consider the morphism $\phi_{\chi}=\phi_{t_\chi}$ (see loc. cit. section 1.3). When $\iota_j(t_\chi\delta)\ne0$, this is a formal immersion at the point $P_{/{\bf F}_p}$, according to loc. cit., Proposition 4. The hypothesis that $L(t_\chi J_0(p),\chi,1)\ne0$implies that the $\chi$-isotypical component of $t_\chi J_0(p)({\bf Q}(\mu_p))$ 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 ${\bf Q}(\sqrt p)$-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 1modulo 4. Let E be an elliptic curve over $\bar{\bf Q}$. There exists a cyclic subgroup C of order p of $E(\bar{\bf Q})[p]$, such that for all elliptic curves E' over ${\bf Q}(\sqrt p)$equipped with a ${\bf Q}(\sqrt p)$-rational subgroup C', the pairs (E,C) and (E', C') are not $\bar{\bf Q}$-isomorphic.

Proof. -- We procede by contradiction. Let E0 be an elliptic curve over ${\bf Q}(\sqrt p)$that is $\bar{\bf Q}$ isomorphic to E (it exists by hypothesis). We first show that the subgroup ${\rm Gal}(\bar{\bf Q}/{{\bf Q}(\sqrt p)})$ acts by scalars on the ${\bf F}_p$-vector space $E_0(\bar{\bf Q})[p]$.

Denote by X(p) the algebraic curve over ${\bf Q}$ that parametrizes classes (fine because p>2) of generalized elliptic curves equipped with an embedding $\pi$ : $({\bf Z}/p{\bf Z})^2\longrightarrow E[p]$. Consider the morphism (of algebraic varieties over ${\bf Q}$) $\phi$ : $X(p)\longrightarrow X_0(p)^{{\bf P}^1({\bf F}_p)}$ that to $(E,\pi)$ associates $\prod_{t\in{\bf P}^1({\bf F}_p)}(E,\pi(t))$. Denote by $X_\Delta(p)$ the image of $\phi$. The covering (of algebraic curves over ${\bf Q}$) $\phi'$ : $X(p)\longrightarrow X_\Delta(p)$ is Galois with Galois group isomorphic to ${\bf F}_p^*$ (the action being deduced from the scalar action of ${\bf F}_p^*$ on E[p]).

Let $\pi_0$ be an embedding $({\bf Z}/p{\bf Z})^2\longrightarrow E_0[p]$. Denote by P the $\bar{\bf Q}$-rational point of X(p) deduced from $(E_0,\pi_0)$. Its image by $\phi$ is ${\bf Q}(\sqrt p)$-rational by hypothesis. We have then a character $\alpha$ : ${\rm Gal}(\bar{\bf Q}/{\bf Q}(\sqrt p))\longrightarrow{\bf F}_p^*$ such that $\sigma(P)=\alpha(\sigma).P$ ( $\sigma\in {\rm Gal}(\bar{\bf Q}/{\bf Q}(\sqrt p))$). In other words, ${\rm Gal}(\bar{\bf Q}/{{\bf Q}(\sqrt p)})$ acts by scalars on $E_0(\bar{\bf Q})[p]$ via the character $\alpha$.

Because of the Weil pairing, $\alpha^2$ coincides with the cyclotomic character modulo p, and it factors through ${\rm Gal}({\bf Q}(\mu_p)/{\bf Q}(\sqrt p))$. But, when $p\equiv 1 \pmod 4$, the group ${\rm Gal}({\bf Q}(\mu_p)/{\bf Q}(\sqrt p))$ 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. These computations were done using the C++ program HECKE, with a litte help at the end from the computer algebra systems MAGMA and PARI. We denote by $S_2(\Gamma_0(p))$ the group formed by the cuspidal modular forms of weight 2 for the congruence subgroup $\Gamma_0(p)$ which have integer Fourier coefficients. By a newform we mean a newform in the $\bar{\bf Q}$-vector space generated by $S_2(\Gamma_0(p))$.

We first list the modular j-invariants that present an anomaly; one is only interested in these. Here is a list of the anomalous j-invariants for p less than 1000. This list was created using this very simple PARI program. Let $\chi$ be a Dirichlet character $({\bf Z}/p{\bf Z})^*\longrightarrow{\bf C}^*$ and suppose $j\in{\bf P}^1({\bf F}_p)$ presents an anomaly.

We study the following three ${\bf T}$-modules: ${\bf T}$, $\Delta_S$, and $S_2(\Gamma_0(p))$. After extension of scalars to ${\bf Q}$, these are ${\bf T}\otimes{\bf Q}$-modules that are free of rank 1, of which the irreducible sub-${\bf T}$ modules are the annihilators of the minimal prime ideals of ${\bf T}$. We establish a list of the minimal prime ideals of ${\bf T}$. 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]. A list of conjugacy classes of eigenforms in $\Delta_S$ can be found here; these correspond to the minimal primes.

For each ideal ${\cal P}$ of ${\bf T}$, we can determine if the following three conditions are satisfied:

i) There exists $x\in\Delta_S$ such that ${\cal P}x=0$ and $\iota_j(x)\ne 0$ for all anomalous j.

ii) All newforms f such that ${\cal P}f=0$ satisfy $L(f,\chi,1)\ne 0$. This information can be deduced from this table.

iii) The image of ${\cal P}$ in the ${\bf T}$-module ${\bf T}/p{\bf T}$ is a direct factor.

We study the case when ${\cal P}$ 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 $p\ne 389$, because one of us has verified that the discriminant of ${\bf T}$ is prime to p and so the ring ${\bf T}/p{\bf T}$ is semi-simple. In fact, we verified this for p<10000; this was accomplished by computing discrimininants of characteristic polynomials mod p of T2, T3, T5, and T7 using the method of graphs. If all four characteristic polynomials had discriminant equal to 0 mod p, we resorted to modular symbols to compute several more characteristic polynomials until one is found with nonzero discriminant mod p.

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. Here is the MAGMA source code that was used. The output for primes congruent to 1 modulo 4 gives the corresponding minimal primes for each prime and Dirichlet character. For completeness, here is the output for primes congruent to 3 modulo 4.

In the case when p=389, we find ${\cal P}$ in the following way. There exists two minimal prime ideals ${\cal P}_1$ and ${\cal P}_2$that satisfy the first two conditions. Because the discriminant of ${\bf T}$ has p-adic valuation 1, the image of at least one of the ideals ${\cal P}_1$, ${\cal P}_2$ and ${\cal P}_1\cap{\cal P}_2$ is a direct factor of ${\bf T}/p{\bf T}$. We choose ${\cal P}$ to be one of these that is appropriate.

When our three conditions are satisfied for an ideal ${\cal P}$ of ${\bf T}$, there exists $t_\chi\in {\bf T}$ which is annihilated by ${\cal P}$ and is the inverse image of a projector of ${\bf T}/p{\bf T}$ on the complement of ${\cal P}+p{\bf T}$. Putting $\delta=x$, one has $\iota_j(t_\chi \delta)=\iota_j(\delta)\ne0$ (because $\iota_j$ takes its values in characteristic p, $\delta$ is annihilated by ${\cal P}$ and $t_\chi\in 1+p{\bf T}+{\cal P}$). The assertion $L(t_\chi J_0(p),\chi,1)\ne0$ follows easily from the second condition, since $L(t_\chi J_0(p),\chi,s)$ is the product of $L(f,\chi,s)$, where f runs over the newforms that are not annihilated by $t_\chi$.

The pair $(t_\chi,\delta)$ thus satisfy the required conditions.



Bibliography


[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 and J. Oesterlé, Courbes elliptiques de conducteur premier, Manuscrit non publié.



Printable dvi version of this document

Here is the dvi version of this paper, which has no tables. A French version is also available. Here's the English plain TeX file and the French plain TeX file.



Maintained by William A. Stein.    Last modified January 6, 2000