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\title{The Method of Graphs. Examples and Applications}
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\author{J.-F. Mestre. \\{}Tr. Andrei Jorza}
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\begin{document}
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\maketitle
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\section{Introduction}
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Let $S_k(N,\e)$ be the space of cusp forms of weight $k$, level
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$N$ and character $\e$, where $k$ and $N$ are integers $\geq 1$,
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and $\e$ is a Dirichlet character $N$. There are several ways
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to construct a basis. For example one can use Selberg's trace
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formula. Denote by $\Tr(n)$ the trace of $T_n$, the $n$-th Hecke
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operator. The function \[f=\sum_{n=1}^\infty \Tr(n)q^n\] is in
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$S_k(N,\e)$. The set of $f_i=T_if$ generate this space, and this
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theoretically allows us to construct a basis. For example, if $N$
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is prime, $\e=1$ and $k=2$, then the set of the $f_i$ ($1\leq
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i\leq g$ where $g$ is the genus of $X_0(N)$) is a basis of
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$S_2(N,1)$.
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But, even in that case, which is the most favorable, the
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computations become hard: on an average computer we can only hope
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to treat $N$ of the size of $5000$ (always with $N$ prime, weight
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2 and trivial character); actually, the computation of $\Tr(n)$
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requires the knowledge of many class numbers of imaginary quadratic
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fields of discriminant of order at most~$n$ and
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to obtain a basis
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$\vvn{f}{g}$ one needs to compute $\Tr(n)$ for $n\leq g^2$.
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In the following section we describe the ``method of graphs'', which
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relies on the results of Deuring and Eichler, and developed by J.
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Oestrl\'e and myself, which allows us to obtain a basis for $S_2(N,1)$
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more quickly (at least when $N$ is a prime).
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In the second section, we indicate how this method allows us to prove
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that certain elliptic curves defined over $\Q$ are Weil
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curves\footnote{It is now a theorem that every elliptic curve
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is a Weil curve, i.e., the Shimura-Taniyama-Weil conjecture
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is true. -- William Stein}
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(which, by providing an adequate Weil curve, yields all the
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imaginary quadratic fields of class number at most 3, due to a
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result of Goldfeld and recent works of Gross and Zagier).
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The third section is dedicated to the verification of a conjecture
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of Serre in certain particular cases; this is
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possible because of the method described in the first section. It is
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known that this conjecture, if it is true, has numerous
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consequences (e.g., the Shimura-Taniyama-Weil conjecture, and thus
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Fermat's Last Theorem).
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\section{The method of graphs}
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\subsection{Definitions and notations}
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In the following $p$ is a prime number and $N_1$ is a positive
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integer coprime to~$p$. Set $N=pN_1$.
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Let \[M_N=\oplus_S\Z[S]\]where $S$ is taken over all supersingular
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points of $X_0(N_1)$ in characteristic $p$, i.e., over the set of
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isomorphism classes of pairs $(E,C)$ consisting of an elliptic
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curve $E$ defined over $\overline{\F}_p$ and a cyclic group $C$ of $E$
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of order $N_1$. Two such pairs are identified if they are, in the
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obvious sense, $\overline{\F}_p$-isomorphic.
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Let \[\a_S=\frac{|\Aut(S)|}{2},\]where $\Aut(S)$ is the group of
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$\overline{\F}_p$-automorphisms of $S$. We always have $\a_S\leq
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12$, and if $p$ does not divide 6 then $\a_S\leq 3$.
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Therefore we can define a scalar product on $M_N$ by
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$\s{S,S}=\a_S$ and $\s{S,S'}=0$ if $S\neq S'$. Let
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$\Eis=\sum\a_S^{-1}[S]$ and let
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\[
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M_N^0=\left\{\sum x_S[S] : \sum x_S=0\right\}
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\]
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be the subspace orthogonal to $\Eis$.
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For all integers $n\geq 1$ coprime with $p$ we define an operator
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$T_n$ on $M$ by \[T_n(E,C)=\sum_{C_n}(E/C_n,(C+C_n)/C_n)\]
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where $C_n$ runs over all the cyclic subgroups of
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order~$n$ such that $C\i C_n=0$.
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For all $q\mid{}N_1$ and coprime with $q'=N_1/q$, we define the same
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way the Atkin-Lehner involutions $W_q$ by
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\[W_q(E,C)=(E/q'C,(E_q+C)/q'C),\]where $E_q$ is the group of
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points of order dividing $q$ of $E$.
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Finally we define an involution $W_p$ by $W_p=-\Frob_p$, where
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$\Frob_p$ is the endomorphism of $M_N$ that transforms $(E,C)$ to
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$(E^p,C^p)$. (The fact that it is an involution reflects that the
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supersingular points are defined over $\F_{p^2}$.)
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These operators have the following properties: the set of $W_q$
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and $T_n$ ($n$ coprime with $N$) generate an abelian semigroup of
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hermitian operators with respect to the scalar product $\s{\cdot,
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\cdot}$. The $T_n$ commute with each other for all $n$ coprime to $p$. If
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$q=q_1q_2$ ($q_1,q_2$ coprime) and if $n=n_1n_2$ ($n_1,n_2$
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coprime with each other and with $p$) then $W_q=W_{q_1}W_{q_2}$
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and $T_n=T_{n_1}T_{n_2}$.
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For all $d\mid N_1$ we have a homomorphism $\phi_d:M_N\to M_{N/d}$
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that transforms $(E,C)$ to $(E,dC)$. This
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homomorphism commutes with $T_n$ ($n$ coprime with $N$), and with
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$W_q$ (for $q\mid{}N/d$). For $d\mid{}N_1$ and coprime with $N_1/d$ we have
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\[T_d\phi_d=\phi_d(T_d+W_d)\]
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\subsection{An isomorphism with $S_2(N)$}
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We consider here the space $S_2(N)$ of cusp forms of weight 2 over
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$\Gamma_0(N)$, with its natural structure of $\T$-module, where~$\T$
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is the Hecke algebra \cite{1}.
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%% NEWFORMS
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\begin{theorem} There exists an isomorphism compatible with the
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action of the Hecke operators, between $M_N^0\otimes \CC$ and the
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subspace of $S_2(N)$ generated by the newforms of level $N$ and
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oldforms coming from the cusp forms of weight 2 and level $pd$,
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$d\mid{}N_1$.
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\end{theorem}
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\begin{remark}Assume $N$ (or without loss $N_1$) is
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square-free. We can determine efficiently the subspace $M_N^0$
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corresponding to newforms in $S_2(N)$; it is the subspace
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formed by all~$x$ so that for all divisors~$d$ of $N_1$ we have
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\[
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\phi_d(x)=\phi_d(W_d(x))=0.
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\]
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In particular if $N=pq$, $q$ prime, it is the subspace of $M_{pq}$
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intersection of the kernel of $\phi_q$ and of $\phi_qW_q$.
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\end{remark}
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\subsection{Relation to the quaternion algebra}
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The matrices of the operators $T_n$ acting on $M_N$ are the same
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as the classical Brandt matrices \cite{15}, constructed using
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quaternion algebras.
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Let $B_{p,\infty}$ be the quaternion algebra over~$\Q$ ramified exactly
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at~$p$ and infinity, and let~$\O$ is an Eichler order of level $N_1$
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(defined by Eichler \cite{6} in the case when $N_1$ is
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square-free, and defined in general by Pizer \cite{14}), and let
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$\vvn{I}{h}$ be representatives of the left ideal classes of
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$\O$.
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Let $\O_i$ be the right order (i.e., right normalizers) of the
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ideals $I_i$, and $e_i$ be the number of units of $\O_i$.
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The Brandt matrix $B(n)=(b_{i,j}^{(n)})$ has $i,j$ entry
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\[
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b_{i,j}^{(n)} =
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e_j^{-1}\cdot |\{\a : \a\in
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I_j^{-1}I_i,\,\Nor(\a)\Nor(I_j)/\Nor(I_i)=n\}|
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\]
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where $\Nor$ is
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the norm over $B_{p,\infty}$ (the norm of an ideal being the
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$\gcd$ of the norms of its nonzero elements).
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In the language of supersingular curves of characteristic $p$, we
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may give these matrices (actually their transposes) the following
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interpretation:
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Let $S$ be a supersingular point as in $I.1$, i.e., a
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supersingular elliptic curve $E$ defined over $\overline{\F}_p$
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together with a cyclic group $C$ of order $N_1$. The ring of
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endomorphisms $\O_1$ of $S$ is an Eichler order of level $N_1$. To
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all the other supersingular points $S'=(E',C')$ we associate the
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set $I_{S,S'}$ of homomorphisms from $S$ to $S'$, i.e. the set of
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all homomorphisms $\a$ from $E$ to $E'$ that send $C$ to $C'$.
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This is obviously a left ideal over $\O_1$, and its inverse ideal
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is $I_{S',S}$. We can prove that all the right ideals of $\O_1$
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are obtained in this way, and the whole Eichler order of level
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$N_1$ if the rign of endomorphisms of a supersingular point $S$.
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It is clear that the general term $B_{i,j}^{(n)}$ of the $n$-th
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Brandt matrix is the number of isogenies of $S_i$ to $S_j$ (the
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supersingular points being conveniently indexed,) two such
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isogenies being identified is different by an automorphism of
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$S_j$. We can retrieve the matrix of the operator $T_n$ acting
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over $M_n$.
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On the other hand if for all pairs of supersingular points
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$(S,S')$ we associate the function \[\t_{S,S'}(q)=\sum_\a
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q^{\deg\a}\]where $\a$ goes through all the homomorphisms of $S$
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to $S'$, we retrieve the functions $\t$ classically associated
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with the ideals of the orders of the quaternions, or, if one
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prefers, associated with the positive integer quadratic forms in 4
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variables.
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It is therefore easy to prove that if $\sum{x_S}[S]$ is an
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elements of $M_N\otimes \CC$ eigenvector of all the Hecke
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operators and if $f(q)$ is the corresponding modular form, we
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have, for all $S'$
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\[x_{S'}f(q)=\sum_Sx_S\t_{S,S'}\]which allows, in theory, to find
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the coefficients $a_n$ of $f$, using the $x_S$. In practice,
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unfortunately, the computation of $a_n$ demands the knowledge of
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all the isogenies of degree $n$ to $S'$, and there doesn't seem to
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be a simple algorithm for that.
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Nevertheless, in certain cases, there exists a different method to
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calculate the coefficients of $f$, which is easy as far as
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computation is concerned. Suppose that $N$ is a prime (thus equal
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to $p$), or $N$ is a product of primes $pq$ and $X_0(q)$ is of
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genus $0$ (thus $q=2,3,5,7$ or $13$).
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In the appendix, we give for each such case an equation of
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$X_0(q)$ of the form $xy=p^k$, thus the action of the Hecke
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operators $T_2$ and $T_3$ over $X_0(q)$, which is given by an
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equation much simpler than the equation of modular polynomials
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$\Phi_2(j,j'),\Phi_3(j,j')$ (which give the action of $T_2,T_3$ on
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$X_0(1)$, parametrized by the modular invariant $j$; cf. section
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2.4).
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Let $u=x$ if $N=pq$ and $u=j$ if $N=p$. The Fourier expansion of
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$u$ at infinity is $1/q+\cdots$. Let $f(q)=\sum a_nq^n$ a normalized
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newform of level $N$ and weight 2 corresponding to a vector
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$\sum x_S[S]$ of $M_N^0\otimes K$, where $K$ is the extension of
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$\Q$ generated by the $a_n$. Therefore there exists a prime ideal
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$\wp$ of $K$ over $p$ so that \bean\left(\sum
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x_S \cdot u(S)\right)f(q)\frac{dq}{q}\equiv \sum x_S\frac{du}{u-u(S)} \pmod{\wp}
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.\eean (it is about the congruence between Laurent series in
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$q$).
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Suppose for example that $f$ corresponds to a Weil curve of
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conductor $N$, so that $a_n$ are in $\Z$. The $x_S$ are in $\Z$
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and one can prove that $\sum x_Su(S)\neq 0$. Thus we know $a_n \m
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p$ for all $n$. Hasse's inequality $|a_l|<2\sqrt{l}$ for $l$ prime
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proves that we know the $a_n$ for $n<p^2/16$.
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%% NETWORK
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\subsection{Explicit construction of the net $M_N$}
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In this section we suppose that $N$ is odd. Suppose that given an
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explicit model of the curve $X_0(N_1)$, and so the action of the
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Hecke operator $T_2$ on that model (cf. Appedix).
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%%INERT
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First we need to find a supersingular points. Note that they are
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defined over $\F_{p^2}$. For example suppose that $N=p$. First we
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check to see if $p$ is inert in one of the 9 imaginary quadratic
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fields of class number 1. If yes, then one can take for the
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initial value of $j$ the modular invariant of the curve of complex
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multiplication by the ring of integers of corresponding fields. If
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not, one can know a list of minimal polynomials of modular
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invariants of elliptic curves of complex multiplication by
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imaginary quadratic fields of small class numbers, and apply the
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same method. One needs here to solve over $\F_{p^2}$ a polynomial
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equation, which can be done in $\log p$ operations -- at least
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probabilistically. Finally suppose that all these attempts fail.
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There remains the possibility to enumerate all the values of
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$\F_p$ until finding a supersingular value. We know there must
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exist a supersingular $j$-invariant in $\F_p$,
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but unfortunately only a very small number---on the
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order of the size of the class group of $\Q(\sqrt{-p})$, or
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approximately $\sqrt{p}$.
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So assume we know a supersingular point $S_1$. Knowing the action
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of $T_2$ on the model given by $X_0(N)$ allows us to obtain the
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three supersingular points $S_2,S_3,S_4$ (not necessarily
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distinct) related to $S_1$ by a 2-isogeny. It comes down to
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solving a degree 3 polynomial over $\F_{p^2}$, which needs
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extracting cubic and square roots, operations that need $O(\log
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p)$ operations. Sometimes we may as well exlude this computation.
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Suppose that $n=p$ and that we have, say $p\equiv 2 (\m 3)$. Thus
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$p$ is inert in $\Q(\sqrt{-3})$, so $j=0$ is a supersingular
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value, and we know that the three isogenies of degree 2 send the
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curve of the invariant to the curve of complex multiplication by
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$\Z[\sqrt{-3}]$, for which the invariant is $j=54000$.
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In any case, we have at most one time when we need to solve a 3rd
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degree equation: once $S_2$ is known, we search from $S_i$ ($i\geq
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2$) the three supersingular points which are related, but we
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already know one, so we only need to solve a second degree
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equation, which comes down to square roots over $\F_{p^2}$ which
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is fast (probabilistic methods require $O(\log p)$ operations
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using an algorithm that is very simple to implement).
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To prove that we can find, step by step, all the supersingular
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points of $M_N$ it is enough to prove that the graph of $T_2$ (and
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more generally of $T_n$) is connected. But, as Serre remarked, the
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eigenvalue $a_2=3$ of $T_2$ over $M_N$ has multiplicity equal to
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the number of connected components of the graph of $T_2$. But in
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$M_N$, the space $M_N^0$ corresponding to the cusp forms of
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codimension 1, so 3 is a simple eigenvalue in $M_N$ (because for a
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cusp form we have $|a_2|<2\sqrt{2}$), so the graph of $T_2$ is
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connected.
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In conclusion, an algorithm in $O(N\log N)$ operations gives all
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the supersingular points and the Brandt matrix $B_2$ associated to
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them. One of the advantages of this matrix is that it is very
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sparse; on each line and column there are at most 3 nonzero terms,
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which are integers whose sum is 3. This allows, given an
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eigenvalue, to find very quickly, if $N$ is large, the
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corresponding eigenvectors.
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\subsection{Examples}
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\be
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\item Take for example $N=p=37$. Since $37$ is inert in
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$\Q(\sqrt{-2})$, one can take as the first vertex of our graph the
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curve $E_1$ of complex multiplications by $\Z[\sqrt{-2}]$, for
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which the modular invariant is $j_1=8000\equiv 8 \m 37$. We need
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to find now all the invariants of curves 2-isogenous to this,
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i.e., to solve the equation $\Phi_2(x,8000)\equiv 0 (\m 37)$. But
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$\sqrt{-2}$ is an endomorphism of degree 2 of $E_1$, so $j_1$ is a
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root (over $\Q$) of the polynomial $\Phi_2(x,8000)$. Dividing this
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polynomial by $x-8000$ we get a second degree polynomials with
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roots $j_2,j_3$, the invariants of the other two curves, $E_2,E_3$
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related to $E_1$ by a degree 2 isogeny. Let $\w\in\F_{p^2}$ so
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that $\w^2=-2$. One gets that then $j_2=3+14\w,j_3=3-14\w$.
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Another method to find $j_2,j_3$ consists in remarking that 37 is
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equally inert in the field $K=\Q(\sqrt{-15})$, for which the class
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number is 2. The second degree polynomial giving the values of the
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modular invariants of 2 curves of complex multiplication by the
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ring of integers of $K$ is $x^2+191025x-121287375$, whose roots
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generate $\Q(\sqrt{5})$, so modulo 37 are conjugate in
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$\F_{37^2}$. We can thus find $j_2,j_3$.
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For $N$ prime congruent to 1 mod 12, the number of supersingular
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curves mod $N$ is $(N-1)/12$. For $N=37$ we get 3 supersingular
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curves. It remains to show that the action of $T_2$ on $E_2$ (by
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conjugations we get the action on $E_3$). It is not possible to
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have 2 isogenies of $E_2$ on $E_1$, because then we would have 5
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isogenies of degree 2 starting in $E_1$. Therefore there is one
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2-isogeny of $E_2$ over $E_2$.
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Actually, if there is a 2-isogeny of an elliptic curve of
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invariant $j$ on itself, this invariant is the root of the
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equation $\Phi_2(x,x)=0$, a fourth degree equation that can be
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written as \[(x-1728)(x-8000)(x+3375)^2\]. (To see this, one can
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make the computation of the equation of $\Phi_2(j,j')$ above. One
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can also search which are the curves of complex multiplication
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that admit a degree 2 endomorphism, i.e., which are the imaginary
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quadratic fields that contains an element of norm 2. One finds, by
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multiplication by the units of the (``corps pres?'') the elements
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$1+i,\sqrt{-2}, \frac{1+\sqrt{-7}}{2}$ and $\frac{1-\sqrt{-7}}{2}$
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that are the endomorphisms of degree 2 of the curves of invariant
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$j=1728,j=8000$ and for the last two, $j=-3375$.)
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By order, mod $p$, the graph of $T_2$ cannot contain a loop of a
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supersingular curve on itself -- although this curve is defined
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over $\F_p$ (and, more precisely, it is one of 3 curves described
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above). Therefore, there are 2 isogenies relating $E_2$ to $E_3$
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and the graph of $T_2$ acting on $M_{37}$ is completely
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determined.
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To compute the corresponding eigenvectors, one can evidently
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diagonalize the matrix $(3,3)$ of $T_2$ but there is a simpler
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method:
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the involution $W_{37}=-\Frob_{37}$ separates $M_{37}$ in an
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obvious way into two orthogonal proper subspaces, one generated by
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$u_1=[E_2]-[E_3]$, associated with the eigenvalue 1, and the other
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associated with the eigenvalue -1, generates by
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$\Eis=[E_1]+[E_2]+[E_3]$ and the vector product of $u_1$ and $\Eis$,
386
let it be $u_2=2[E_1]-[E_2]-[E_3]$. One can deduce, without
387
recourse to $T_2$, that there exist 2 newforms for which the
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$q$-expansion has rational coefficient, and thus that $J_0(37)$,
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the jacobian of $X_0(37)$ is isogenous to the product of 2
390
elliptic curves (which is well-known, see for example \cite{9}).
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Formula (1) above allows us to obtain the first 83 terms of their
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function $L$.
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\item $p-37,N=2\cdot 37$.
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To study $X_0(74)$ one uses the homomorphism $\phi_2$ of $M_{74}$
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to $M_{37}$ defined previously. The fibres of reach of the three
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supersingular points $[E_1],[E_2]$ and $[E_3]$ of $X_0(1) \m 37$
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are formed by three distinct supersingular points of $X_0(2) \m
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2$. In a general way, write that if $\vvn{S}{k}$ are the
401
supersingular points of $X_0(qM) \m p$ above a supersingular point
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$S$ of $X_0(M) \m p$ ($p,q$ coprime and coprime with $M$), one has
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the formula
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\[\frac{q+1}{\Aut S}=\sum_1^k\frac{1}{\Aut S_i}.\]
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The equation of $X_0(2)$ used here is that described in the
407
appendix: $uv=2^{12}$, the involution $W_2$ switching $u$ and $v$.
408
Recall that $W_{37}=-\Frob_{37}$ and that $j=(u+16)^3/u$ (where $j$
409
is the invariant of the curve $E$, image of the point $(E,C)$ of
410
$X_0(2)$ via the homomorphism ``oubli -- oblivion?'' of $X_0(2)$ on
411
$X_0(1)$.) From the equation $j=j_1=8$ one gets the values of the
412
three supersingular points of $E_1$, of coordinates
413
$u_1=(-1+\w)/2,u_2=(-1-\w)/2=W_2(u_1)$ and $u_3=27=W_2(u_2)$.
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(Here again, it is possible to guess the value of $u_3$, because
415
it is clear by the action of $T(2)$ on $X_0(1) \m 37$ done
416
previously that one of the above $E_1$ must be invariant relative
417
to $W_2$; or the two solutions of $u^2=2^{12}$ are $u_1,-u_1$.
418
Replacing them in the equation that gives $j$ one can see that it
419
is about $u_1$. To get $u_2,u_3$ it is enough to solve a second
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degree equation.)
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One can compute that $u_4=W_2(u_1)=2^{12}/u_1=-5-5\w$, and one
423
finds that the corresponding invariant $j(u_4)$ is $j_2=3+14\w$.
424
One solves the second degree equation given 2 other points above
425
by $j_2$ and so $u_5=15+17\w,u_6=16-12\w$. Note that
426
$u_7=W_2(u_2)=\bar{u}_4, u_8=W_2(u_5)=\bar{u}_5$ and
427
$u_9=W_2(u_6)=\bar{u}_6$ the $x$-coordinates of three
428
supersingular points over $E_3$ ($x\to \bar{x}$ being the
429
nontrivial automorphism of $\F_{p^2}$.) We get the list of all
430
supersingular points of $X_0(2) \m 37$.
431
432
As said above, the space $M_{74}^{new}$ corresponding to the
433
newforms is the intersection of the kernel of $\phi_2$ and the
434
kernel of $\phi_2W_2$. If we write $[u_i]$, $i=1,\ldots, 9$) the
435
generators of $M_{74}$ corresponding to the supersingular points
436
of $x$-coordinate $u_i$, an examination of the action of $W_{37}$
437
and $W_2$ prove that $M_{74}^{new}$ is the direct sum of two
438
2-dimensional subspaces, one $G_1$, generated by
439
$e_1=[u_1]-[u_2]-[u_4]+[u_7]-[u_9]$ and
440
$e_2=[u_5]-[u_6]-[u_8]+[u_9]$, on which $W_{37}=-W_2=1$ and the
441
other, $G_2$, generated by
442
$e_3=[u_1]+[u_2]-2[u_3]+[u_4]-[u_6]+[u_7]-[u_9]$, on which
443
$W_2=-W_{37}=1$.
444
445
Using the equation of $T_3$ acting on $X_0(2)$ (cf. appendix), one
446
can prove that the matrix of $T_3$ acting on $G_1$ (respectively
447
$G_2$) in the basis $(e_1,e_2)$ (respectively $(e_3,e_4)$) is $\left(%
448
\begin{array}{cc}
449
-1 & 1 \\
450
1 & 0 \\
451
\end{array}%
452
\right)$, of characteristic polynomial $x^2+x-1$ (respectively $\left(%
453
\begin{array}{cc}
454
3 & 1 \\
455
1 & 0 \\
456
\end{array}%
457
\right)$, of characteristic polynomial $x^2-3x-1$).
458
459
One deduces that $J_0^{new}(74)$ is isogenous to the product of
460
two abelian simple varieties, $A_1$ (resp. $A_2$), of real
461
multiplication by the ring of integers of $\Q(\sqrt{5}),$
462
(respectively $\Q(\sqrt{13})$.)
463
464
If $\l=\frac{-1+\sqrt{5}}{2},\mu=\frac{3+\sqrt{13}}{2}$, then the
465
vectors $v_1=e_1+(\l+1)e_2,v_2=e_1-\l e_2,v_3=\mu
466
e_3+e_4,v_4=(3-\mu)e_3+e_4$ corresponding to the 4 newforms
467
$f_1,f_2,f_3,f_4$ of weights 2 and level $74$. Using (1) one gets
468
the first 83 values of the coefficients of these newforms. For
469
example for $f_1$ the list of the first values of $a_l$ is
470
\[\begin{array}{ccccccc}
471
l & 2 & 3 & 5 & 7 & 11 & 13 \\
472
a_l & 1 & \frac{-1+\sqrt{5}}{2} & \frac{1-3\sqrt{5}}{2} & -1+\sqrt{5} & \frac{-5-\sqrt{5}}{2} & \frac{1+3\sqrt{5}}{2}\\
473
\end{array}\]
474
475
and for $f_3$ one gets
476
\[\begin{array}{ccccccc}
477
l & 2 & 3 & 5 & 7 & 11 & 13 \\
478
a_l & -1 & \frac{3+\sqrt{13}}{2} & -1-\sqrt{13} & \frac{1-\sqrt{13}}{2} & \frac{-1-\sqrt{13}}{2} & \frac{-1+\sqrt{13}}{2}\\
479
\end{array}\]
480
481
\ee
482
483
\section{Application to the study of Weil curves}
484
485
Let $f=\sum a_nq^n$ be a newform of weight 2 and level $N$ with
486
integer coefficients. They correspond to a strong Weil curve $\E$
487
of conductor $N$. Unfortunately the coefficients $a_n$ don't give
488
too much information on $\E$ and do not allow us to obtain a
489
simple equation for $\E$. (In \cite{10} there is a method due to
490
Serre that sometimes allows us to get such an equation, but that
491
method is not systematic.) Here we give a method that at least
492
when $N=p$ is a prime, one can solve the problem.
493
494
From now on let $N$ be a prime. According to the last section, for
495
a newform $f$ there is associated a vector $v_f=\sum x_S[S]$,
496
$x_S\in\Z$, an eigenvector of the Hecke operators defined in 2.1.
497
Theorem 1 doesn't describe the isomorphism (which is not
498
canonical) between $S_2(N)$ and $M_N^0\otimes\CC$. But suppose
499
known the terms $a_n$ of $f$ ($a_2$ is sufficient in general). The
500
construction of section 2.4 gives us both the supersingular values
501
$\m N$ and the graph of $T_2$ acting on $M_N$. We can determine
502
the eigenspace $V_2$ associated with the eigenvalue $a_2$. If it
503
is of dimension 1, we also have $v_f$, or at least the space it
504
generates. Otherwise, we apply $T_3$ to $V_2$ (which is of
505
tentatively small dimension---for conductors $< 80000$, $\dim V_2$
506
doesn't goes beyond 6), until finding a 1-dimensional space,
507
corresponding to the same eigenvalues of the operators $T_l$ as
508
$f$. Choose in this space a vector $r_f=\sum x_E[E]$ with integer
509
$x_E$ coprime in pairs\footnote{That seems to strong to me; do we
510
just mean that the gcd of coefficients is $1$?};
511
then $r_f$ is determined up to sign.
512
513
To go further, we need a geometric interpretation of the $x_E$.
514
Let $\d=\pm N^\delta$, the discriminant of the minimal Weierstrass
515
model of $\E$, let $\phi:X_0(N)\to\E$ be a minimal cover of $\E$
516
of degree $n=\deg \phi$.
517
518
According to Deligne-Rapoport \cite{5}, there exists a model
519
$X_0(N)_{/\Z}$ of $X_0(N)$ defined over $\Z$ for which reduction
520
mod $N$ is the union of two projective lines, one $C_\infty$
521
classifying the elliptic curves of characteristic $N$ provided
522
with the group scheme kernel of the Frobenius (this
523
corresponding to inseparable isogenies), the other one, $C_0$,
524
classifying the curves provided with Verschiebung. These two
525
lines intersect at supersingular points. As far as the curve $\E$
526
is concerned, reduction mod $N$ of its Neron model
527
has identity component $\E^0_{/\F_N}$ isomorphic to $\F_{N^2}$ of the
528
multiplicative group $G_m$. One can prove that the cover extends
529
to $X_0(N)_{/\Z}-\S$ where $\S$ is the set of all supersingular
530
points of characteristic $N$, and define by restriction a regular
531
``application ?'' of $C_\infty$ on $\E^0_{/\overline{\F}_N}$, of a
532
rational function $\phi$ over $C_\infty$, for which the poles and
533
zeros are in $\E$. The divisor $\sum \l_E[E]$ of $\phi$, $E$ going
534
through all the supersingular curves $\m N$, and thus an element
535
of $M_N^0$, defined up to sign (depending on the choice of
536
isomorphism of $\E^0_{/\F_N}$ over $G_m$.)
537
538
\begin{proposition} In the above notation the divisor $(\Phi)=\sum
539
\l_E[E]$ is equal to $\pm r_f$.
540
\end{proposition}
541
542
It is not difficult to see that $(\Phi)$ is proportional to $r_f$.
543
By contradiction, the fact that the $l_E$ are coprime with one
544
another is obtained from the result of Ribet which says that if
545
$l$ is a prime different from 2, 3 then all cusp forms mod $l$ of
546
weight 2 and level $Np$ (where $Np$ is square-free) for which the
547
associated representation mod $l$ is irreducible and not ramified
548
at $p$, comes from a cusp form mod $l$ of weight 2 and level $N$
549
(this result was conjectured by Serre in 1985. This also shows
550
that the Taniyama-Weil conjecture implies the Fermat theorem.)
551
552
To prove the previous theorem, one proves first that $\delta$ is
553
related to $\l_E$ by $\delta=\gcd (\l_E\w_E-\l_F\w_F)$ where
554
$\w_E$ is the number of automorphisms of $E$. Suppose that a prime
555
number $l$ divides the gcd is $\l_E$. It also divides $\delta$,
556
and one deduces from here that $p$ is not ramified in the field of
557
points of order $l$ or $\E$. If $l$ is coprime with 6 Ribet's
558
\footnote{K.Ribet, {\it Lectures on Serre's conjectures}, MSRI,
559
Fall 1986} theorem shows that the modular form $f$ associated to
560
$\E$ is congruent mod $l$ to a modular form of weight 2 and level
561
1, which cannot be but the Eisenstein series. The curve $\E$ is
562
semi-stable, which implies (\cite{16}, p.306) that $\E$ or a curve
563
$\Q$-isogenous to it has a point of finite order $l$. If $l=2,3$
564
we get the same result due to \cite{4}, Appendix. Now, we know
565
explicitly the curves of prime conductor with torsion \cite{11}
566
namely the curves 11A and 11B of \cite{19}, which have a point of
567
order 5, curves 17A,17B,17C (point of order 4), 17D (point of
568
order 2), 19A and 19B (point of order 3), 37B, 37C (point of order
569
3) and the curves of Setzer-Neumann \cite{18}, which have a point
570
of order 2. In each of these cases, we know $\delta$, which is
571
equal to the number of finite points rational over $\Q$ of the
572
considered curves, and one can verify that the $\l_E$ are coprime
573
with one another. This proves the proposition. Note that along the
574
proof we showed that Ribet's theorem implies the following
575
576
\begin{theorem}
577
Let $E$ be a strong Weil curve of prime conductor $N$. The
578
valuation of its discriminant in $N$ is equal to the number of
579
torsion points of $E(\Q)$.
580
\end{theorem}
581
582
We state without proof the theorem that allows us to get an
583
explicit equation for $\E$ once we know the $\l_E$.
584
585
\begin{theorem}
586
Let $\E$ be a strong Weil curve of prime conductor $N$, and $\sum
587
\l_E[E]$ the element of $M_N^0$ associated to $\E$ via the
588
constructions above. There exists an equation of $\E$
589
\[y^2=x^3-\frac{c_4}{48}x-\frac{c_6}{864}\]
590
with $c_4,c_6\in\Z$ so that, if $H=\max
591
(\sqrt{|c_4|},\sqrt[3]{|c_6|})$ we have: \be
592
593
\item $H\leq \frac{8n}{\sqrt{N}-2}(\log (H^6/1728)+b)$, where
594
$b=(\Gamma(1/3)/\Gamma(2/3))^3=7.74316962\ldots$.
595
596
\item Let $\d'=(c_4^3-c_6^2)/1728$. Then $\d'=\d$ if $\E$ is
597
supersingular in characteristic 2, and $\d'=\d$ or $2^{12}\d$ otherwise.
598
599
\item $c_4\equiv (\sum\l_Ej_E)^4 \m N$.
600
601
\item $c_6\equiv -(\sum\l_Ej_E)^6 \m N$.
602
603
\item $n\delta=\sum \l_E^2\w_E$.
604
605
\ee
606
\end{theorem}
607
608
If the $\l_E$ are known then 5 allows us to get $n$ and 1 allows
609
us to find a bound on $H$, thus on $c_4,c_6$. By 2 we have
610
$c_4^3-c_6^2=1728\d',$ which allows us to find $c_4,c_6$. The
611
congruences 3 and 4 allow us to reduce the number of computations
612
significantly. Thus we have found an equation of a strong Weil
613
curve corresponding to the initial newform $f$.
614
615
This method also allows us to prove that an elliptic curve of
616
small prime conductor is a Weil curve. Suppose that we are given
617
such a curve by its equation. Then we may compute the number of
618
its points $N_l$ mod $l$ for $l=2,3,\ldots$. Next we search, by
619
the method of graphs, whether $a_2=3-N_2$ is the eigenvalue of
620
$T_2$ acting on $M_N$. If not then the Taniyama-Weil conjecture is
621
false. If yes, then continue with $T_3$ acting on the found
622
eigenspace, if it is not of dimension 1, until we get an
623
eigenspace of dimension 1 for the Hecke operators, with integers
624
eigenvalues. If there is no such thing, then we get a
625
counterexample to the Taniyama-Weil conjecture. If there is one,
626
we compute the equation of a corresponding Weil curve. If this
627
curve is isogenous to the initial curve, we are done. Otherwise,
628
the initial curve is not a Weil curve.
629
630
In particular, this allows us to prove that the elliptic equation
631
\[y^2+y=x^3-7x+6\] of conductor 5077, is a Weil curve.
632
633
This curve seems to be the smallest curve (ordering the curves by
634
their conductors) having a Mordell-Weil rank $\geq 3$ \cite{3}.
635
The interest in it is the following:
636
637
Let $f(z)=\sum a_nq^n$ ($q=e^{2\pi iz}$), a newform of weight 2
638
and conductor $N$, and let $L(s)=\sum a_nn^{-s}$, the associated
639
$L$ function. If the order of $L$ in 1 is $\geq 3$ then Goldfeld
640
proved that there exists a computable constant $C_f$ so that
641
\[\log p<C_fh(-p),\]where $p\equiv 3 (\m 4)$ is a prime number coprime with $N$ and
642
$h(-p)$ is the number of classes of imaginary quadratic fields of
643
discriminant $-p$. We have other formulas, but more complicated,
644
in the case of imaginary quadratic fields of non-prime
645
discriminant (see \cite{13} for example).
646
647
If the Birch and Swinnerton-Dyer conjecture is true, all the Weil
648
curves for which the Mordell-Weil group over $\Q$ is of rank $\geq
649
3$ have to be given by such modular forms, but until the work of
650
Gross and Zagier \cite{8}, there was no way to verify that the
651
derivative at 1 of the $L$ function of a Weil curve is indeed 0.
652
The results of Gross and Zagier allow to write $L'(1)$ as the
653
product of a non-zero factor easily computable and the
654
N\'eron-Tate height of a Heegner point (cf. \cite{8} for more
655
details.) It is therefore possible, by decreasing the height of
656
rational points on the curve and increasing $L'(1)$ by a careful
657
computation, to prove that $L$ is of order $\geq 3$ at $s=1$. (In
658
all the previous, we considered odd Weil curves, i.e., for which
659
the $L$ function has an odd order at 1 -- or if one prefers for
660
which the sign of the functional equation is -1.)
661
662
One has several method to construct Weil curves for which the
663
Mordell-Weil group is of rank $\geq 3$ (and which are good
664
candidates for the preceding question: by the method of
665
Gross-Zagier, one may compute $L'(1)$. If it is zero, one has an
666
$L$ function which allows to obtain an increase of the absolute
667
value of the discriminant of imaginary quadratic fields of given
668
class numbers; if it is non-zero, the conjecture of Birch and
669
Swinnerton-Dyer is false.) One can, for example, search for curves
670
of complex multiplication of rank 3 (we know that they are Weil
671
curves), but the constant $C_f$ is very large. One can deform\footnote{Twist?} a
672
Weil curve (for example the curve 37C of \cite{19} until getting a
673
rank 3 curve (for the curve 37C, one can deform by
674
$\Q(\sqrt{-139})$, as shown by Gross and Zagier \cite{8}.) This
675
leads to a constant $C_f$ of order of 7000
676
677
One may choose some elliptic curve defined over $\Q$, or rank 3,
678
and try to prove that it is a Weil curve. This was done in
679
\cite{10} for the mentioned curve of conductor 5077, using the
680
trace formula. But the computation is very long. The method of
681
graphs allows us to do it in about 5 seconds an a computer that
682
needed 5 hours with the mentioned method.
683
684
For this curve, one has $C_f<50$: all imaginary quadratic curves
685
of discriminant $d$ with $|d|>e^{150}$ therefore has a class
686
number $\geq 4$. On the other hand, there is no imaginary
687
quadratic field of discriminant $d$ and class number 3 for
688
$907<|d|<10^{2500}$ \cite {12}. Therefore (after an examination of
689
a table of class numbers of the first quadratic fields):
690
691
\begin{theorem}
692
The imaginary quadratic fields of class number 3 are the 16 fields
693
of discriminant:
694
$-23,-31,-59.-83,-107,-139,-211,-283,-307,-331,-379,-499,-547,-643,-883,-907$.
695
\end{theorem}
696
697
\section{Application to a conjecture of Serre}
698
699
Let $\r$ be a continuous representation of $Gal(\overline{\Q}/\Q)$
700
in $GL_2(V)$ where $V$ is a dimension 2 vector space over a finite
701
field $\F_q$ of characteristic $p$. Assume this is an odd
702
representation, i.e., that $\r(c)$ the image of the complex
703
conjugation, seen as an element of $Gal(\overline{\Q}/\Q)$ has
704
eigenvalues 1 and -1. In that case put $G=Im\r$.
705
706
In \cite{17} Serre defines the level, the character and the weight
707
of such a representation:
708
709
\be
710
711
\item The level.
712
713
Let $l$ be a prime number different from $p$. Write $G_i$
714
($i=0,\ldots$) the groups of ramifications of $\r$ at $l$. Let
715
\[n(l)=\sum_{i=0}^\infty \frac{g_i}{g_0}\codim V^{G_i},\]where
716
$g_i=|G_i|$.
717
718
The conductor of the representation $\r$ is defined as
719
\[N=\prod_{l\neq p}l^{n(l)}.\]
720
721
\item The character.
722
723
The determinant of $\r$ yields a character of
724
$Gal(\overline{\Q}/\Q)$ in $\F_q^*$, for which the conductor
725
divides $pN$. Therefore, one can write
726
\[\det\r=\e\chi^{k-1},\]where $\chi$ is the cyclotomic character
727
of conductor $p$ and $\e$ is the character $(\Z/N\Z)^*\to\F_q^*.$
728
The integer $k$ is defined mod $(p-1)$, and the fact that the
729
representation is odd implies that $\e(-1)=(-1)^k$.
730
731
By definition, $\e$ is the character of the representation $\r$.
732
733
\item The weight.
734
735
The integer $k$ above is defined mod $(p-1)$. Read Serre's article
736
for the definition of the weight $k\in\Z$ of the representation
737
$\r$. As the conductor $N$ depends only on the behavior of $\r$ ar
738
places coprime with $p$, the definition of weight only uses the
739
local properties at $p$ of the representation $\r$.
740
741
\ee
742
743
Then Serre's conjecture is:
744
745
\begin{conjecture}
746
Let $\r$ be a representation as above, of weight $k$, level $N$
747
and character $\e$. Assume this representation is irreducible.
748
Then it comes from a cusp form $\m p$ of weight $k$, level $N$ and
749
character $\e$.
750
\end{conjecture}
751
752
This conjectures, if true, has numerous consequences: it implies
753
the Taniyama-Weil conjecture and Fermat's theorem.
754
755
Many such representations $\r$ are modular, either by
756
construction, or because they are part of classical conjectures
757
(Langlands, Artin, $\ldots$) that carry on the conjecture (but
758
sometimes in a weak form, i.e., with a weight or conductor bigger
759
than those defined in \cite{17}.)
760
761
In order to verify (or contradict) Serre's conjecture, we need to
762
find the extensions $K/\Q$ of Galois group subgroup of
763
$GL_2(\F_q)$ of odd determinant and $p\neq 2$. It is in general
764
not difficult to calculate, for $l$ prime and not too large, the
765
trace $a_l$ of $\Frob_l$ in $GL_2(\F_q)$: if $P(x)$ is a polynomial
766
whose roots generate $K$ the decomposition of $P \m l$ usually
767
will suffice.
768
769
It is however, much harder to find modular forms $\m p$, if they
770
exist, that correspond to the representation $\r$ given by the
771
field $K$: the discriminant of $K$ is usually large, thus so is
772
the conductor of $\r$, which is related to it, so it is not easy
773
to make the computations.
774
775
\subsection{The case $SL_2(\mathbb{F}_4)$}
776
777
A troubling case is that of $p=2$, because, since $-1\equiv 1 (\m
778
2)$ all representations are odd.
779
780
The representations of $Gal(\overline{\Q}/\Q)$ in $GL_2(\F_2)=S_3$
781
(although altogether real, cf. \cite{17}) come from weight 1
782
modular forms; the group $S_3$ can be realized as a subgroup of
783
$GL_2(\CC)$. One can hope that by multiplication with convenient
784
Eisenstein series, one can obtain a modular form of weight and
785
level predicted by the Serre conjecture (cf. \cite{17} for
786
examples.)
787
788
In order to obtain the most interesting case for characteristic 2,
789
one considers the representations with values in $GL_2(\F_4)$. The
790
isomorphism $A_5\simeq SL_2(\F_4)$ allows us to obtain several
791
examples. Let $K$ be an extension of $\Q$ of Galois group $A_5$.
792
Since $A_5$ ``immerses ?'' into $PGL_2(\CC)$, if the field is not
793
completely real, the associated representation $\r$ comes from a
794
weight 1 modular form (module Artin's conjecture, cf. \cite{2}).
795
Suppose now that $K$ is real. None of the classical conjectures
796
allow us to suspect that $\r$ comes from a modular form, even if
797
of higher weight or level. It is this case that we will study in
798
what follows. The method of graphs here is indispensable, the
799
modular forms that we look at having a conductor too large to be
800
studied with the Eichler-Selberg trace formula.
801
802
Let $P(x)=x^5+a_1x^4+a_2x^3+a_3x^2+a_4x+a_5$ be a rational
803
polynomial of discriminant $D$. In order that the field of roots
804
of $P$ be $A_5$ it is sufficient and necessary that $P$ be
805
irreducible, that $D$ be square-free, and that there exist a prime
806
number $l$ not dividing $D$ so that $P \m l$ having exactly two
807
roots in $\F_l$ (this last condition assuring that the group is
808
all of $A_5$).
809
810
It is clear that $\e=1$. If $p\mid{}D$, $p$ coprime with 30, $n(p)=1$
811
if it ``seulment si l'inertie en p ==?'' is of order 2, and thus the
812
polynomial $P$ has at most double roots mod $p$. As far as the
813
weight $k$ is concerned, it is either 2 or 4 according to the
814
ramification of $K$ at 2. To simplify the computation, we have
815
limited to searching examples among the representations of prime
816
level and weight 2.
817
818
On the other hand, since it is about representations in
819
$SL_2(\F_4)$m the coefficient $a_2$ of the sought modular form, if
820
it exists, cannot be in $\F_4$, but in $\F_{16}$. This comes from
821
the fact that the coefficient $a_l$ of a modular form $\m l$ is
822
equal to an eigenvalue of $\Frob_l$, and not to its trace. Now, if
823
a matrix in $SL_2(\F_4)$ is of order 5, its eigenvalues are in
824
$\F_{16}$ not in $\F_4$.
825
826
The examples treated above were obtained by making a systematic
827
search on a computer of convenient polynomials (totally real, of
828
type $A_5$, for which the conductor of the associated
829
representation is a prime $N$, and for which the weight is 2).
830
831
Thereafter, for each such polynomial $P$, one computes the
832
corresponding eigenvalue $a_2$ (in $\F_{16}$), and one tries to
833
find whether there exists a modular form mod 2 of level $N$ and
834
weight 2 so that $T_2$ has $a_2$ as an eigenvalue. In all the
835
cases considered, we have thereafter found an eigenspace of
836
dimension 1 or 2. Using the operators $T_3,T_5$, one calculates
837
the coefficients $a_3,a_5$, and verifies that they correspond to
838
the values predicted by the decomposition of $P$ in 3 and 5.
839
840
Clearly, this doesn't really prove that the representation $\r$
841
associated to $P$ is modular: we have only exhibited a modular
842
form mod 2 of proper level and weight for which the terms
843
$a_2,a_3,a_5$ are convenient. But there is a good indication of
844
the truthfulness of the conjecture of Serre in the considered
845
cases: an exhaustive search over numerous primes $N$ of the
846
coefficients $a_2$ of modular forms of weight 2 and level $N$
847
proves that it is rare that there are fields of small degree.
848
(Actually, is seems that 2, and in general the small primes, are
849
the most ``inert'' possible in the fields that appear in the Hecke
850
algebra of modular forms, fields which themselves in general
851
appear to have the largest degree possible, taking into account
852
constraints such as the Atkin-Lehner involutions, primes of
853
Eisenstein, etc. One gets that one has small factors, --
854
corresponding for example to elliptic curves with prime conductor
855
-- but this is apparently rare.)
856
857
\subsection{A few examples}
858
859
\be
860
861
\item $P(x)=x^5-10x^3+2x^2+19x-6$.
862
863
The discriminant is $(2^3887)^2$. This polynomial is irreducible
864
mod 5, thus irreducible over $\Q$. Its roots are all real (apply
865
Sturm's algorithm). One has that \[P(x)\equiv x(x-1)(x^3+x^2-1) \m
866
3,\] which gives a cycle of order 3; the Galois group of $K$, the
867
field of roots of $P$, is thus $A_5$.
868
869
%%
870
%%
871
%% THESE COMPUTATIONS ARE NOT CORRECT.
872
%% I AM REDOING THEM
873
%%
874
%%
875
876
%ORIGINAL: From $P(x)\equiv (x-446)(x-126)^2(x-538)^2 \m 887$ one gets that
877
878
From $P(x)\equiv (x-462)(x-755)^2(x-788)^2 \m 887$ one gets that
879
the conductor $N$ of the associated representation is $N=887$. One
880
can also prove that 2 is ``little ramified'' in the sense of
881
\cite{17}, thus $\r$ has weight 2. Examining the reduction mod 2
882
of $P$ proves that the coefficients $a_2,a_3,a_5$ of the modular
883
form mod 2 of level 887 (which must correspond to $\r$ via the
884
Serre conjecture) are 1, 1, j (where $j\in\F_4$ has the property
885
that $j^2+j+1=0$).
886
887
One therefore applies the method of graphs: the space of modular
888
forms mod 2 of weight 2 and level 887 has dimension 73, and
889
computation shows that the eigenspace $G_1$ of $T_2$ corresponding
890
to the eigenvalue 1 has dimension 2; $T_3$ acts as the identity on
891
$G_1$, and $j,j^2$ are the eigenvalues of $T_5$ acting on $G_1$,
892
from where get a basis of $G_1$ formed by
893
$f_1=q+q^2+q^3+q^4+jq^5+\cdots$ and
894
$f_2=q+q^2+q^3+q^4+j^2q^5+\cdots$, eigenvectors of Hecke
895
operators. These corroborate the conjecture.
896
897
\item $P(x)=x^5-23x^3+55x^2-33x-1$.
898
899
Then $D=13613^2,P(x)\equiv (x-6308)(x-2211)^2(x-8248)^2 \m 13613$,
900
$N=13613$; $P$ being irreducible mod 2, $\Frob_2$ is a cycle of
901
order 5, and $a_2=\zeta_5$ is a fifth root of unity, viewed as an
902
element of $\F_{16}$. Computation also shows that in the space of
903
modular forms mod 2 of level 13613 and weight 2, which has
904
dimension 1134, $\zeta_5$ is a simple eigenvalue of $T_2$. The
905
coefficients $a_3,a_5$ are respectively equal to
906
$1+\zeta_5^2+\zeta_5^3=j$ and $\zeta_5^2+\zeta_5^3=j^2$, which are
907
the traces of $\Frob_3,\Frob_5$ in $SL_2(\F_4)$.
908
909
\item We write the other found polynomials; in each case there
910
exists a modular form of weight 2 and appropriate level, for which
911
the first terms $a_n$ correspond to those values predicted by the
912
Serre conjecture.
913
\[P(x)=x^5+x^4-16x^3-7x^2+57x-35,N=8311,\sqrt{D}=N\]
914
\[P(x)=x^5+2x^4-43x^3+29x^2+2x-3,N=8447,\sqrt{D}=2^2N\]
915
\[P(x)=x^5+x^4-13x^3-14x^2+18x+14,N=15233,\sqrt{D}=2N\]
916
\[P(x)=x^5+x^4-37x^3+67x^2+21x+1,N=24077,\sqrt{D}=2^2N\]
917
918
\ee
919
920
\section{Appendix: The curves $X_0(p)$ of genus 0}
921
922
In \cite{5}, it is proven that if $p$ is a prime number then the
923
curve $X_0(p)$ over $\Z_p$ is formally isomorphic to the curve of
924
equation $xy=p^k$, in the neighborhood of each point reducing mod
925
$p$ to a supersingular point $S$, $k$ being one half the number of
926
automorphisms of $S$.
927
928
If $X_0(p)$ has genus 0 (i.e., $p=2,3,5,7,13$) one has such a
929
model over $\Z$, given by the function \bean
930
x=\left(\frac{\eta(z)}{\eta(pz)}\right)^\frac{24}{p-1},\eean where
931
$\eta(z)=q^{1/24}\prod_{i=1}^\infty (1-q^n)$ and $q=e^{2\pi iz}$.
932
933
This results from Fricke \cite{7}, who gives for each of the above
934
$p$'s an expression of the ``oubli ?'' homomorphism $j:X_0(p)\to
935
X_0(1)$, which associates to each point $(E,C)$ of $X_0(p)$ the
936
point $(E)$ of $X_0(1)$, parametrized by the modular invariant
937
$j$.
938
939
In the following we recall these equations and give the
940
expressions of the correspondences $T_2,T_3$ over these curves.
941
The variable $x$ is the one given by equation (2), the involution
942
$W_p$ switches $x$ and $y$ and the divisor of $x$ is
943
$(0)-(\infty)$, where $0$ and $\infty$ are two points of $X_0(p)$.
944
945
\be
946
947
\item $p=2$ The equations given by Fricke (modified to give the
948
model of $X_0(2)$ over $\Z$) are:
949
\[xy=2^{12}\]
950
\[j=\frac{(x+16)^3}{x}\]
951
952
$T_2$ is given by \[y^2-y(x^2+2^43x)-2^{12}x=0\] (to each point
953
$x$ is associated by $T_2$ the formal sum of points of coordinate
954
$y$ that are roots of this polynomial.)
955
956
$T_3$ is given by
957
\[x^4+y^4-x^3y^3-2^33^2x^2y^2(x+y)-2^23^25^2xy(x^2+y^2)+2\cdot
958
3^21579x^2y^2-2^{15}3^2xy(x+y)-2^{24}xy=0\]
959
960
\item $p=3$.
961
\[xy=3^6\]
962
\[j=\frac{(x+27)(x+3)^3}{x}\]
963
\[T_2:x^3+y^3-2^33xy(x+y)-x^2y^2-3^6xy=0\]
964
\[T_3:y^3-y^2(x^3+2^23^2x^2+2\cdot 3^25y)-3^6yx
965
(x+2^23^2)-3^{12}x=0\]
966
967
\item $p=5$.
968
\[xy=5^3\]
969
\[j=\frac{(x^2+10x+5)^3}{x}\]
970
\[T_2:x^3+y^3-x^2y^2-2^3xy(x+y)-7^2xy=0\]
971
\[T_3:x^4+y^4-x^3y^3-2\cdot 3^2x^2y^2(x+y)-3^4xy(x^2+y^2)-2\cdot
972
3^223x^2y^2-2250xy(x+y)-5^6xy=0\]
973
974
\item $p=7$.
975
\[xy=7^2\]
976
\[j=\frac{(x^2+13x+49)(x^2+5x+1)^3}{x}\]
977
\[T_2:x^3+y^3-x^2y^2-2^3xy(x+y)-7^2xy=0\]
978
\[T_3:x^4+y^4-x^3y^3-2^23x^2y^2(x+y)-2\cdot 3\cdot 7xy(x^2+y^2)-3\cdot
979
53x^2y^2-2^23\cdot 7^2xy(x+y)-7^4xy=0\]
980
981
\item $p=13$.
982
\[xy=13\]
983
\[j=\frac{(x^2+5x+13)(x^4+7x^3+20x^2+19x+1)^3}{x}\]
984
\[T_2:x^3+y^3-x^2y^2-2^2xy(x+y)-13xy=0\]
985
\[T_3:x^4+y^4-x^3y^3-2\cdot 3x^2y^2(x+y)- 3\cdot 5xy(x^2+y^2)-3\cdot
986
11x^2y^2-2\cdot 3\cdot 13xy(x+y)-13^2xy=0\]
987
988
\ee
989
990
The polynomials above that give $T_2,T_3$ are of simpler form than
991
the classical modular equations $\Phi_2(j,j')$ and $\Phi_3(j,j')$
992
(that correspond to the action of $T_2$ and $T_3$ on $X_0(1)$).
993
For comparison, we recall their expressions:
994
995
\bea \Phi_2(j,j') &=& j^3+j'^3-j^2j'^2+2^43\cdot
996
31jj'(j+j')-2^43^45^3(j^2+j'^2)\\
997
&& + 3^45^34027jj'+2^83^75^6(j+j')-2^{12}3^95^9 \eea
998
999
\bea \Phi_3(j,j') &=&
1000
j^4+j'^4-j^3j'^3-2^23^39907jj'(j^2+j'^2)+2^33^231j^2j'^2(j+j')\\
1001
&&-2^{16}5^33^517\cdot 263jj'(j+j')+2^{15}3^25^3(j^3+j'^3)+2\cdot
1002
3^413\cdot 193\cdot 6367j^2j'^2\\
1003
&& - 2^{31}5^622973jj'+2^{30}3^35^6(j^2+j'^2)+2^{45}3^35^9(j+j')
1004
\eea
1005
1006
1007
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1085
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\end{document}
1096