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\newcommand{\s}[1]{\langle #1 \rangle}



\title{The Method of Graphs. Examples and Applications}
\author{J.-F. Mestre. \\{}Tr. Andrei Jorza}



Let $S_k(N,\e)$ be the space of cusp forms of weight $k$, level
$N$ and character $\e$, where $k$ and $N$ are integers $\geq 1$,
and $\e$ is a Dirichlet character $N$. There are several ways
to construct a basis. For example one can use Selberg's trace
formula.  Denote by $\Tr(n)$ the trace of $T_n$, the $n$-th Hecke
operator.  The function \[f=\sum_{n=1}^\infty \Tr(n)q^n\] is in
$S_k(N,\e)$. The set of $f_i=T_if$ generate this space, and this
theoretically allows us to construct a basis. For example, if $N$
is prime, $\e=1$ and $k=2$, then the set of the $f_i$ ($1\leq
i\leq g$ where $g$ is the genus of $X_0(N)$) is a basis of

But, even in that case, which is the most favorable, the
computations become hard: on an average computer we can only hope
to treat $N$ of the size of $5000$ (always with $N$ prime, weight
2 and trivial character); actually, the computation of $\Tr(n)$
requires the knowledge of many class numbers of imaginary quadratic
fields of discriminant of order at most~$n$ and
to obtain a basis 
$\vvn{f}{g}$ one needs to compute $\Tr(n)$ for $n\leq g^2$.

In the following section we describe the ``method of graphs'', which
relies on the results of Deuring and Eichler, and developed by J.
Oestrl\'e and myself, which allows us to obtain a basis for $S_2(N,1)$
more quickly (at least when $N$ is a prime).

In the second section, we indicate how this method allows us to prove
that certain elliptic curves defined over $\Q$ are Weil 
curves\footnote{It is now a theorem that every elliptic curve
is a Weil curve, i.e., the Shimura-Taniyama-Weil conjecture
is true. -- William Stein}
(which, by providing an adequate Weil curve, yields all the
imaginary quadratic fields of class number at most 3, due to a
result of Goldfeld and recent works of Gross and Zagier).

The third section is dedicated to the verification of a conjecture
of Serre in certain particular cases; this is
possible because of the method described in the first section. It is
known that this conjecture, if it is true, has numerous
consequences (e.g., the Shimura-Taniyama-Weil conjecture, and thus
Fermat's Last Theorem).

\section{The method of graphs}

\subsection{Definitions and notations}

In the following $p$ is a prime number and $N_1$ is a positive
integer coprime to~$p$. Set $N=pN_1$.

Let \[M_N=\oplus_S\Z[S]\]where $S$ is taken over all supersingular
points of $X_0(N_1)$ in characteristic $p$, i.e., over the set of
isomorphism classes of pairs $(E,C)$ consisting of an elliptic 
curve $E$ defined over $\overline{\F}_p$ and a cyclic group $C$ of $E$
of order $N_1$. Two such pairs are identified if they are, in the
obvious sense, $\overline{\F}_p$-isomorphic.

Let \[\a_S=\frac{|\Aut(S)|}{2},\]where $\Aut(S)$ is the group of
$\overline{\F}_p$-automorphisms of $S$. We always have $\a_S\leq
12$, and if $p$ does not divide 6 then $\a_S\leq 3$.

Therefore we can define a scalar product on $M_N$ by
$\s{S,S}=\a_S$ and $\s{S,S'}=0$ if $S\neq S'$. Let
$\Eis=\sum\a_S^{-1}[S]$ and let 
  M_N^0=\left\{\sum x_S[S] : \sum x_S=0\right\}
be the subspace orthogonal to $\Eis$.

For all integers $n\geq 1$ coprime with $p$ we define an operator
$T_n$ on $M$ by \[T_n(E,C)=\sum_{C_n}(E/C_n,(C+C_n)/C_n)\]
where $C_n$ runs over all the cyclic subgroups of 
order~$n$ such that $C\i C_n=0$.

For all $q\mid{}N_1$ and coprime with $q'=N_1/q$, we define the same
way the Atkin-Lehner involutions $W_q$ by
\[W_q(E,C)=(E/q'C,(E_q+C)/q'C),\]where $E_q$ is the group of
points of order dividing $q$ of $E$.

Finally we define an involution $W_p$ by $W_p=-\Frob_p$, where
$\Frob_p$ is the endomorphism of $M_N$ that transforms $(E,C)$ to
$(E^p,C^p)$. (The fact that it is an involution reflects that the
supersingular points are defined over $\F_{p^2}$.)

These operators have the following properties: the set of $W_q$
and $T_n$ ($n$ coprime with $N$) generate an abelian semigroup of
hermitian operators with respect to the scalar product $\s{\cdot,
\cdot}$. The $T_n$ commute with each other for all $n$ coprime to $p$. If
$q=q_1q_2$ ($q_1,q_2$ coprime) and if $n=n_1n_2$ ($n_1,n_2$
coprime with each other and with $p$) then $W_q=W_{q_1}W_{q_2}$
and $T_n=T_{n_1}T_{n_2}$.

For all $d\mid N_1$ we have a homomorphism $\phi_d:M_N\to M_{N/d}$ 
that transforms $(E,C)$ to $(E,dC)$. This
homomorphism commutes with $T_n$ ($n$ coprime with $N$), and with
$W_q$ (for $q\mid{}N/d$). For $d\mid{}N_1$ and coprime with $N_1/d$ we have

\subsection{An isomorphism with $S_2(N)$}

We consider here the space $S_2(N)$ of cusp forms of weight 2 over
$\Gamma_0(N)$, with its natural structure of $\T$-module, where~$\T$
is the Hecke algebra \cite{1}.


\begin{theorem} There exists an isomorphism compatible with the
action of the Hecke operators, between $M_N^0\otimes \CC$ and the
subspace of $S_2(N)$ generated by the newforms of level $N$ and
oldforms coming from the cusp forms of weight 2 and level $pd$,
\begin{remark}Assume $N$ (or without loss $N_1$) is
square-free. We can determine efficiently the subspace $M_N^0$
corresponding to newforms in $S_2(N)$; it is the subspace
formed by all~$x$ so that for all divisors~$d$ of $N_1$ we have
In particular if $N=pq$, $q$ prime, it is the subspace of $M_{pq}$
intersection of the kernel of $\phi_q$ and of $\phi_qW_q$.

\subsection{Relation to the quaternion algebra}

The matrices of the operators $T_n$ acting on $M_N$ are the same
as the classical Brandt matrices \cite{15}, constructed using 
quaternion algebras.

Let $B_{p,\infty}$ be the quaternion algebra over~$\Q$ ramified exactly
at~$p$ and infinity, and let~$\O$ is an Eichler order of level $N_1$
(defined by Eichler \cite{6} in the case when $N_1$ is
square-free, and defined in general by Pizer \cite{14}), and let
$\vvn{I}{h}$ be representatives of the left ideal classes of

Let $\O_i$ be the right order (i.e., right normalizers) of the
ideals $I_i$, and $e_i$ be the number of units of $\O_i$. 
The Brandt matrix $B(n)=(b_{i,j}^{(n)})$ has $i,j$ entry
b_{i,j}^{(n)} =
 e_j^{-1}\cdot |\{\a : \a\in
where $\Nor$ is
the norm over $B_{p,\infty}$ (the norm of an ideal being the
$\gcd$ of the norms of its nonzero elements).

In the language of supersingular curves of characteristic $p$, we
may give these matrices (actually their transposes) the following

Let $S$ be a supersingular point as in $I.1$, i.e., a
supersingular elliptic curve $E$ defined over $\overline{\F}_p$
together with a cyclic group $C$ of order $N_1$. The ring of
endomorphisms $\O_1$ of $S$ is an Eichler order of level $N_1$. To
all the other supersingular points $S'=(E',C')$ we associate the
set $I_{S,S'}$ of homomorphisms from $S$ to $S'$, i.e. the set of
all homomorphisms $\a$ from $E$ to $E'$ that send $C$ to $C'$.
This is obviously a left ideal over $\O_1$, and its inverse ideal
is $I_{S',S}$. We can prove that all the right ideals of $\O_1$
are obtained in this way, and the whole Eichler order of level
$N_1$ if the rign of endomorphisms of a supersingular point $S$.
It is clear that the general term $B_{i,j}^{(n)}$ of the $n$-th
Brandt matrix is the number of isogenies of $S_i$ to $S_j$ (the
supersingular points being conveniently indexed,) two such
isogenies being identified is different by an automorphism of
$S_j$. We can retrieve the matrix of the operator $T_n$ acting
over $M_n$.

On the other hand if for all pairs of supersingular points
$(S,S')$ we associate the function \[\t_{S,S'}(q)=\sum_\a
q^{\deg\a}\]where $\a$ goes through all the homomorphisms of $S$
to $S'$, we retrieve the functions $\t$ classically associated
with the ideals of the orders of the quaternions, or, if one
prefers, associated with the positive integer quadratic forms in 4

It is therefore easy to prove that if $\sum{x_S}[S]$ is an
elements of $M_N\otimes \CC$ eigenvector of all the Hecke
operators and if $f(q)$ is the corresponding modular form, we
have, for all $S'$
\[x_{S'}f(q)=\sum_Sx_S\t_{S,S'}\]which allows, in theory, to find
the coefficients $a_n$ of $f$, using the $x_S$. In practice,
unfortunately, the computation of $a_n$ demands the knowledge of
all the isogenies of degree $n$ to $S'$, and there doesn't seem to
be a simple algorithm for that.

Nevertheless, in certain cases, there exists a different method to
calculate the coefficients of $f$, which is easy as far as
computation is concerned. Suppose that $N$ is a prime (thus equal
to $p$), or $N$ is a product of primes $pq$ and $X_0(q)$ is of
genus $0$ (thus $q=2,3,5,7$ or $13$).

In the appendix, we give for each such case an equation of
$X_0(q)$ of the form $xy=p^k$, thus the action of the Hecke
operators $T_2$ and $T_3$ over $X_0(q)$, which is given by an
equation much simpler than the equation of modular polynomials
$\Phi_2(j,j'),\Phi_3(j,j')$ (which give the action of $T_2,T_3$ on
$X_0(1)$, parametrized by the modular invariant $j$; cf. section

Let $u=x$ if $N=pq$ and $u=j$ if $N=p$. The Fourier expansion of
$u$ at infinity is $1/q+\cdots$. Let $f(q)=\sum a_nq^n$ a normalized 
newform of level $N$ and weight 2 corresponding to a vector
$\sum x_S[S]$ of $M_N^0\otimes K$, where $K$ is the extension of
$\Q$ generated by the $a_n$. Therefore there exists a prime ideal
$\wp$ of $K$ over $p$ so that \bean\left(\sum
x_S \cdot u(S)\right)f(q)\frac{dq}{q}\equiv \sum x_S\frac{du}{u-u(S)} \pmod{\wp}
.\eean (it is about the congruence between Laurent series in

Suppose for example that $f$ corresponds to a Weil curve of
conductor $N$, so that $a_n$ are in $\Z$. The $x_S$ are in $\Z$
and one can prove that $\sum  x_Su(S)\neq 0$. Thus we know $a_n \m
p$ for all $n$. Hasse's inequality $|a_l|<2\sqrt{l}$ for $l$ prime
proves that we know the $a_n$ for $n<p^2/16$.


\subsection{Explicit construction of the net $M_N$}

In this section we suppose that $N$ is odd. Suppose that given an
explicit model of the curve $X_0(N_1)$, and so the action of the
Hecke operator $T_2$ on that model (cf. Appedix).


First we need to find a supersingular points. Note that they are
defined over $\F_{p^2}$. For example suppose that $N=p$. First we
check to see if $p$ is inert in one of the 9 imaginary quadratic
fields of class number 1. If yes, then one can take for the
initial value of $j$ the modular invariant of the curve of complex
multiplication by the ring of integers of corresponding fields. If
not, one can know a list of minimal polynomials of modular
invariants of elliptic curves of complex multiplication by
imaginary quadratic fields of small class numbers, and apply the
same method. One needs here to solve over $\F_{p^2}$ a polynomial
equation, which can be done in $\log p$ operations -- at least
probabilistically. Finally suppose that all these attempts fail.
There remains the possibility to enumerate all the values of
$\F_p$ until finding a supersingular value. We know there must
exist a supersingular $j$-invariant in $\F_p$, 
but unfortunately only a very small number---on the 
order of the size of the class group of $\Q(\sqrt{-p})$, or
approximately $\sqrt{p}$.

So assume we know a supersingular point $S_1$. Knowing the action
of $T_2$ on the model given by $X_0(N)$ allows us to obtain the
three supersingular points $S_2,S_3,S_4$ (not necessarily
distinct) related to $S_1$ by a 2-isogeny. It comes down to
solving a degree 3 polynomial over $\F_{p^2}$, which needs
extracting cubic and square roots, operations that need $O(\log
p)$ operations. Sometimes we may as well exlude this computation.
Suppose that $n=p$ and that we have, say $p\equiv 2 (\m 3)$. Thus
$p$ is inert in $\Q(\sqrt{-3})$, so $j=0$ is a supersingular
value, and we know that the three isogenies of degree 2 send the
curve of the invariant to the curve of complex multiplication by
$\Z[\sqrt{-3}]$, for which the invariant is $j=54000$.

In any case, we have at most one time when we need to solve a 3rd
degree equation: once $S_2$ is known, we search from $S_i$ ($i\geq
2$) the three supersingular points which are related, but we
already know one, so we only need to solve a second degree
equation, which comes down to square roots over $\F_{p^2}$ which
is fast (probabilistic methods require $O(\log p)$ operations
using an algorithm that is very simple to implement).

To prove that we can find, step by step, all the supersingular
points of $M_N$ it is enough to prove that the graph of $T_2$ (and
more generally of $T_n$) is connected. But, as Serre remarked, the
eigenvalue $a_2=3$ of $T_2$ over $M_N$ has multiplicity equal to
the number of connected components of the graph of $T_2$. But in
$M_N$, the space $M_N^0$ corresponding to the cusp forms of
codimension 1, so 3 is a simple eigenvalue in $M_N$ (because for a
cusp form we have $|a_2|<2\sqrt{2}$), so the graph of $T_2$ is

In conclusion, an algorithm in $O(N\log N)$ operations gives all
the supersingular points and the Brandt matrix $B_2$ associated to
them. One of the advantages of this matrix is that it is very
sparse; on each line and column there are at most 3 nonzero terms,
which are integers whose sum is 3. This allows, given an
eigenvalue, to find very quickly, if $N$ is large, the
corresponding eigenvectors.



\item Take for example $N=p=37$. Since $37$ is inert in
$\Q(\sqrt{-2})$, one can take as the first vertex of our graph the
curve $E_1$ of complex multiplications by $\Z[\sqrt{-2}]$, for
which the modular invariant is $j_1=8000\equiv 8 \m 37$. We need
to find now all the invariants of curves 2-isogenous to this,
i.e., to solve the equation $\Phi_2(x,8000)\equiv 0 (\m 37)$. But
$\sqrt{-2}$ is an endomorphism of degree 2 of $E_1$, so $j_1$ is a
root (over $\Q$) of the polynomial $\Phi_2(x,8000)$. Dividing this
polynomial by $x-8000$ we get a second degree polynomials with
roots $j_2,j_3$, the invariants of the other two curves, $E_2,E_3$
related to $E_1$ by a degree 2 isogeny. Let $\w\in\F_{p^2}$ so
that $\w^2=-2$. One gets that then $j_2=3+14\w,j_3=3-14\w$.

Another method to find $j_2,j_3$ consists in remarking that 37 is
equally inert in the field $K=\Q(\sqrt{-15})$, for which the class
number is 2. The second degree polynomial giving the values of the
modular invariants of 2 curves of complex multiplication by the
ring of integers of $K$ is $x^2+191025x-121287375$, whose roots
generate $\Q(\sqrt{5})$,  so modulo 37 are conjugate in
$\F_{37^2}$. We can thus find $j_2,j_3$.

For $N$ prime congruent to 1 mod 12, the number of supersingular
curves mod $N$ is $(N-1)/12$. For $N=37$ we get 3 supersingular
curves. It remains to show that the action of $T_2$ on $E_2$ (by
conjugations we get the action on $E_3$). It is not possible to
have 2 isogenies of $E_2$ on $E_1$, because then we would have 5
isogenies of degree 2 starting in $E_1$. Therefore there is one
2-isogeny of $E_2$ over $E_2$.

Actually, if there is a 2-isogeny of an elliptic curve of
invariant $j$ on itself, this invariant is the root of the
equation $\Phi_2(x,x)=0$, a fourth degree equation that can be
written as \[(x-1728)(x-8000)(x+3375)^2\]. (To see this, one can
make the computation of the equation of $\Phi_2(j,j')$ above. One
can also search which are the curves of complex multiplication
that admit a degree 2 endomorphism, i.e., which are the imaginary
quadratic fields that contains an element of norm 2. One finds, by
multiplication by the units of the (``corps pres?'') the elements
$1+i,\sqrt{-2}, \frac{1+\sqrt{-7}}{2}$ and $\frac{1-\sqrt{-7}}{2}$
that are the endomorphisms of degree 2 of the curves of invariant
$j=1728,j=8000$ and for the last two, $j=-3375$.)

By order, mod $p$, the graph of $T_2$ cannot contain a loop of a
supersingular curve on itself -- although this curve is defined
over $\F_p$ (and, more precisely, it is one of 3 curves described
above). Therefore, there are 2 isogenies relating $E_2$ to $E_3$
and the graph of $T_2$ acting on $M_{37}$ is completely

To compute the corresponding eigenvectors, one can evidently
diagonalize the matrix $(3,3)$ of $T_2$ but there is a simpler

the involution $W_{37}=-\Frob_{37}$ separates $M_{37}$ in an
obvious way into two orthogonal proper subspaces, one generated by
$u_1=[E_2]-[E_3]$, associated with the eigenvalue 1, and the other
associated with the eigenvalue -1, generates by
$\Eis=[E_1]+[E_2]+[E_3]$ and the vector product of $u_1$ and $\Eis$,
let it be $u_2=2[E_1]-[E_2]-[E_3]$. One can deduce, without
recourse to $T_2$, that there exist 2 newforms for which the
$q$-expansion has rational coefficient, and thus that $J_0(37)$,
the jacobian of $X_0(37)$ is isogenous to the product of 2
elliptic curves (which is well-known, see for example \cite{9}).
Formula (1) above allows us to obtain the first 83 terms of their
function $L$.

\item $p-37,N=2\cdot 37$.

To study $X_0(74)$ one uses the homomorphism $\phi_2$ of $M_{74}$
to $M_{37}$ defined previously. The fibres of reach of the three
supersingular points $[E_1],[E_2]$ and $[E_3]$ of $X_0(1) \m 37$
are formed by three distinct supersingular points of $X_0(2) \m
2$. In a general way, write that if $\vvn{S}{k}$ are the
supersingular points of $X_0(qM) \m p$ above a supersingular point
$S$ of $X_0(M) \m p$ ($p,q$ coprime and coprime with $M$), one has
the formula
\[\frac{q+1}{\Aut S}=\sum_1^k\frac{1}{\Aut S_i}.\]

The equation of $X_0(2)$ used here is that described in the
appendix: $uv=2^{12}$, the involution $W_2$ switching $u$ and $v$.
Recall that $W_{37}=-\Frob_{37}$ and that $j=(u+16)^3/u$ (where $j$
is the invariant of the curve $E$, image of the point $(E,C)$ of
$X_0(2)$ via the homomorphism ``oubli -- oblivion?'' of $X_0(2)$ on
$X_0(1)$.) From the equation $j=j_1=8$ one gets the values of the
three supersingular points of $E_1$, of coordinates
$u_1=(-1+\w)/2,u_2=(-1-\w)/2=W_2(u_1)$ and $u_3=27=W_2(u_2)$.
(Here again, it is possible to guess the value of $u_3$, because
it is clear by the action of $T(2)$ on $X_0(1) \m 37$ done
previously that one of the above $E_1$ must be invariant relative
to $W_2$; or the two solutions of $u^2=2^{12}$ are $u_1,-u_1$.
Replacing them in the equation that gives $j$ one can see that it
is about $u_1$. To get $u_2,u_3$ it is enough to solve a second
degree equation.)

One can compute that $u_4=W_2(u_1)=2^{12}/u_1=-5-5\w$, and one
finds that the corresponding invariant $j(u_4)$ is $j_2=3+14\w$.
One solves the second degree equation given 2 other points above
by $j_2$ and so $u_5=15+17\w,u_6=16-12\w$. Note that
$u_7=W_2(u_2)=\bar{u}_4, u_8=W_2(u_5)=\bar{u}_5$ and
$u_9=W_2(u_6)=\bar{u}_6$ the $x$-coordinates of three
supersingular points over $E_3$ ($x\to \bar{x}$ being the
nontrivial automorphism of $\F_{p^2}$.) We get the list of all
supersingular points of $X_0(2) \m 37$.

As said above, the space $M_{74}^{new}$ corresponding to the
newforms is the intersection of the kernel of $\phi_2$ and the
kernel of $\phi_2W_2$. If we write $[u_i]$, $i=1,\ldots, 9$) the
generators of $M_{74}$ corresponding to the supersingular points
of $x$-coordinate $u_i$, an examination of the action of $W_{37}$
and $W_2$ prove that $M_{74}^{new}$ is the direct sum of two
2-dimensional subspaces, one $G_1$, generated by
$e_1=[u_1]-[u_2]-[u_4]+[u_7]-[u_9]$ and
$e_2=[u_5]-[u_6]-[u_8]+[u_9]$, on which $W_{37}=-W_2=1$ and the
other, $G_2$, generated by
$e_3=[u_1]+[u_2]-2[u_3]+[u_4]-[u_6]+[u_7]-[u_9]$, on which

Using the equation of $T_3$ acting on $X_0(2)$ (cf. appendix), one
can prove that the matrix of $T_3$ acting on $G_1$ (respectively
$G_2$) in the basis $(e_1,e_2)$ (respectively $(e_3,e_4)$) is $\left(%
  -1 & 1 \\
  1 & 0 \\
\right)$, of characteristic polynomial $x^2+x-1$ (respectively $\left(%
  3 & 1 \\
  1 & 0 \\
\right)$, of characteristic polynomial $x^2-3x-1$).

One deduces that $J_0^{new}(74)$ is isogenous to the product of
two abelian simple varieties, $A_1$ (resp. $A_2$), of real
multiplication by the ring of integers of $\Q(\sqrt{5}),$
(respectively $\Q(\sqrt{13})$.)

If $\l=\frac{-1+\sqrt{5}}{2},\mu=\frac{3+\sqrt{13}}{2}$, then the
vectors $v_1=e_1+(\l+1)e_2,v_2=e_1-\l e_2,v_3=\mu
e_3+e_4,v_4=(3-\mu)e_3+e_4$ corresponding to the 4 newforms
$f_1,f_2,f_3,f_4$ of weights 2 and level $74$. Using (1) one gets
the first 83 values of the coefficients of these newforms. For
example for $f_1$ the list of the first values of $a_l$ is
  l & 2 & 3 & 5 & 7 & 11 & 13 \\
  a_l & 1 & \frac{-1+\sqrt{5}}{2} & \frac{1-3\sqrt{5}}{2} & -1+\sqrt{5} & \frac{-5-\sqrt{5}}{2} & \frac{1+3\sqrt{5}}{2}\\

and for $f_3$ one gets
  l & 2 & 3 & 5 & 7 & 11 & 13 \\
  a_l & -1 & \frac{3+\sqrt{13}}{2} & -1-\sqrt{13} & \frac{1-\sqrt{13}}{2} & \frac{-1-\sqrt{13}}{2} & \frac{-1+\sqrt{13}}{2}\\


\section{Application to the study of Weil curves}

Let $f=\sum a_nq^n$ be a newform of weight 2 and level $N$ with
integer coefficients. They correspond to a strong Weil curve $\E$
of conductor $N$. Unfortunately the coefficients $a_n$ don't give
too much information on $\E$ and do not allow us to obtain a
simple equation for $\E$. (In \cite{10} there is a method due to
Serre that sometimes allows us to get such an equation, but that
method is not systematic.) Here we give a method that at least
when $N=p$ is a prime, one can solve the problem.

From now on let $N$ be a prime. According to the last section, for
a newform $f$ there is associated a vector $v_f=\sum x_S[S]$,
$x_S\in\Z$, an eigenvector of the Hecke operators defined in 2.1.
Theorem 1 doesn't describe the isomorphism (which is not
canonical) between $S_2(N)$ and $M_N^0\otimes\CC$. But suppose
known the terms $a_n$ of $f$ ($a_2$ is sufficient in general). The
construction of section 2.4 gives us both the supersingular values
$\m N$ and the graph of $T_2$ acting on $M_N$. We can determine
the eigenspace $V_2$ associated with the eigenvalue $a_2$. If it
is of dimension 1, we also have $v_f$, or at least the space it
generates. Otherwise, we apply $T_3$ to $V_2$ (which is of
tentatively small dimension---for conductors $< 80000$, $\dim V_2$
doesn't goes beyond 6), until finding a 1-dimensional space,
corresponding to the same eigenvalues of the operators $T_l$ as
$f$. Choose in this space a vector $r_f=\sum x_E[E]$ with integer
$x_E$ coprime in pairs\footnote{That seems to strong to me; do we
just mean that the gcd of coefficients is $1$?}; 
then $r_f$ is determined up to sign.

To go further, we need a geometric interpretation of the $x_E$.
Let $\d=\pm N^\delta$, the discriminant of the minimal Weierstrass
model of $\E$, let $\phi:X_0(N)\to\E$ be a minimal cover of $\E$
 of degree $n=\deg \phi$.

According to Deligne-Rapoport \cite{5}, there exists a model
$X_0(N)_{/\Z}$ of $X_0(N)$ defined over $\Z$ for which reduction
mod $N$ is the union of two projective lines, one $C_\infty$
classifying the elliptic curves of characteristic $N$ provided
with the group scheme kernel of the Frobenius (this
corresponding to inseparable isogenies), the other one, $C_0$,
classifying the curves provided with Verschiebung. These two
lines intersect at supersingular points. As far as the curve $\E$
is concerned, reduction mod $N$ of its Neron model 
has identity component  $\E^0_{/\F_N}$ isomorphic to $\F_{N^2}$ of the
multiplicative group $G_m$. One can prove that the cover extends
to $X_0(N)_{/\Z}-\S$ where $\S$ is the set of all supersingular
points of characteristic $N$, and define by restriction a regular
``application ?'' of $C_\infty$ on $\E^0_{/\overline{\F}_N}$, of a
rational function $\phi$ over $C_\infty$, for which the poles and
zeros are in $\E$. The divisor $\sum \l_E[E]$ of $\phi$, $E$ going
through all the supersingular curves $\m N$, and thus an element
of $M_N^0$, defined up to sign (depending on the choice of
isomorphism of $\E^0_{/\F_N}$ over $G_m$.)

\begin{proposition} In the above notation the divisor $(\Phi)=\sum
\l_E[E]$ is equal to $\pm r_f$.

It is not difficult to see that $(\Phi)$ is proportional to $r_f$.
By contradiction, the fact that the $l_E$ are coprime with one
another is obtained from the result of Ribet which says that if
$l$ is a prime different from 2, 3 then all cusp forms mod $l$ of
weight 2 and level $Np$ (where $Np$ is square-free) for which the
associated representation mod $l$ is irreducible and not ramified
at $p$, comes from a cusp form mod $l$ of weight 2 and level $N$
(this result was conjectured by Serre in 1985. This also shows
that the Taniyama-Weil conjecture implies the Fermat theorem.)

To prove the previous theorem, one proves first that $\delta$ is
related to $\l_E$ by $\delta=\gcd (\l_E\w_E-\l_F\w_F)$ where
$\w_E$ is the number of automorphisms of $E$. Suppose that a prime
number $l$ divides the gcd is $\l_E$. It also divides $\delta$,
and one deduces from here that $p$ is not ramified in the field of
points of order $l$ or $\E$. If $l$ is coprime with 6 Ribet's
\footnote{K.Ribet, {\it Lectures on Serre's conjectures}, MSRI,
Fall 1986} theorem shows that the modular form $f$ associated to
$\E$ is congruent mod $l$ to a modular form of weight 2 and level
1, which cannot be but the Eisenstein series. The curve $\E$ is
semi-stable, which implies (\cite{16}, p.306) that $\E$ or a curve
$\Q$-isogenous to it has a point of finite order $l$. If $l=2,3$
we get the same result due to \cite{4}, Appendix. Now, we know
explicitly the curves of prime conductor with torsion \cite{11}
namely the curves 11A and 11B of \cite{19}, which have a point of
order 5, curves 17A,17B,17C (point of order 4), 17D (point of
order 2), 19A and 19B (point of order 3), 37B, 37C (point of order
3) and the curves of Setzer-Neumann \cite{18}, which have a point
of order 2. In each of these cases, we know $\delta$, which is
equal to the number of finite points rational over $\Q$ of the
considered curves, and one can verify that the $\l_E$ are coprime
with one another. This proves the proposition. Note that along the
proof we showed that Ribet's theorem implies the following

Let $E$ be a strong Weil curve of prime conductor $N$. The
valuation of its discriminant in $N$ is equal to the number of
torsion points of $E(\Q)$.

We state without proof the theorem that allows us to get an
explicit equation for $\E$ once we know the $\l_E$.

Let $\E$ be a strong Weil curve of prime conductor $N$, and $\sum
\l_E[E]$ the element of $M_N^0$ associated to $\E$ via the
constructions above. There exists an equation of $\E$
with $c_4,c_6\in\Z$ so that, if $H=\max
(\sqrt{|c_4|},\sqrt[3]{|c_6|})$ we have: \be

\item $H\leq \frac{8n}{\sqrt{N}-2}(\log (H^6/1728)+b)$, where

\item Let $\d'=(c_4^3-c_6^2)/1728$. Then $\d'=\d$ if $\E$ is
supersingular in characteristic 2, and $\d'=\d$ or $2^{12}\d$ otherwise.

\item $c_4\equiv (\sum\l_Ej_E)^4 \m N$.

\item $c_6\equiv -(\sum\l_Ej_E)^6 \m N$.

\item $n\delta=\sum \l_E^2\w_E$.


If the $\l_E$ are known then 5 allows us to get $n$ and 1 allows
us to find a bound on $H$, thus on $c_4,c_6$. By 2 we have
$c_4^3-c_6^2=1728\d',$ which allows us to find $c_4,c_6$. The
congruences 3 and 4 allow us to reduce the number of computations
significantly. Thus we have found an equation of a strong Weil
curve corresponding to the initial newform $f$.

This method also allows us to prove that an elliptic curve of
small prime conductor is a Weil curve. Suppose that we are given
such a curve by its equation. Then we may compute the number of
its points $N_l$ mod $l$ for $l=2,3,\ldots$. Next we search, by
the method of graphs, whether $a_2=3-N_2$ is the eigenvalue of
$T_2$ acting on $M_N$. If not then the Taniyama-Weil conjecture is
false. If yes, then continue with $T_3$ acting on the found
eigenspace, if it is not of dimension 1, until we get an
eigenspace of dimension 1 for the Hecke operators, with integers
eigenvalues. If there is no such thing, then we get a
counterexample to the Taniyama-Weil conjecture. If there is one,
we compute the equation of a corresponding Weil curve. If this
curve is isogenous to the initial curve, we are done. Otherwise,
the initial curve is not a Weil curve.

In particular, this allows us to prove that the elliptic equation
\[y^2+y=x^3-7x+6\] of conductor 5077, is a Weil curve.

This curve seems to be the smallest curve (ordering the curves by
their conductors) having a Mordell-Weil rank $\geq 3$ \cite{3}.
The interest in it is the following:

Let $f(z)=\sum a_nq^n$ ($q=e^{2\pi iz}$), a newform of weight 2
and conductor $N$, and let $L(s)=\sum a_nn^{-s}$, the associated
$L$ function. If the order of $L$ in 1 is $\geq 3$ then Goldfeld
proved that there exists a computable constant $C_f$ so that
\[\log p<C_fh(-p),\]where $p\equiv 3 (\m 4)$ is a prime number coprime with $N$ and
$h(-p)$ is the number of classes of imaginary quadratic fields of
discriminant $-p$. We have other formulas, but more complicated,
in the case of imaginary quadratic fields of non-prime
discriminant (see \cite{13} for example).

If the Birch and Swinnerton-Dyer conjecture is true, all the Weil
curves for which the Mordell-Weil group over $\Q$ is of rank $\geq
3$ have to be given by such modular forms, but until the work of
Gross and Zagier \cite{8}, there was no way to verify that the
derivative at 1 of the $L$ function of a Weil curve is indeed 0.
The results of Gross and Zagier allow to write $L'(1)$ as the
product of a non-zero factor easily computable and the
N\'eron-Tate height of a Heegner point (cf. \cite{8} for more
details.) It is therefore possible, by decreasing the height of
rational points on the curve and increasing $L'(1)$ by a careful
computation, to prove that $L$ is of order $\geq 3$ at $s=1$. (In
all the previous, we considered odd Weil curves, i.e., for which
the $L$ function has an odd order at 1 -- or if one prefers for
which the sign of the functional equation is -1.)

One has several method to construct Weil curves for which the
Mordell-Weil group is of rank $\geq 3$ (and which are good
candidates for the preceding question: by the method of
Gross-Zagier, one may compute $L'(1)$. If it is zero, one has an
$L$ function which allows to obtain an increase of the absolute
value of the discriminant of imaginary quadratic fields of given
class numbers; if it is non-zero, the conjecture of Birch and
Swinnerton-Dyer is false.) One can, for example, search for curves
of complex multiplication of rank 3 (we know that they are Weil
curves), but the constant $C_f$ is very large. One can deform\footnote{Twist?} a
Weil curve (for example the curve 37C of \cite{19} until getting a
rank 3 curve (for the curve 37C, one can deform by
$\Q(\sqrt{-139})$, as shown by Gross and Zagier \cite{8}.) This
leads to a constant $C_f$ of order of 7000

One may choose some elliptic curve defined over $\Q$, or rank 3,
and try to prove that it is a Weil curve. This was done in
\cite{10} for the mentioned curve of conductor 5077, using the
trace formula. But the computation is very long. The method of
graphs allows us to do it in about 5 seconds an a computer that
needed 5 hours with the mentioned method.

For this curve, one has $C_f<50$: all imaginary quadratic curves
of discriminant $d$ with $|d|>e^{150}$ therefore has a class
number $\geq 4$. On the other hand, there is no imaginary
quadratic field of discriminant $d$ and class number 3 for
$907<|d|<10^{2500}$ \cite {12}. Therefore (after an examination of
a table of class numbers of the first quadratic fields):

The imaginary quadratic fields of class number 3 are the 16 fields
of discriminant:

\section{Application to a conjecture of Serre}

Let $\r$ be a continuous representation of $Gal(\overline{\Q}/\Q)$
in $GL_2(V)$ where $V$ is a dimension 2 vector space over a finite
field $\F_q$ of characteristic $p$. Assume this is an odd
representation, i.e., that $\r(c)$ the image of the complex
conjugation, seen as an element of $Gal(\overline{\Q}/\Q)$ has
eigenvalues 1 and -1. In that case put $G=Im\r$.

In \cite{17} Serre defines the level, the character and the weight
of such a representation:


\item The level.

Let $l$ be a prime number different from $p$. Write $G_i$
($i=0,\ldots$) the groups of ramifications of $\r$ at $l$. Let
\[n(l)=\sum_{i=0}^\infty \frac{g_i}{g_0}\codim V^{G_i},\]where

The conductor of the representation $\r$ is defined as
\[N=\prod_{l\neq p}l^{n(l)}.\]

\item The character.

The determinant of $\r$ yields a character of
$Gal(\overline{\Q}/\Q)$ in $\F_q^*$, for which the conductor
divides $pN$. Therefore, one can write
\[\det\r=\e\chi^{k-1},\]where $\chi$ is the cyclotomic character
of conductor $p$ and $\e$ is the character $(\Z/N\Z)^*\to\F_q^*.$
The integer $k$ is defined mod $(p-1)$, and the fact that the
representation is odd implies that $\e(-1)=(-1)^k$.

By definition, $\e$ is the character of the representation $\r$.

\item The weight.

The integer $k$ above is defined mod $(p-1)$. Read Serre's article
for the definition of the weight $k\in\Z$ of the representation
$\r$. As the conductor $N$ depends only on the behavior of $\r$ ar
places coprime with $p$, the definition of weight only uses the
local properties at $p$ of the representation $\r$.


Then Serre's conjecture is:

Let $\r$ be a representation as above, of weight $k$, level $N$
and character $\e$. Assume this representation is irreducible.
Then it comes from a cusp form $\m p$ of weight $k$, level $N$ and
character $\e$.

This conjectures, if true, has numerous consequences: it implies
the Taniyama-Weil conjecture and Fermat's theorem.

Many such representations $\r$ are modular, either by
construction, or because they are part of classical conjectures
(Langlands, Artin, $\ldots$) that carry on the conjecture (but
sometimes in a weak form, i.e., with a weight or conductor bigger
than those defined in \cite{17}.)

In order to verify (or contradict) Serre's conjecture, we need to
find the extensions $K/\Q$ of Galois group subgroup of
$GL_2(\F_q)$ of odd determinant and $p\neq 2$. It is in general
not difficult to calculate, for $l$ prime and not too large, the
trace $a_l$ of $\Frob_l$ in $GL_2(\F_q)$: if $P(x)$ is a polynomial
whose roots generate $K$ the decomposition of $P \m l$ usually
will suffice.

It is however, much harder to find modular forms $\m p$, if they
exist, that correspond to the representation $\r$ given by the
field $K$: the discriminant of $K$ is usually large, thus so is
the conductor of $\r$, which is related to it, so it is not easy
to make the computations.

\subsection{The case $SL_2(\mathbb{F}_4)$}

A troubling case is that of $p=2$, because, since $-1\equiv 1 (\m
2)$ all representations are odd.

The representations of $Gal(\overline{\Q}/\Q)$ in $GL_2(\F_2)=S_3$
(although altogether real, cf. \cite{17}) come from weight 1
modular forms; the group $S_3$ can be realized as a subgroup of
$GL_2(\CC)$. One can hope that by multiplication with convenient
Eisenstein series, one can obtain a modular form of weight and
level predicted by the Serre conjecture (cf. \cite{17} for

In order to obtain the most interesting case for characteristic 2,
one considers the representations with values in $GL_2(\F_4)$. The
isomorphism $A_5\simeq SL_2(\F_4)$ allows us to obtain several
examples. Let $K$ be an extension of $\Q$ of Galois group $A_5$.
Since $A_5$ ``immerses ?'' into $PGL_2(\CC)$, if the field is not
completely real, the associated representation $\r$ comes from a
weight 1 modular form (module Artin's conjecture, cf. \cite{2}).
Suppose now that $K$ is real. None of the classical conjectures
allow us to suspect that $\r$ comes from a modular form, even if
of higher weight or level. It is this case that we will study in
what follows. The method of graphs here is indispensable, the
modular forms that we look at having a conductor too large to be
studied with the Eichler-Selberg trace formula.

Let $P(x)=x^5+a_1x^4+a_2x^3+a_3x^2+a_4x+a_5$ be a rational
polynomial of discriminant $D$. In order that the field of roots
of $P$ be $A_5$ it is sufficient and necessary that $P$ be
irreducible, that $D$ be square-free, and that there exist a prime
number $l$ not dividing $D$ so that $P \m l$ having exactly two
roots in $\F_l$ (this last condition assuring that the group is
all of $A_5$).

It is clear that $\e=1$. If $p\mid{}D$, $p$ coprime with 30, $n(p)=1$
if it ``seulment si l'inertie en p ==?'' is of order 2, and thus the
polynomial $P$ has at most double roots mod $p$. As far as the
weight $k$ is concerned, it is either 2 or 4 according to the
ramification of $K$ at 2. To simplify the computation, we have
limited to searching examples among the representations of prime
level and weight 2.

On the other hand, since it is about representations in
$SL_2(\F_4)$m the coefficient $a_2$ of the sought modular form, if
it exists, cannot be in $\F_4$, but in $\F_{16}$. This comes from
the fact that the coefficient $a_l$ of a modular form $\m l$ is
equal to an eigenvalue of $\Frob_l$, and not to its trace. Now, if
a matrix in $SL_2(\F_4)$ is of order 5, its eigenvalues are in
$\F_{16}$ not in $\F_4$.

The examples treated above were obtained by making a systematic
search on a computer of convenient polynomials (totally real, of
type $A_5$, for which the conductor of the associated
representation is a prime $N$, and for which the weight is 2).

Thereafter, for each such polynomial $P$, one computes the
corresponding eigenvalue $a_2$ (in $\F_{16}$), and one tries to
find whether there exists a modular form mod 2 of level $N$ and
weight 2 so that $T_2$ has $a_2$ as an eigenvalue. In all the
cases considered, we have thereafter found an eigenspace of
dimension 1 or 2. Using the operators $T_3,T_5$, one calculates
the coefficients $a_3,a_5$, and verifies that they correspond to
the values predicted by the decomposition of $P$ in 3 and 5.

Clearly, this doesn't really prove that the representation $\r$
associated to $P$ is modular: we have only exhibited a modular
form mod 2 of proper level and weight for which the terms
$a_2,a_3,a_5$ are convenient. But there is a good indication of
the truthfulness of the conjecture of Serre in the considered
cases: an exhaustive search over numerous primes $N$ of the
coefficients $a_2$ of modular forms of weight 2 and level $N$
proves that it is rare that there are fields of small degree.
(Actually, is seems that 2, and in general the small primes, are
the most ``inert'' possible in the fields that appear in the Hecke
algebra of modular forms, fields which themselves in general
appear to have the largest degree possible, taking into account
constraints such as the Atkin-Lehner involutions, primes of
Eisenstein, etc. One gets that one has small factors, --
corresponding for example to elliptic curves with prime conductor
-- but this is apparently rare.)

\subsection{A few examples}


\item $P(x)=x^5-10x^3+2x^2+19x-6$.

The discriminant is $(2^3887)^2$. This polynomial is irreducible
mod 5, thus irreducible over $\Q$. Its roots are all real (apply
Sturm's algorithm). One has that \[P(x)\equiv x(x-1)(x^3+x^2-1) \m
3,\] which gives a cycle of order 3; the Galois group of $K$, the
field of roots of $P$, is thus $A_5$.


%ORIGINAL: From $P(x)\equiv (x-446)(x-126)^2(x-538)^2 \m 887$ one gets that

From $P(x)\equiv (x-462)(x-755)^2(x-788)^2 \m 887$ one gets that
the conductor $N$ of the associated representation is $N=887$. One
can also prove that 2 is ``little ramified'' in the sense of
\cite{17}, thus $\r$ has weight 2. Examining the reduction mod 2
of $P$ proves that the coefficients $a_2,a_3,a_5$ of the modular
form mod 2 of level 887 (which must correspond to $\r$ via the
Serre conjecture) are 1, 1, j (where $j\in\F_4$ has the property
that $j^2+j+1=0$).

One therefore applies the method of graphs: the space of modular
forms mod 2 of weight 2 and level 887 has dimension 73, and
computation shows that the eigenspace $G_1$ of $T_2$ corresponding
to the eigenvalue 1 has dimension 2; $T_3$ acts as the identity on
$G_1$, and $j,j^2$ are the eigenvalues of $T_5$ acting on $G_1$,
from where get a basis of $G_1$ formed by
$f_1=q+q^2+q^3+q^4+jq^5+\cdots$ and
$f_2=q+q^2+q^3+q^4+j^2q^5+\cdots$, eigenvectors of Hecke
operators. These corroborate the conjecture.

\item $P(x)=x^5-23x^3+55x^2-33x-1$.

Then $D=13613^2,P(x)\equiv (x-6308)(x-2211)^2(x-8248)^2 \m 13613$,
$N=13613$; $P$ being irreducible mod 2, $\Frob_2$ is a cycle of
order 5, and $a_2=\zeta_5$ is a fifth root of unity, viewed as an
element of $\F_{16}$. Computation also shows that in the space of
modular forms mod 2 of level 13613 and weight 2, which has
dimension 1134, $\zeta_5$ is a simple eigenvalue of $T_2$. The
coefficients $a_3,a_5$ are respectively equal to
$1+\zeta_5^2+\zeta_5^3=j$ and $\zeta_5^2+\zeta_5^3=j^2$, which are
the traces of $\Frob_3,\Frob_5$ in $SL_2(\F_4)$.

\item We write the other found polynomials; in each case there
exists a modular form of weight 2 and appropriate level, for which
the first terms $a_n$ correspond to those values predicted by the
Serre conjecture.


\section{Appendix: The curves $X_0(p)$ of genus 0}

In \cite{5}, it is proven that if $p$ is a prime number then the
curve $X_0(p)$ over $\Z_p$ is formally isomorphic to the curve of
equation $xy=p^k$, in the neighborhood of each point reducing mod
$p$ to a supersingular point $S$, $k$ being one half the number of
automorphisms of $S$.

If $X_0(p)$ has genus 0 (i.e., $p=2,3,5,7,13$) one has such a
model over $\Z$, given by the function \bean
x=\left(\frac{\eta(z)}{\eta(pz)}\right)^\frac{24}{p-1},\eean where
$\eta(z)=q^{1/24}\prod_{i=1}^\infty (1-q^n)$ and $q=e^{2\pi iz}$.

This results from Fricke \cite{7}, who gives for each of the above
$p$'s an expression of the ``oubli ?'' homomorphism $j:X_0(p)\to
X_0(1)$, which associates to each point $(E,C)$ of $X_0(p)$ the
point $(E)$ of $X_0(1)$, parametrized by the modular invariant

In the following we recall these equations and give the
expressions of the correspondences $T_2,T_3$ over these curves.
The variable $x$ is the one given by equation (2), the involution
$W_p$ switches $x$ and $y$ and the divisor of $x$ is
$(0)-(\infty)$, where $0$ and $\infty$ are two points of $X_0(p)$.


\item $p=2$ The equations given by Fricke (modified to give the
model of $X_0(2)$ over $\Z$) are:

$T_2$ is given by \[y^2-y(x^2+2^43x)-2^{12}x=0\] (to each point
$x$ is associated by $T_2$ the formal sum of points of coordinate
$y$ that are roots of this polynomial.)

$T_3$ is given by

\item $p=3$.
\[T_3:y^3-y^2(x^3+2^23^2x^2+2\cdot 3^25y)-3^6yx

\item $p=5$.
\[T_3:x^4+y^4-x^3y^3-2\cdot 3^2x^2y^2(x+y)-3^4xy(x^2+y^2)-2\cdot

\item $p=7$.
\[T_3:x^4+y^4-x^3y^3-2^23x^2y^2(x+y)-2\cdot 3\cdot 7xy(x^2+y^2)-3\cdot
53x^2y^2-2^23\cdot 7^2xy(x+y)-7^4xy=0\]

\item $p=13$.
\[T_3:x^4+y^4-x^3y^3-2\cdot 3x^2y^2(x+y)- 3\cdot 5xy(x^2+y^2)-3\cdot
11x^2y^2-2\cdot 3\cdot 13xy(x+y)-13^2xy=0\]


The polynomials above that give $T_2,T_3$ are of simpler form than
the classical modular equations $\Phi_2(j,j')$ and $\Phi_3(j,j')$
(that correspond to the action of $T_2$ and $T_3$ on $X_0(1)$).
For comparison, we recall their expressions:

\bea \Phi_2(j,j') &=& j^3+j'^3-j^2j'^2+2^43\cdot
&& + 3^45^34027jj'+2^83^75^6(j+j')-2^{12}3^95^9 \eea

\bea \Phi_3(j,j') &=&
&&-2^{16}5^33^517\cdot 263jj'(j+j')+2^{15}3^25^3(j^3+j'^3)+2\cdot
3^413\cdot 193\cdot 6367j^2j'^2\\
&& - 2^{31}5^622973jj'+2^{30}3^35^6(j^2+j'^2)+2^{45}3^35^9(j+j')


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