Let *p* be a prime number.
Let
be an algebraic closure of .
Denote by
the cyclotomic subfield of
generated by the *p*th roots of unity.
Let *E* be an elliptic curve over
, such that the
points of order *p* of
are all
-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
is given in [2].
We propose to give the details that permit our treating the
more difficult case in which
.
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).

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 of
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 the group formed
by the divisors of degree 0 with support on *S*. It is equipped with
a structure of -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 ).

Let
, where *J*_{S} denotes the set of supersingular modular
invariants. We denote by the homomorphism of
groups
that associates to
the quantity
, where *j*(*E*) denotes the modular invariant of *E*.

One says that an element
*presents an anomaly*
if there exists an elliptic curve over with modular invariant *j*
that possesses an -rational point of order *p*
(then necessarily
).

PROPOSITION 1. -- *Suppose that **p** is congruent to *1* modulo *4*.
Suppose that for all
*
* that present an anomaly and all
non-quadratic Dirichlet characters ** : *
*,
there exists *
* and
*
* such that *
* and
*
*.
*

*Then for all subgroups **C** of order **p** of *
*, there exists an
elliptic curve **E*_{C}* over *
* equipped with a
*
*-rational subgroup **D*_{C}* or order **p**, and
the pairs *(*E*,*C*)* and *(*E*_{C},*D*_{C})* are *
*-isomorphic.*

*Proof*. -- We indicate how this is deduced from [2].
The hypothesis
forces
and, *a fortiori*,
; in addition,
the non-vanishing hypothesis on the *L*-series
forces the hypothesis of *loc. cit*, introduction.

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

Denote by *j* the modular invariant of the fiber at 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
of order *p*.
By assumption *E* is an elliptic curve over
whose points
of order *p* are all
-rational, so
the pair (*E*,*C*) defines a
-rational point *P*
of the modular curve *X*_{0}(*p*).

Consider the morphism
(see *loc. cit.*
section 1.3). When
, this is a formal
immersion at the point
, according to *loc. cit.*,
Proposition 4. The hypothesis that
implies that the -isotypical component of
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
-rational; this translates into the conclusion of Proposition 1.

PROPOSITION 2. -- *Let **p** be a prime number that is congruent to *1*modulo *4*. Let **E** be an elliptic curve over *
*.
There exists a cyclic subgroup **C** of order **p** of
*
*, such that for all elliptic curves **E*'* over
*
*equipped with a *
*-rational subgroup **C*'*, the
pairs *(*E*,*C*)* and *(*E*', *C*')* are not *
*-isomorphic.
*

*Proof*. --
We procede by contradiction.
Let *E*_{0} be an elliptic curve over
that is
isomorphic to *E* (it exists by hypothesis).
We first show that the subgroup
acts by scalars
on the -vector space
.

Denote by *X*(*p*) the algebraic curve over that parametrizes classes
(fine because *p*>2) of generalized elliptic curves equipped with an
embedding
:
.
Consider the morphism (of algebraic varieties
over ) :
that to associates
.
Denote by
the image of .
The covering (of algebraic curves over )
:
is Galois with Galois group isomorphic to
(the action being deduced from the scalar action of
on *E*[*p*]).

Let be an embedding
.
Denote by *P* the
-rational point of *X*(*p*) deduced from
.
Its image by is
-rational by hypothesis.
We have then a character :
such that
(
). In other words,
acts
by scalars on
via the character .

Because of the Weil pairing, coincides
with the cyclotomic character modulo *p*, and it factors through
. But, when
, the group
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.

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
the group formed by the cuspidal
modular forms of weight 2 for the congruence subgroup
which have integer Fourier coefficients.
By a newform we mean a newform in the
-vector space generated by
.

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 be a Dirichlet character
and
suppose
presents an anomaly.

We study the following three -modules: , , and . After extension of scalars to , these are -modules that are free of rank 1, of which the irreducible sub- modules are the annihilators of the minimal prime ideals of . We establish a list of the minimal prime ideals of . 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 can be found here; these correspond to the minimal primes.

For each ideal of , we can determine if the following three conditions are satisfied:

i)
There exists
such that
and
for all anomalous *j*.

ii) All newforms *f* such that
satisfy
.
This information can be deduced from this table.

iii) The image of in the -module is a direct factor.

We study the case when 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 ,
because one of us has verified that the discriminant of
is prime to *p* and so the ring
is semi-simple.
In fact, we verified this for *p<10000*; this was accomplished by
computing discrimininants of
characteristic polynomials mod *p* of *T _{2}*,

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 in the following way.
There exists two minimal prime ideals
and
that satisfy the first two conditions.
Because the discriminant of has *p*-adic valuation 1,
the image of at least one of the ideals
,
and
is a direct factor of
. We choose to be one of these that is
appropriate.

When our three conditions are satisfied for an ideal
of , there exists
which is annihilated by and
is the inverse image of a projector of
on the complement of
.
Putting , one has
(because takes its values in characteristic *p*,
is annihilated by and
).
The assertion
follows
easily from the second condition,
since
is the product of
, where *f* runs over the
newforms that are not annihilated by .

The pair thus satisfy the required conditions.

[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.

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