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\centerline
{\titchap The field generated by the points of small}
\centerline
{\titchap prime order on an elliptic curve}
\medskip

\bigskip\bigskip\bigskip
\centerline{\pc Lo\"\i c Merel {\rm and} William A.~Stein}
\bigskip\bigskip\bigskip

\bigskip

\bigskip\bigskip\noindent
{\bf Introduction}
\bigskip\bigskip

Let~$p$ be a prime number.
Let $\bar\B Q$ be an algebraic closure of $\B Q$.
Denote by $\B Q(\mu_p)$ the cyclotomic subfield of $\bar\B Q$
generated by the $p$th roots of unity.
Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the
points of order~$p$ of
$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.
\bigskip
\th{}{\it One has $p>1000$, $p<6$ or $p=13$.}
\bigskip

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~\Merel. 
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 {\it loc. cit.}

It may be possible to exclude the case when $p=13$ 
by studying the modular curve $X_1(13)$ and its Jacobian $J_1(13)$. 

\bigskip\noindent
{\bf 1. We recall the results of \Merel}
\bigskip
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 $\B T$ of ${\rm End}\,J_0(p)$
generated by the Hecke operators. We refer the reader to~\Mazur{}
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 $\B T$-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 $\B Z$).

Let $j\in\bar\B F_p-J_S$, where $J_S$ denotes the set of supersingular modular
invariants. We denote by  $\iota_j$ the homomorphism of
groups $\Delta_S\raw \bar\B F_p$ 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  $j\in\B F_p$ {\it presents an anomaly}
if there exists an elliptic curve over $\B F_p$ with modular invariant~$j$ 
that possesses an $\B F_p$-rational point  of order~$p$ 
(then necessarily $j\notin{}J_S$).

\bigskip
\prop{1}{\it Suppose that~$p$ is congruent to~$1$ modulo~$4$. 
Suppose that for all
$j\in\B F_p$ that present an anomaly and all
non-quadratic Dirichlet characters $\chi$ : $(\B Z/p\B Z)^*\raw \B C$,
there exists $t_\chi\in \B T$ 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 $E(\bar \B Q)$, there exists an
elliptic curve $E_C$ over $\B Q(\sqrt p)$ equipped with a 
$\B Q(\sqrt p)$-rational subgroup $D_C$ or order~$p$, and
the pairs $(E,C)$ and $(E_C,D_C)$ are $\bar \B Q$-isomorphic.}

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

According to Corollary~3 of Proposition~6 of {\it loc. cit},~$E$ has
potentially good reduction at the prime ideal 
$\cP$ of $\B Z[\mu_p]$ 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~$\cP$ of the 
N\'eron model of~$E$.
According to the corollary of Proposition~15 of {\it loc. cit.},
$j$~presents an anomaly.

Let~$C$ be a subgroup of $E(\bar\B Q)$ of order~$p$.
By assumption~$E$ is an elliptic curve over~$\B Q(\mu_p)$ whose points
of order~$p$ are all $\B Q(\mu_p)$-rational, so 
the pair $(E,C)$ defines a $\B Q(\mu_p)$-rational point~$P$ 
of the modular curve $X_0(p)$.

Consider the morphism $\phi_{\chi}=\phi_{t_\chi}$ (see {\it loc. cit.}
section 1.3). When $\iota_j(t_\chi\delta)\ne0$, this is a formal
immersion at the point $P_{/\B F_p}$, according to {\it loc. cit.},
Proposition~4.  The hypothesis that $L(t_\chi J_0(p),\chi,1)\ne0$
implies that the $\chi$-isotypical component of $t_\chi J_0(p)(\B
Q(\mu_p))$ is finite (this is Kato's theorem, see the discussion in
{\it loc. cit.} section 1.5).  We can then apply Corollary~1 of
Proposition~6 of {\it loc. cit}.  This proves that $P$ is $\B Q(\sqrt
p)$-rational; this translates into the conclusion of Proposition~1.



\bigskip\bigskip\noindent
{\bf 2. A lemma about elliptic curves}
\bigskip
\prop{2}{\it Let~$p$ be a prime number that is congruent to~$1$
modulo~$4$. Let~$E$ be an elliptic curve over $\bar\B Q$. 
There exists a cyclic subgroup~$C$ of order~$p$ of
$E(\bar\B Q)[p]$, such that for all elliptic curves~$E'$ over 
${\B Q(\sqrt{p})}$
equipped with a ${\B Q(\sqrt p)}$-rational subgroup~$C'$, the
pairs $(E,C)$ and $(E', C')$ are not $\bar\B Q$-isomorphic.
}

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

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

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

Because of the Weil pairing, $\alpha^2$ coincides
with the cyclotomic character modulo~$p$, and it factors through
$\Gal(\B Q(\mu_p)/\B Q(\sqrt p))$. But, when
$p\equiv 1\pmod 4$, the group $\Gal(\B Q(\mu_p)/{\B Q(\sqrt p)})$ 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.

\bigskip\bigskip\noindent
{\bf 3. Verification of the hypothesis of Proposition~1}
\bigskip
Let $p$ be a prime number.
We now indicate how to computationally 
verify that the hypothesis of Proposition~1 are 
satisfied.  We denote by $S_2(\Gamma_0(p))$ 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
$\bar\B Q$-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.
Let~$\chi$ be a Dirichlet character $(\B Z/p\B Z)^*\raw \B C^*$ and
suppose $j\in\B P^1(\B F_p)$ presents an anomaly.

We study the following three $\B T$-modules: $\B T$, $\Delta_S$, and
$S_2(\Gamma_0(p))$. After extension of scalars to~$\B Q$,
these are $\B T\otimes\B Q$-modules that are free of rank~$1$, of which
the irreducible sub-$\B T$ modules are the annihilators of the 
minimal prime ideals of  $\B T$. 
We establish a list of the minimal prime ideals of $\B T$.
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 Oesterl\'e \MO.

For each ideal $\cP$ of $\B T$, we can determine if
the following three conditions are satisfied:

\noindent i)
There exists $x\in\Delta_S$ such that
 $\cP x=0$ and  $\iota_j(x)\ne 0$
for all anomalous~$j$.

\noindent
ii) All newforms~$f$ such that $\cP f=0$ satisfy $L(f,\chi,1)\ne 0$.

\noindent
iii) The image of~$\cP$ in the $\B T$-module $\B T/p\B T$ 
is a direct factor.

We study the case when~$\cP$ is a minimal prime.
The first condition is studied by the graph method of Mestre and Oesterl\'e.
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
$\B T$ is prime to~$p$ and so the ring  $\B T/p\B T$ is semi-simple. 
In fact, we verified this for $p<10000$; this was accomplished by
computing discrimininants of 
characteristic polynomials mod~$p$ of $T_p$, for $p\leq 7$,
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 thus
verified the existence of a minimal prime ideal 
that satisfies the three conditions given above.

In the case when $p=389$,
we find $\cP$ in the following way.
There exists two minimal prime ideals $\cP_1$ and $\cP_2$
that satisfy the first two conditions.
Because the discriminant of~$\B T$ has $p$-adic valuation~$1$,
the image of at least one of the ideals
$\cP_1$, $\cP_2$ and $\cP_1\cap\cP_2$ is a direct factor of
$\B T/p\B T$.  We choose~$\cP$ to be one of these that is 
appropriate.

When our three conditions are satisfied for an ideal~$\cP$ 
of~$\B T$, there exists
$t_\chi\in\B T$ which is annihilated by $\cP$ and
is the inverse image of a projector of 
$\B T/p\B T$ on the complement of $\cP+p\B T$.
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~$\cP$ and $t_\chi\in 1+p\B T+\cP$).
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.

\bigskip\bigskip



\centerline{\pc Bibliography}
\bigskip

\item{\Mazur}{\pc B. Mazur},
{\it Modular curves and the Eisenstein ideal},
Pub. math. de l'IHES {\bf 47}, 33--186, {\oldstyle 1977}.

\item{\Merel}{\pc L. Merel},
{\it Sur la nature non cyclotomique des points d'ordre fini des courbes
elliptiques},
Pr\'epublication , {\oldstyle
1999}.

\item{\MO}{\pc J.-F. Mestre} and {\pc J. Oesterl\'e},
{\it Courbes elliptiques de conducteur premier}, Manuscrit non publi\'e.
\end