1\include{mestre-en-header}23\renewcommand{\a}{\alpha}4\newcommand{\e}{\varepsilon}5\newcommand{\F}{\mathbb{F}}6\renewcommand{\t}{\theta}7\renewcommand{\l}{\lambda}8\newcommand{\w}{\omega}9\newcommand{\E}{\mathcal{E}}10\renewcommand{\r}{\rho}1112\newcommand{\codim}{\textrm{codim}}1314\newcommand{\s}[1]{\langle #1 \rangle}15\renewcommand{\O}{\mathcal{O}}161718\usepackage{amsmath}19\DeclareMathOperator{\Aut}{Aut}20\DeclareMathOperator{\Gal}{Gal}21\DeclareMathOperator{\Eis}{Eis}22\DeclareMathOperator{\Frob}{Frob}23\DeclareMathOperator{\Nor}{Nor}24\DeclareMathOperator{\Tr}{Tr}2526\newcommand{\T}{\mathbf{T}}2728\title{The Method of Graphs. Examples and Applications}29\author{J.-F. Mestre. \\{}Tr. Andrei Jorza}303132\begin{document}33\maketitle3435\section{Introduction}3637Let $S_k(N,\e)$ be the space of cusp forms of weight $k$, level38$N$ and character $\e$, where $k$ and $N$ are integers $\geq 1$,39and $\e$ is a Dirichlet character $N$. There are several ways40to construct a basis. For example one can use Selberg's trace41formula. Denote by $\Tr(n)$ the trace of $T_n$, the $n$-th Hecke42operator. The function \[f=\sum_{n=1}^\infty \Tr(n)q^n\] is in43$S_k(N,\e)$. The set of $f_i=T_if$ generate this space, and this44theoretically allows us to construct a basis. For example, if $N$45is prime, $\e=1$ and $k=2$, then the set of the $f_i$ ($1\leq46i\leq g$ where $g$ is the genus of $X_0(N)$) is a basis of47$S_2(N,1)$.4849But, even in that case, which is the most favorable, the50computations become hard: on an average computer we can only hope51to treat $N$ of the size of $5000$ (always with $N$ prime, weight522 and trivial character); actually, the computation of $\Tr(n)$53requires the knowledge of many class numbers of imaginary quadratic54fields of discriminant of order at most~$n$ and55to obtain a basis56$\vvn{f}{g}$ one needs to compute $\Tr(n)$ for $n\leq g^2$.5758In the following section we describe the ``method of graphs'', which59relies on the results of Deuring and Eichler, and developed by J.60Oestrl\'e and myself, which allows us to obtain a basis for $S_2(N,1)$61more quickly (at least when $N$ is a prime).6263In the second section, we indicate how this method allows us to prove64that certain elliptic curves defined over $\Q$ are Weil65curves\footnote{It is now a theorem that every elliptic curve66is a Weil curve, i.e., the Shimura-Taniyama-Weil conjecture67is true. -- William Stein}68(which, by providing an adequate Weil curve, yields all the69imaginary quadratic fields of class number at most 3, due to a70result of Goldfeld and recent works of Gross and Zagier).7172The third section is dedicated to the verification of a conjecture73of Serre in certain particular cases; this is74possible because of the method described in the first section. It is75known that this conjecture, if it is true, has numerous76consequences (e.g., the Shimura-Taniyama-Weil conjecture, and thus77Fermat's Last Theorem).7879\section{The method of graphs}8081\subsection{Definitions and notations}8283In the following $p$ is a prime number and $N_1$ is a positive84integer coprime to~$p$. Set $N=pN_1$.8586Let \[M_N=\oplus_S\Z[S]\]where $S$ is taken over all supersingular87points of $X_0(N_1)$ in characteristic $p$, i.e., over the set of88isomorphism classes of pairs $(E,C)$ consisting of an elliptic89curve $E$ defined over $\overline{\F}_p$ and a cyclic group $C$ of $E$90of order $N_1$. Two such pairs are identified if they are, in the91obvious sense, $\overline{\F}_p$-isomorphic.9293Let \[\a_S=\frac{|\Aut(S)|}{2},\]where $\Aut(S)$ is the group of94$\overline{\F}_p$-automorphisms of $S$. We always have $\a_S\leq9512$, and if $p$ does not divide 6 then $\a_S\leq 3$.9697Therefore we can define a scalar product on $M_N$ by98$\s{S,S}=\a_S$ and $\s{S,S'}=0$ if $S\neq S'$. Let99$\Eis=\sum\a_S^{-1}[S]$ and let100\[101M_N^0=\left\{\sum x_S[S] : \sum x_S=0\right\}102\]103be the subspace orthogonal to $\Eis$.104105For all integers $n\geq 1$ coprime with $p$ we define an operator106$T_n$ on $M$ by \[T_n(E,C)=\sum_{C_n}(E/C_n,(C+C_n)/C_n)\]107where $C_n$ runs over all the cyclic subgroups of108order~$n$ such that $C\i C_n=0$.109110For all $q\mid{}N_1$ and coprime with $q'=N_1/q$, we define the same111way the Atkin-Lehner involutions $W_q$ by112\[W_q(E,C)=(E/q'C,(E_q+C)/q'C),\]where $E_q$ is the group of113points of order dividing $q$ of $E$.114115Finally we define an involution $W_p$ by $W_p=-\Frob_p$, where116$\Frob_p$ is the endomorphism of $M_N$ that transforms $(E,C)$ to117$(E^p,C^p)$. (The fact that it is an involution reflects that the118supersingular points are defined over $\F_{p^2}$.)119120These operators have the following properties: the set of $W_q$121and $T_n$ ($n$ coprime with $N$) generate an abelian semigroup of122hermitian operators with respect to the scalar product $\s{\cdot,123\cdot}$. The $T_n$ commute with each other for all $n$ coprime to $p$. If124$q=q_1q_2$ ($q_1,q_2$ coprime) and if $n=n_1n_2$ ($n_1,n_2$125coprime with each other and with $p$) then $W_q=W_{q_1}W_{q_2}$126and $T_n=T_{n_1}T_{n_2}$.127128For all $d\mid N_1$ we have a homomorphism $\phi_d:M_N\to M_{N/d}$129that transforms $(E,C)$ to $(E,dC)$. This130homomorphism commutes with $T_n$ ($n$ coprime with $N$), and with131$W_q$ (for $q\mid{}N/d$). For $d\mid{}N_1$ and coprime with $N_1/d$ we have132\[T_d\phi_d=\phi_d(T_d+W_d)\]133134\subsection{An isomorphism with $S_2(N)$}135136We consider here the space $S_2(N)$ of cusp forms of weight 2 over137$\Gamma_0(N)$, with its natural structure of $\T$-module, where~$\T$138is the Hecke algebra \cite{1}.139140%% NEWFORMS141142\begin{theorem} There exists an isomorphism compatible with the143action of the Hecke operators, between $M_N^0\otimes \CC$ and the144subspace of $S_2(N)$ generated by the newforms of level $N$ and145oldforms coming from the cusp forms of weight 2 and level $pd$,146$d\mid{}N_1$.147\end{theorem}148\begin{remark}Assume $N$ (or without loss $N_1$) is149square-free. We can determine efficiently the subspace $M_N^0$150corresponding to newforms in $S_2(N)$; it is the subspace151formed by all~$x$ so that for all divisors~$d$ of $N_1$ we have152\[153\phi_d(x)=\phi_d(W_d(x))=0.154\]155In particular if $N=pq$, $q$ prime, it is the subspace of $M_{pq}$156intersection of the kernel of $\phi_q$ and of $\phi_qW_q$.157\end{remark}158159\subsection{Relation to the quaternion algebra}160161The matrices of the operators $T_n$ acting on $M_N$ are the same162as the classical Brandt matrices \cite{15}, constructed using163quaternion algebras.164165Let $B_{p,\infty}$ be the quaternion algebra over~$\Q$ ramified exactly166at~$p$ and infinity, and let~$\O$ is an Eichler order of level $N_1$167(defined by Eichler \cite{6} in the case when $N_1$ is168square-free, and defined in general by Pizer \cite{14}), and let169$\vvn{I}{h}$ be representatives of the left ideal classes of170$\O$.171172Let $\O_i$ be the right order (i.e., right normalizers) of the173ideals $I_i$, and $e_i$ be the number of units of $\O_i$.174The Brandt matrix $B(n)=(b_{i,j}^{(n)})$ has $i,j$ entry175\[176b_{i,j}^{(n)} =177e_j^{-1}\cdot |\{\a : \a\in178I_j^{-1}I_i,\,\Nor(\a)\Nor(I_j)/\Nor(I_i)=n\}|179\]180where $\Nor$ is181the norm over $B_{p,\infty}$ (the norm of an ideal being the182$\gcd$ of the norms of its nonzero elements).183184In the language of supersingular curves of characteristic $p$, we185may give these matrices (actually their transposes) the following186interpretation:187188Let $S$ be a supersingular point as in $I.1$, i.e., a189supersingular elliptic curve $E$ defined over $\overline{\F}_p$190together with a cyclic group $C$ of order $N_1$. The ring of191endomorphisms $\O_1$ of $S$ is an Eichler order of level $N_1$. To192all the other supersingular points $S'=(E',C')$ we associate the193set $I_{S,S'}$ of homomorphisms from $S$ to $S'$, i.e. the set of194all homomorphisms $\a$ from $E$ to $E'$ that send $C$ to $C'$.195This is obviously a left ideal over $\O_1$, and its inverse ideal196is $I_{S',S}$. We can prove that all the right ideals of $\O_1$197are obtained in this way, and the whole Eichler order of level198$N_1$ if the rign of endomorphisms of a supersingular point $S$.199It is clear that the general term $B_{i,j}^{(n)}$ of the $n$-th200Brandt matrix is the number of isogenies of $S_i$ to $S_j$ (the201supersingular points being conveniently indexed,) two such202isogenies being identified is different by an automorphism of203$S_j$. We can retrieve the matrix of the operator $T_n$ acting204over $M_n$.205206On the other hand if for all pairs of supersingular points207$(S,S')$ we associate the function \[\t_{S,S'}(q)=\sum_\a208q^{\deg\a}\]where $\a$ goes through all the homomorphisms of $S$209to $S'$, we retrieve the functions $\t$ classically associated210with the ideals of the orders of the quaternions, or, if one211prefers, associated with the positive integer quadratic forms in 4212variables.213214It is therefore easy to prove that if $\sum{x_S}[S]$ is an215elements of $M_N\otimes \CC$ eigenvector of all the Hecke216operators and if $f(q)$ is the corresponding modular form, we217have, for all $S'$218\[x_{S'}f(q)=\sum_Sx_S\t_{S,S'}\]which allows, in theory, to find219the coefficients $a_n$ of $f$, using the $x_S$. In practice,220unfortunately, the computation of $a_n$ demands the knowledge of221all the isogenies of degree $n$ to $S'$, and there doesn't seem to222be a simple algorithm for that.223224Nevertheless, in certain cases, there exists a different method to225calculate the coefficients of $f$, which is easy as far as226computation is concerned. Suppose that $N$ is a prime (thus equal227to $p$), or $N$ is a product of primes $pq$ and $X_0(q)$ is of228genus $0$ (thus $q=2,3,5,7$ or $13$).229230In the appendix, we give for each such case an equation of231$X_0(q)$ of the form $xy=p^k$, thus the action of the Hecke232operators $T_2$ and $T_3$ over $X_0(q)$, which is given by an233equation much simpler than the equation of modular polynomials234$\Phi_2(j,j'),\Phi_3(j,j')$ (which give the action of $T_2,T_3$ on235$X_0(1)$, parametrized by the modular invariant $j$; cf. section2362.4).237238Let $u=x$ if $N=pq$ and $u=j$ if $N=p$. The Fourier expansion of239$u$ at infinity is $1/q+\cdots$. Let $f(q)=\sum a_nq^n$ a normalized240newform of level $N$ and weight 2 corresponding to a vector241$\sum x_S[S]$ of $M_N^0\otimes K$, where $K$ is the extension of242$\Q$ generated by the $a_n$. Therefore there exists a prime ideal243$\wp$ of $K$ over $p$ so that \bean\left(\sum244x_S \cdot u(S)\right)f(q)\frac{dq}{q}\equiv \sum x_S\frac{du}{u-u(S)} \pmod{\wp}245.\eean (it is about the congruence between Laurent series in246$q$).247248Suppose for example that $f$ corresponds to a Weil curve of249conductor $N$, so that $a_n$ are in $\Z$. The $x_S$ are in $\Z$250and one can prove that $\sum x_Su(S)\neq 0$. Thus we know $a_n \m251p$ for all $n$. Hasse's inequality $|a_l|<2\sqrt{l}$ for $l$ prime252proves that we know the $a_n$ for $n<p^2/16$.253254%% NETWORK255256257\subsection{Explicit construction of the net $M_N$}258259In this section we suppose that $N$ is odd. Suppose that given an260explicit model of the curve $X_0(N_1)$, and so the action of the261Hecke operator $T_2$ on that model (cf. Appedix).262263%%INERT264265First we need to find a supersingular points. Note that they are266defined over $\F_{p^2}$. For example suppose that $N=p$. First we267check to see if $p$ is inert in one of the 9 imaginary quadratic268fields of class number 1. If yes, then one can take for the269initial value of $j$ the modular invariant of the curve of complex270multiplication by the ring of integers of corresponding fields. If271not, one can know a list of minimal polynomials of modular272invariants of elliptic curves of complex multiplication by273imaginary quadratic fields of small class numbers, and apply the274same method. One needs here to solve over $\F_{p^2}$ a polynomial275equation, which can be done in $\log p$ operations -- at least276probabilistically. Finally suppose that all these attempts fail.277There remains the possibility to enumerate all the values of278$\F_p$ until finding a supersingular value. We know there must279exist a supersingular $j$-invariant in $\F_p$,280but unfortunately only a very small number---on the281order of the size of the class group of $\Q(\sqrt{-p})$, or282approximately $\sqrt{p}$.283284So assume we know a supersingular point $S_1$. Knowing the action285of $T_2$ on the model given by $X_0(N)$ allows us to obtain the286three supersingular points $S_2,S_3,S_4$ (not necessarily287distinct) related to $S_1$ by a 2-isogeny. It comes down to288solving a degree 3 polynomial over $\F_{p^2}$, which needs289extracting cubic and square roots, operations that need $O(\log290p)$ operations. Sometimes we may as well exlude this computation.291Suppose that $n=p$ and that we have, say $p\equiv 2 (\m 3)$. Thus292$p$ is inert in $\Q(\sqrt{-3})$, so $j=0$ is a supersingular293value, and we know that the three isogenies of degree 2 send the294curve of the invariant to the curve of complex multiplication by295$\Z[\sqrt{-3}]$, for which the invariant is $j=54000$.296297In any case, we have at most one time when we need to solve a 3rd298degree equation: once $S_2$ is known, we search from $S_i$ ($i\geq2992$) the three supersingular points which are related, but we300already know one, so we only need to solve a second degree301equation, which comes down to square roots over $\F_{p^2}$ which302is fast (probabilistic methods require $O(\log p)$ operations303using an algorithm that is very simple to implement).304305To prove that we can find, step by step, all the supersingular306points of $M_N$ it is enough to prove that the graph of $T_2$ (and307more generally of $T_n$) is connected. But, as Serre remarked, the308eigenvalue $a_2=3$ of $T_2$ over $M_N$ has multiplicity equal to309the number of connected components of the graph of $T_2$. But in310$M_N$, the space $M_N^0$ corresponding to the cusp forms of311codimension 1, so 3 is a simple eigenvalue in $M_N$ (because for a312cusp form we have $|a_2|<2\sqrt{2}$), so the graph of $T_2$ is313connected.314315In conclusion, an algorithm in $O(N\log N)$ operations gives all316the supersingular points and the Brandt matrix $B_2$ associated to317them. One of the advantages of this matrix is that it is very318sparse; on each line and column there are at most 3 nonzero terms,319which are integers whose sum is 3. This allows, given an320eigenvalue, to find very quickly, if $N$ is large, the321corresponding eigenvectors.322323\subsection{Examples}324325\be326327\item Take for example $N=p=37$. Since $37$ is inert in328$\Q(\sqrt{-2})$, one can take as the first vertex of our graph the329curve $E_1$ of complex multiplications by $\Z[\sqrt{-2}]$, for330which the modular invariant is $j_1=8000\equiv 8 \m 37$. We need331to find now all the invariants of curves 2-isogenous to this,332i.e., to solve the equation $\Phi_2(x,8000)\equiv 0 (\m 37)$. But333$\sqrt{-2}$ is an endomorphism of degree 2 of $E_1$, so $j_1$ is a334root (over $\Q$) of the polynomial $\Phi_2(x,8000)$. Dividing this335polynomial by $x-8000$ we get a second degree polynomials with336roots $j_2,j_3$, the invariants of the other two curves, $E_2,E_3$337related to $E_1$ by a degree 2 isogeny. Let $\w\in\F_{p^2}$ so338that $\w^2=-2$. One gets that then $j_2=3+14\w,j_3=3-14\w$.339340Another method to find $j_2,j_3$ consists in remarking that 37 is341equally inert in the field $K=\Q(\sqrt{-15})$, for which the class342number is 2. The second degree polynomial giving the values of the343modular invariants of 2 curves of complex multiplication by the344ring of integers of $K$ is $x^2+191025x-121287375$, whose roots345generate $\Q(\sqrt{5})$, so modulo 37 are conjugate in346$\F_{37^2}$. We can thus find $j_2,j_3$.347348For $N$ prime congruent to 1 mod 12, the number of supersingular349curves mod $N$ is $(N-1)/12$. For $N=37$ we get 3 supersingular350curves. It remains to show that the action of $T_2$ on $E_2$ (by351conjugations we get the action on $E_3$). It is not possible to352have 2 isogenies of $E_2$ on $E_1$, because then we would have 5353isogenies of degree 2 starting in $E_1$. Therefore there is one3542-isogeny of $E_2$ over $E_2$.355356Actually, if there is a 2-isogeny of an elliptic curve of357invariant $j$ on itself, this invariant is the root of the358equation $\Phi_2(x,x)=0$, a fourth degree equation that can be359written as \[(x-1728)(x-8000)(x+3375)^2\]. (To see this, one can360make the computation of the equation of $\Phi_2(j,j')$ above. One361can also search which are the curves of complex multiplication362that admit a degree 2 endomorphism, i.e., which are the imaginary363quadratic fields that contains an element of norm 2. One finds, by364multiplication by the units of the (``corps pres?'') the elements365$1+i,\sqrt{-2}, \frac{1+\sqrt{-7}}{2}$ and $\frac{1-\sqrt{-7}}{2}$366that are the endomorphisms of degree 2 of the curves of invariant367$j=1728,j=8000$ and for the last two, $j=-3375$.)368369By order, mod $p$, the graph of $T_2$ cannot contain a loop of a370supersingular curve on itself -- although this curve is defined371over $\F_p$ (and, more precisely, it is one of 3 curves described372above). Therefore, there are 2 isogenies relating $E_2$ to $E_3$373and the graph of $T_2$ acting on $M_{37}$ is completely374determined.375376To compute the corresponding eigenvectors, one can evidently377diagonalize the matrix $(3,3)$ of $T_2$ but there is a simpler378method:379380the involution $W_{37}=-\Frob_{37}$ separates $M_{37}$ in an381obvious way into two orthogonal proper subspaces, one generated by382$u_1=[E_2]-[E_3]$, associated with the eigenvalue 1, and the other383associated with the eigenvalue -1, generates by384$\Eis=[E_1]+[E_2]+[E_3]$ and the vector product of $u_1$ and $\Eis$,385let it be $u_2=2[E_1]-[E_2]-[E_3]$. One can deduce, without386recourse to $T_2$, that there exist 2 newforms for which the387$q$-expansion has rational coefficient, and thus that $J_0(37)$,388the jacobian of $X_0(37)$ is isogenous to the product of 2389elliptic curves (which is well-known, see for example \cite{9}).390Formula (1) above allows us to obtain the first 83 terms of their391function $L$.392393\item $p-37,N=2\cdot 37$.394395To study $X_0(74)$ one uses the homomorphism $\phi_2$ of $M_{74}$396to $M_{37}$ defined previously. The fibres of reach of the three397supersingular points $[E_1],[E_2]$ and $[E_3]$ of $X_0(1) \m 37$398are formed by three distinct supersingular points of $X_0(2) \m3992$. In a general way, write that if $\vvn{S}{k}$ are the400supersingular points of $X_0(qM) \m p$ above a supersingular point401$S$ of $X_0(M) \m p$ ($p,q$ coprime and coprime with $M$), one has402the formula403\[\frac{q+1}{\Aut S}=\sum_1^k\frac{1}{\Aut S_i}.\]404405The equation of $X_0(2)$ used here is that described in the406appendix: $uv=2^{12}$, the involution $W_2$ switching $u$ and $v$.407Recall that $W_{37}=-\Frob_{37}$ and that $j=(u+16)^3/u$ (where $j$408is the invariant of the curve $E$, image of the point $(E,C)$ of409$X_0(2)$ via the homomorphism ``oubli -- oblivion?'' of $X_0(2)$ on410$X_0(1)$.) From the equation $j=j_1=8$ one gets the values of the411three supersingular points of $E_1$, of coordinates412$u_1=(-1+\w)/2,u_2=(-1-\w)/2=W_2(u_1)$ and $u_3=27=W_2(u_2)$.413(Here again, it is possible to guess the value of $u_3$, because414it is clear by the action of $T(2)$ on $X_0(1) \m 37$ done415previously that one of the above $E_1$ must be invariant relative416to $W_2$; or the two solutions of $u^2=2^{12}$ are $u_1,-u_1$.417Replacing them in the equation that gives $j$ one can see that it418is about $u_1$. To get $u_2,u_3$ it is enough to solve a second419degree equation.)420421One can compute that $u_4=W_2(u_1)=2^{12}/u_1=-5-5\w$, and one422finds that the corresponding invariant $j(u_4)$ is $j_2=3+14\w$.423One solves the second degree equation given 2 other points above424by $j_2$ and so $u_5=15+17\w,u_6=16-12\w$. Note that425$u_7=W_2(u_2)=\bar{u}_4, u_8=W_2(u_5)=\bar{u}_5$ and426$u_9=W_2(u_6)=\bar{u}_6$ the $x$-coordinates of three427supersingular points over $E_3$ ($x\to \bar{x}$ being the428nontrivial automorphism of $\F_{p^2}$.) We get the list of all429supersingular points of $X_0(2) \m 37$.430431As said above, the space $M_{74}^{new}$ corresponding to the432newforms is the intersection of the kernel of $\phi_2$ and the433kernel of $\phi_2W_2$. If we write $[u_i]$, $i=1,\ldots, 9$) the434generators of $M_{74}$ corresponding to the supersingular points435of $x$-coordinate $u_i$, an examination of the action of $W_{37}$436and $W_2$ prove that $M_{74}^{new}$ is the direct sum of two4372-dimensional subspaces, one $G_1$, generated by438$e_1=[u_1]-[u_2]-[u_4]+[u_7]-[u_9]$ and439$e_2=[u_5]-[u_6]-[u_8]+[u_9]$, on which $W_{37}=-W_2=1$ and the440other, $G_2$, generated by441$e_3=[u_1]+[u_2]-2[u_3]+[u_4]-[u_6]+[u_7]-[u_9]$, on which442$W_2=-W_{37}=1$.443444Using the equation of $T_3$ acting on $X_0(2)$ (cf. appendix), one445can prove that the matrix of $T_3$ acting on $G_1$ (respectively446$G_2$) in the basis $(e_1,e_2)$ (respectively $(e_3,e_4)$) is $\left(%447\begin{array}{cc}448-1 & 1 \\4491 & 0 \\450\end{array}%451\right)$, of characteristic polynomial $x^2+x-1$ (respectively $\left(%452\begin{array}{cc}4533 & 1 \\4541 & 0 \\455\end{array}%456\right)$, of characteristic polynomial $x^2-3x-1$).457458One deduces that $J_0^{new}(74)$ is isogenous to the product of459two abelian simple varieties, $A_1$ (resp. $A_2$), of real460multiplication by the ring of integers of $\Q(\sqrt{5}),$461(respectively $\Q(\sqrt{13})$.)462463If $\l=\frac{-1+\sqrt{5}}{2},\mu=\frac{3+\sqrt{13}}{2}$, then the464vectors $v_1=e_1+(\l+1)e_2,v_2=e_1-\l e_2,v_3=\mu465e_3+e_4,v_4=(3-\mu)e_3+e_4$ corresponding to the 4 newforms466$f_1,f_2,f_3,f_4$ of weights 2 and level $74$. Using (1) one gets467the first 83 values of the coefficients of these newforms. For468example for $f_1$ the list of the first values of $a_l$ is469\[\begin{array}{ccccccc}470l & 2 & 3 & 5 & 7 & 11 & 13 \\471a_l & 1 & \frac{-1+\sqrt{5}}{2} & \frac{1-3\sqrt{5}}{2} & -1+\sqrt{5} & \frac{-5-\sqrt{5}}{2} & \frac{1+3\sqrt{5}}{2}\\472\end{array}\]473474and for $f_3$ one gets475\[\begin{array}{ccccccc}476l & 2 & 3 & 5 & 7 & 11 & 13 \\477a_l & -1 & \frac{3+\sqrt{13}}{2} & -1-\sqrt{13} & \frac{1-\sqrt{13}}{2} & \frac{-1-\sqrt{13}}{2} & \frac{-1+\sqrt{13}}{2}\\478\end{array}\]479480\ee481482\section{Application to the study of Weil curves}483484Let $f=\sum a_nq^n$ be a newform of weight 2 and level $N$ with485integer coefficients. They correspond to a strong Weil curve $\E$486of conductor $N$. Unfortunately the coefficients $a_n$ don't give487too much information on $\E$ and do not allow us to obtain a488simple equation for $\E$. (In \cite{10} there is a method due to489Serre that sometimes allows us to get such an equation, but that490method is not systematic.) Here we give a method that at least491when $N=p$ is a prime, one can solve the problem.492493From now on let $N$ be a prime. According to the last section, for494a newform $f$ there is associated a vector $v_f=\sum x_S[S]$,495$x_S\in\Z$, an eigenvector of the Hecke operators defined in 2.1.496Theorem 1 doesn't describe the isomorphism (which is not497canonical) between $S_2(N)$ and $M_N^0\otimes\CC$. But suppose498known the terms $a_n$ of $f$ ($a_2$ is sufficient in general). The499construction of section 2.4 gives us both the supersingular values500$\m N$ and the graph of $T_2$ acting on $M_N$. We can determine501the eigenspace $V_2$ associated with the eigenvalue $a_2$. If it502is of dimension 1, we also have $v_f$, or at least the space it503generates. Otherwise, we apply $T_3$ to $V_2$ (which is of504tentatively small dimension---for conductors $< 80000$, $\dim V_2$505doesn't goes beyond 6), until finding a 1-dimensional space,506corresponding to the same eigenvalues of the operators $T_l$ as507$f$. Choose in this space a vector $r_f=\sum x_E[E]$ with integer508$x_E$ coprime in pairs\footnote{That seems to strong to me; do we509just mean that the gcd of coefficients is $1$?};510then $r_f$ is determined up to sign.511512To go further, we need a geometric interpretation of the $x_E$.513Let $\d=\pm N^\delta$, the discriminant of the minimal Weierstrass514model of $\E$, let $\phi:X_0(N)\to\E$ be a minimal cover of $\E$515of degree $n=\deg \phi$.516517According to Deligne-Rapoport \cite{5}, there exists a model518$X_0(N)_{/\Z}$ of $X_0(N)$ defined over $\Z$ for which reduction519mod $N$ is the union of two projective lines, one $C_\infty$520classifying the elliptic curves of characteristic $N$ provided521with the group scheme kernel of the Frobenius (this522corresponding to inseparable isogenies), the other one, $C_0$,523classifying the curves provided with Verschiebung. These two524lines intersect at supersingular points. As far as the curve $\E$525is concerned, reduction mod $N$ of its Neron model526has identity component $\E^0_{/\F_N}$ isomorphic to $\F_{N^2}$ of the527multiplicative group $G_m$. One can prove that the cover extends528to $X_0(N)_{/\Z}-\S$ where $\S$ is the set of all supersingular529points of characteristic $N$, and define by restriction a regular530``application ?'' of $C_\infty$ on $\E^0_{/\overline{\F}_N}$, of a531rational function $\phi$ over $C_\infty$, for which the poles and532zeros are in $\E$. The divisor $\sum \l_E[E]$ of $\phi$, $E$ going533through all the supersingular curves $\m N$, and thus an element534of $M_N^0$, defined up to sign (depending on the choice of535isomorphism of $\E^0_{/\F_N}$ over $G_m$.)536537\begin{proposition} In the above notation the divisor $(\Phi)=\sum538\l_E[E]$ is equal to $\pm r_f$.539\end{proposition}540541It is not difficult to see that $(\Phi)$ is proportional to $r_f$.542By contradiction, the fact that the $l_E$ are coprime with one543another is obtained from the result of Ribet which says that if544$l$ is a prime different from 2, 3 then all cusp forms mod $l$ of545weight 2 and level $Np$ (where $Np$ is square-free) for which the546associated representation mod $l$ is irreducible and not ramified547at $p$, comes from a cusp form mod $l$ of weight 2 and level $N$548(this result was conjectured by Serre in 1985. This also shows549that the Taniyama-Weil conjecture implies the Fermat theorem.)550551To prove the previous theorem, one proves first that $\delta$ is552related to $\l_E$ by $\delta=\gcd (\l_E\w_E-\l_F\w_F)$ where553$\w_E$ is the number of automorphisms of $E$. Suppose that a prime554number $l$ divides the gcd is $\l_E$. It also divides $\delta$,555and one deduces from here that $p$ is not ramified in the field of556points of order $l$ or $\E$. If $l$ is coprime with 6 Ribet's557\footnote{K.Ribet, {\it Lectures on Serre's conjectures}, MSRI,558Fall 1986} theorem shows that the modular form $f$ associated to559$\E$ is congruent mod $l$ to a modular form of weight 2 and level5601, which cannot be but the Eisenstein series. The curve $\E$ is561semi-stable, which implies (\cite{16}, p.306) that $\E$ or a curve562$\Q$-isogenous to it has a point of finite order $l$. If $l=2,3$563we get the same result due to \cite{4}, Appendix. Now, we know564explicitly the curves of prime conductor with torsion \cite{11}565namely the curves 11A and 11B of \cite{19}, which have a point of566order 5, curves 17A,17B,17C (point of order 4), 17D (point of567order 2), 19A and 19B (point of order 3), 37B, 37C (point of order5683) and the curves of Setzer-Neumann \cite{18}, which have a point569of order 2. In each of these cases, we know $\delta$, which is570equal to the number of finite points rational over $\Q$ of the571considered curves, and one can verify that the $\l_E$ are coprime572with one another. This proves the proposition. Note that along the573proof we showed that Ribet's theorem implies the following574575\begin{theorem}576Let $E$ be a strong Weil curve of prime conductor $N$. The577valuation of its discriminant in $N$ is equal to the number of578torsion points of $E(\Q)$.579\end{theorem}580581We state without proof the theorem that allows us to get an582explicit equation for $\E$ once we know the $\l_E$.583584\begin{theorem}585Let $\E$ be a strong Weil curve of prime conductor $N$, and $\sum586\l_E[E]$ the element of $M_N^0$ associated to $\E$ via the587constructions above. There exists an equation of $\E$588\[y^2=x^3-\frac{c_4}{48}x-\frac{c_6}{864}\]589with $c_4,c_6\in\Z$ so that, if $H=\max590(\sqrt{|c_4|},\sqrt[3]{|c_6|})$ we have: \be591592\item $H\leq \frac{8n}{\sqrt{N}-2}(\log (H^6/1728)+b)$, where593$b=(\Gamma(1/3)/\Gamma(2/3))^3=7.74316962\ldots$.594595\item Let $\d'=(c_4^3-c_6^2)/1728$. Then $\d'=\d$ if $\E$ is596supersingular in characteristic 2, and $\d'=\d$ or $2^{12}\d$ otherwise.597598\item $c_4\equiv (\sum\l_Ej_E)^4 \m N$.599600\item $c_6\equiv -(\sum\l_Ej_E)^6 \m N$.601602\item $n\delta=\sum \l_E^2\w_E$.603604\ee605\end{theorem}606607If the $\l_E$ are known then 5 allows us to get $n$ and 1 allows608us to find a bound on $H$, thus on $c_4,c_6$. By 2 we have609$c_4^3-c_6^2=1728\d',$ which allows us to find $c_4,c_6$. The610congruences 3 and 4 allow us to reduce the number of computations611significantly. Thus we have found an equation of a strong Weil612curve corresponding to the initial newform $f$.613614This method also allows us to prove that an elliptic curve of615small prime conductor is a Weil curve. Suppose that we are given616such a curve by its equation. Then we may compute the number of617its points $N_l$ mod $l$ for $l=2,3,\ldots$. Next we search, by618the method of graphs, whether $a_2=3-N_2$ is the eigenvalue of619$T_2$ acting on $M_N$. If not then the Taniyama-Weil conjecture is620false. If yes, then continue with $T_3$ acting on the found621eigenspace, if it is not of dimension 1, until we get an622eigenspace of dimension 1 for the Hecke operators, with integers623eigenvalues. If there is no such thing, then we get a624counterexample to the Taniyama-Weil conjecture. If there is one,625we compute the equation of a corresponding Weil curve. If this626curve is isogenous to the initial curve, we are done. Otherwise,627the initial curve is not a Weil curve.628629In particular, this allows us to prove that the elliptic equation630\[y^2+y=x^3-7x+6\] of conductor 5077, is a Weil curve.631632This curve seems to be the smallest curve (ordering the curves by633their conductors) having a Mordell-Weil rank $\geq 3$ \cite{3}.634The interest in it is the following:635636Let $f(z)=\sum a_nq^n$ ($q=e^{2\pi iz}$), a newform of weight 2637and conductor $N$, and let $L(s)=\sum a_nn^{-s}$, the associated638$L$ function. If the order of $L$ in 1 is $\geq 3$ then Goldfeld639proved that there exists a computable constant $C_f$ so that640\[\log p<C_fh(-p),\]where $p\equiv 3 (\m 4)$ is a prime number coprime with $N$ and641$h(-p)$ is the number of classes of imaginary quadratic fields of642discriminant $-p$. We have other formulas, but more complicated,643in the case of imaginary quadratic fields of non-prime644discriminant (see \cite{13} for example).645646If the Birch and Swinnerton-Dyer conjecture is true, all the Weil647curves for which the Mordell-Weil group over $\Q$ is of rank $\geq6483$ have to be given by such modular forms, but until the work of649Gross and Zagier \cite{8}, there was no way to verify that the650derivative at 1 of the $L$ function of a Weil curve is indeed 0.651The results of Gross and Zagier allow to write $L'(1)$ as the652product of a non-zero factor easily computable and the653N\'eron-Tate height of a Heegner point (cf. \cite{8} for more654details.) It is therefore possible, by decreasing the height of655rational points on the curve and increasing $L'(1)$ by a careful656computation, to prove that $L$ is of order $\geq 3$ at $s=1$. (In657all the previous, we considered odd Weil curves, i.e., for which658the $L$ function has an odd order at 1 -- or if one prefers for659which the sign of the functional equation is -1.)660661One has several method to construct Weil curves for which the662Mordell-Weil group is of rank $\geq 3$ (and which are good663candidates for the preceding question: by the method of664Gross-Zagier, one may compute $L'(1)$. If it is zero, one has an665$L$ function which allows to obtain an increase of the absolute666value of the discriminant of imaginary quadratic fields of given667class numbers; if it is non-zero, the conjecture of Birch and668Swinnerton-Dyer is false.) One can, for example, search for curves669of complex multiplication of rank 3 (we know that they are Weil670curves), but the constant $C_f$ is very large. One can deform\footnote{Twist?} a671Weil curve (for example the curve 37C of \cite{19} until getting a672rank 3 curve (for the curve 37C, one can deform by673$\Q(\sqrt{-139})$, as shown by Gross and Zagier \cite{8}.) This674leads to a constant $C_f$ of order of 7000675676One may choose some elliptic curve defined over $\Q$, or rank 3,677and try to prove that it is a Weil curve. This was done in678\cite{10} for the mentioned curve of conductor 5077, using the679trace formula. But the computation is very long. The method of680graphs allows us to do it in about 5 seconds an a computer that681needed 5 hours with the mentioned method.682683For this curve, one has $C_f<50$: all imaginary quadratic curves684of discriminant $d$ with $|d|>e^{150}$ therefore has a class685number $\geq 4$. On the other hand, there is no imaginary686quadratic field of discriminant $d$ and class number 3 for687$907<|d|<10^{2500}$ \cite {12}. Therefore (after an examination of688a table of class numbers of the first quadratic fields):689690\begin{theorem}691The imaginary quadratic fields of class number 3 are the 16 fields692of discriminant:693$-23,-31,-59.-83,-107,-139,-211,-283,-307,-331,-379,-499,-547,-643,-883,-907$.694\end{theorem}695696\section{Application to a conjecture of Serre}697698Let $\r$ be a continuous representation of $Gal(\overline{\Q}/\Q)$699in $GL_2(V)$ where $V$ is a dimension 2 vector space over a finite700field $\F_q$ of characteristic $p$. Assume this is an odd701representation, i.e., that $\r(c)$ the image of the complex702conjugation, seen as an element of $Gal(\overline{\Q}/\Q)$ has703eigenvalues 1 and -1. In that case put $G=Im\r$.704705In \cite{17} Serre defines the level, the character and the weight706of such a representation:707708\be709710\item The level.711712Let $l$ be a prime number different from $p$. Write $G_i$713($i=0,\ldots$) the groups of ramifications of $\r$ at $l$. Let714\[n(l)=\sum_{i=0}^\infty \frac{g_i}{g_0}\codim V^{G_i},\]where715$g_i=|G_i|$.716717The conductor of the representation $\r$ is defined as718\[N=\prod_{l\neq p}l^{n(l)}.\]719720\item The character.721722The determinant of $\r$ yields a character of723$Gal(\overline{\Q}/\Q)$ in $\F_q^*$, for which the conductor724divides $pN$. Therefore, one can write725\[\det\r=\e\chi^{k-1},\]where $\chi$ is the cyclotomic character726of conductor $p$ and $\e$ is the character $(\Z/N\Z)^*\to\F_q^*.$727The integer $k$ is defined mod $(p-1)$, and the fact that the728representation is odd implies that $\e(-1)=(-1)^k$.729730By definition, $\e$ is the character of the representation $\r$.731732\item The weight.733734The integer $k$ above is defined mod $(p-1)$. Read Serre's article735for the definition of the weight $k\in\Z$ of the representation736$\r$. As the conductor $N$ depends only on the behavior of $\r$ ar737places coprime with $p$, the definition of weight only uses the738local properties at $p$ of the representation $\r$.739740\ee741742Then Serre's conjecture is:743744\begin{conjecture}745Let $\r$ be a representation as above, of weight $k$, level $N$746and character $\e$. Assume this representation is irreducible.747Then it comes from a cusp form $\m p$ of weight $k$, level $N$ and748character $\e$.749\end{conjecture}750751This conjectures, if true, has numerous consequences: it implies752the Taniyama-Weil conjecture and Fermat's theorem.753754Many such representations $\r$ are modular, either by755construction, or because they are part of classical conjectures756(Langlands, Artin, $\ldots$) that carry on the conjecture (but757sometimes in a weak form, i.e., with a weight or conductor bigger758than those defined in \cite{17}.)759760In order to verify (or contradict) Serre's conjecture, we need to761find the extensions $K/\Q$ of Galois group subgroup of762$GL_2(\F_q)$ of odd determinant and $p\neq 2$. It is in general763not difficult to calculate, for $l$ prime and not too large, the764trace $a_l$ of $\Frob_l$ in $GL_2(\F_q)$: if $P(x)$ is a polynomial765whose roots generate $K$ the decomposition of $P \m l$ usually766will suffice.767768It is however, much harder to find modular forms $\m p$, if they769exist, that correspond to the representation $\r$ given by the770field $K$: the discriminant of $K$ is usually large, thus so is771the conductor of $\r$, which is related to it, so it is not easy772to make the computations.773774\subsection{The case $SL_2(\mathbb{F}_4)$}775776A troubling case is that of $p=2$, because, since $-1\equiv 1 (\m7772)$ all representations are odd.778779The representations of $Gal(\overline{\Q}/\Q)$ in $GL_2(\F_2)=S_3$780(although altogether real, cf. \cite{17}) come from weight 1781modular forms; the group $S_3$ can be realized as a subgroup of782$GL_2(\CC)$. One can hope that by multiplication with convenient783Eisenstein series, one can obtain a modular form of weight and784level predicted by the Serre conjecture (cf. \cite{17} for785examples.)786787In order to obtain the most interesting case for characteristic 2,788one considers the representations with values in $GL_2(\F_4)$. The789isomorphism $A_5\simeq SL_2(\F_4)$ allows us to obtain several790examples. Let $K$ be an extension of $\Q$ of Galois group $A_5$.791Since $A_5$ ``immerses ?'' into $PGL_2(\CC)$, if the field is not792completely real, the associated representation $\r$ comes from a793weight 1 modular form (module Artin's conjecture, cf. \cite{2}).794Suppose now that $K$ is real. None of the classical conjectures795allow us to suspect that $\r$ comes from a modular form, even if796of higher weight or level. It is this case that we will study in797what follows. The method of graphs here is indispensable, the798modular forms that we look at having a conductor too large to be799studied with the Eichler-Selberg trace formula.800801Let $P(x)=x^5+a_1x^4+a_2x^3+a_3x^2+a_4x+a_5$ be a rational802polynomial of discriminant $D$. In order that the field of roots803of $P$ be $A_5$ it is sufficient and necessary that $P$ be804irreducible, that $D$ be square-free, and that there exist a prime805number $l$ not dividing $D$ so that $P \m l$ having exactly two806roots in $\F_l$ (this last condition assuring that the group is807all of $A_5$).808809It is clear that $\e=1$. If $p\mid{}D$, $p$ coprime with 30, $n(p)=1$810if it ``seulment si l'inertie en p ==?'' is of order 2, and thus the811polynomial $P$ has at most double roots mod $p$. As far as the812weight $k$ is concerned, it is either 2 or 4 according to the813ramification of $K$ at 2. To simplify the computation, we have814limited to searching examples among the representations of prime815level and weight 2.816817On the other hand, since it is about representations in818$SL_2(\F_4)$m the coefficient $a_2$ of the sought modular form, if819it exists, cannot be in $\F_4$, but in $\F_{16}$. This comes from820the fact that the coefficient $a_l$ of a modular form $\m l$ is821equal to an eigenvalue of $\Frob_l$, and not to its trace. Now, if822a matrix in $SL_2(\F_4)$ is of order 5, its eigenvalues are in823$\F_{16}$ not in $\F_4$.824825The examples treated above were obtained by making a systematic826search on a computer of convenient polynomials (totally real, of827type $A_5$, for which the conductor of the associated828representation is a prime $N$, and for which the weight is 2).829830Thereafter, for each such polynomial $P$, one computes the831corresponding eigenvalue $a_2$ (in $\F_{16}$), and one tries to832find whether there exists a modular form mod 2 of level $N$ and833weight 2 so that $T_2$ has $a_2$ as an eigenvalue. In all the834cases considered, we have thereafter found an eigenspace of835dimension 1 or 2. Using the operators $T_3,T_5$, one calculates836the coefficients $a_3,a_5$, and verifies that they correspond to837the values predicted by the decomposition of $P$ in 3 and 5.838839Clearly, this doesn't really prove that the representation $\r$840associated to $P$ is modular: we have only exhibited a modular841form mod 2 of proper level and weight for which the terms842$a_2,a_3,a_5$ are convenient. But there is a good indication of843the truthfulness of the conjecture of Serre in the considered844cases: an exhaustive search over numerous primes $N$ of the845coefficients $a_2$ of modular forms of weight 2 and level $N$846proves that it is rare that there are fields of small degree.847(Actually, is seems that 2, and in general the small primes, are848the most ``inert'' possible in the fields that appear in the Hecke849algebra of modular forms, fields which themselves in general850appear to have the largest degree possible, taking into account851constraints such as the Atkin-Lehner involutions, primes of852Eisenstein, etc. One gets that one has small factors, --853corresponding for example to elliptic curves with prime conductor854-- but this is apparently rare.)855856\subsection{A few examples}857858\be859860\item $P(x)=x^5-10x^3+2x^2+19x-6$.861862The discriminant is $(2^3887)^2$. This polynomial is irreducible863mod 5, thus irreducible over $\Q$. Its roots are all real (apply864Sturm's algorithm). One has that \[P(x)\equiv x(x-1)(x^3+x^2-1) \m8653,\] which gives a cycle of order 3; the Galois group of $K$, the866field of roots of $P$, is thus $A_5$.867868%%869%%870%% THESE COMPUTATIONS ARE NOT CORRECT.871%% I AM REDOING THEM872%%873%%874875%ORIGINAL: From $P(x)\equiv (x-446)(x-126)^2(x-538)^2 \m 887$ one gets that876877From $P(x)\equiv (x-462)(x-755)^2(x-788)^2 \m 887$ one gets that878the conductor $N$ of the associated representation is $N=887$. One879can also prove that 2 is ``little ramified'' in the sense of880\cite{17}, thus $\r$ has weight 2. Examining the reduction mod 2881of $P$ proves that the coefficients $a_2,a_3,a_5$ of the modular882form mod 2 of level 887 (which must correspond to $\r$ via the883Serre conjecture) are 1, 1, j (where $j\in\F_4$ has the property884that $j^2+j+1=0$).885886One therefore applies the method of graphs: the space of modular887forms mod 2 of weight 2 and level 887 has dimension 73, and888computation shows that the eigenspace $G_1$ of $T_2$ corresponding889to the eigenvalue 1 has dimension 2; $T_3$ acts as the identity on890$G_1$, and $j,j^2$ are the eigenvalues of $T_5$ acting on $G_1$,891from where get a basis of $G_1$ formed by892$f_1=q+q^2+q^3+q^4+jq^5+\cdots$ and893$f_2=q+q^2+q^3+q^4+j^2q^5+\cdots$, eigenvectors of Hecke894operators. These corroborate the conjecture.895896\item $P(x)=x^5-23x^3+55x^2-33x-1$.897898Then $D=13613^2,P(x)\equiv (x-6308)(x-2211)^2(x-8248)^2 \m 13613$,899$N=13613$; $P$ being irreducible mod 2, $\Frob_2$ is a cycle of900order 5, and $a_2=\zeta_5$ is a fifth root of unity, viewed as an901element of $\F_{16}$. Computation also shows that in the space of902modular forms mod 2 of level 13613 and weight 2, which has903dimension 1134, $\zeta_5$ is a simple eigenvalue of $T_2$. The904coefficients $a_3,a_5$ are respectively equal to905$1+\zeta_5^2+\zeta_5^3=j$ and $\zeta_5^2+\zeta_5^3=j^2$, which are906the traces of $\Frob_3,\Frob_5$ in $SL_2(\F_4)$.907908\item We write the other found polynomials; in each case there909exists a modular form of weight 2 and appropriate level, for which910the first terms $a_n$ correspond to those values predicted by the911Serre conjecture.912\[P(x)=x^5+x^4-16x^3-7x^2+57x-35,N=8311,\sqrt{D}=N\]913\[P(x)=x^5+2x^4-43x^3+29x^2+2x-3,N=8447,\sqrt{D}=2^2N\]914\[P(x)=x^5+x^4-13x^3-14x^2+18x+14,N=15233,\sqrt{D}=2N\]915\[P(x)=x^5+x^4-37x^3+67x^2+21x+1,N=24077,\sqrt{D}=2^2N\]916917\ee918919\section{Appendix: The curves $X_0(p)$ of genus 0}920921In \cite{5}, it is proven that if $p$ is a prime number then the922curve $X_0(p)$ over $\Z_p$ is formally isomorphic to the curve of923equation $xy=p^k$, in the neighborhood of each point reducing mod924$p$ to a supersingular point $S$, $k$ being one half the number of925automorphisms of $S$.926927If $X_0(p)$ has genus 0 (i.e., $p=2,3,5,7,13$) one has such a928model over $\Z$, given by the function \bean929x=\left(\frac{\eta(z)}{\eta(pz)}\right)^\frac{24}{p-1},\eean where930$\eta(z)=q^{1/24}\prod_{i=1}^\infty (1-q^n)$ and $q=e^{2\pi iz}$.931932This results from Fricke \cite{7}, who gives for each of the above933$p$'s an expression of the ``oubli ?'' homomorphism $j:X_0(p)\to934X_0(1)$, which associates to each point $(E,C)$ of $X_0(p)$ the935point $(E)$ of $X_0(1)$, parametrized by the modular invariant936$j$.937938In the following we recall these equations and give the939expressions of the correspondences $T_2,T_3$ over these curves.940The variable $x$ is the one given by equation (2), the involution941$W_p$ switches $x$ and $y$ and the divisor of $x$ is942$(0)-(\infty)$, where $0$ and $\infty$ are two points of $X_0(p)$.943944\be945946\item $p=2$ The equations given by Fricke (modified to give the947model of $X_0(2)$ over $\Z$) are:948\[xy=2^{12}\]949\[j=\frac{(x+16)^3}{x}\]950951$T_2$ is given by \[y^2-y(x^2+2^43x)-2^{12}x=0\] (to each point952$x$ is associated by $T_2$ the formal sum of points of coordinate953$y$ that are roots of this polynomial.)954955$T_3$ is given by956\[x^4+y^4-x^3y^3-2^33^2x^2y^2(x+y)-2^23^25^2xy(x^2+y^2)+2\cdot9573^21579x^2y^2-2^{15}3^2xy(x+y)-2^{24}xy=0\]958959\item $p=3$.960\[xy=3^6\]961\[j=\frac{(x+27)(x+3)^3}{x}\]962\[T_2:x^3+y^3-2^33xy(x+y)-x^2y^2-3^6xy=0\]963\[T_3:y^3-y^2(x^3+2^23^2x^2+2\cdot 3^25y)-3^6yx964(x+2^23^2)-3^{12}x=0\]965966\item $p=5$.967\[xy=5^3\]968\[j=\frac{(x^2+10x+5)^3}{x}\]969\[T_2:x^3+y^3-x^2y^2-2^3xy(x+y)-7^2xy=0\]970\[T_3:x^4+y^4-x^3y^3-2\cdot 3^2x^2y^2(x+y)-3^4xy(x^2+y^2)-2\cdot9713^223x^2y^2-2250xy(x+y)-5^6xy=0\]972973\item $p=7$.974\[xy=7^2\]975\[j=\frac{(x^2+13x+49)(x^2+5x+1)^3}{x}\]976\[T_2:x^3+y^3-x^2y^2-2^3xy(x+y)-7^2xy=0\]977\[T_3:x^4+y^4-x^3y^3-2^23x^2y^2(x+y)-2\cdot 3\cdot 7xy(x^2+y^2)-3\cdot97853x^2y^2-2^23\cdot 7^2xy(x+y)-7^4xy=0\]979980\item $p=13$.981\[xy=13\]982\[j=\frac{(x^2+5x+13)(x^4+7x^3+20x^2+19x+1)^3}{x}\]983\[T_2:x^3+y^3-x^2y^2-2^2xy(x+y)-13xy=0\]984\[T_3:x^4+y^4-x^3y^3-2\cdot 3x^2y^2(x+y)- 3\cdot 5xy(x^2+y^2)-3\cdot98511x^2y^2-2\cdot 3\cdot 13xy(x+y)-13^2xy=0\]986987\ee988989The polynomials above that give $T_2,T_3$ are of simpler form than990the classical modular equations $\Phi_2(j,j')$ and $\Phi_3(j,j')$991(that correspond to the action of $T_2$ and $T_3$ on $X_0(1)$).992For comparison, we recall their expressions:993994\bea \Phi_2(j,j') &=& j^3+j'^3-j^2j'^2+2^43\cdot99531jj'(j+j')-2^43^45^3(j^2+j'^2)\\996&& + 3^45^34027jj'+2^83^75^6(j+j')-2^{12}3^95^9 \eea997998\bea \Phi_3(j,j') &=&999j^4+j'^4-j^3j'^3-2^23^39907jj'(j^2+j'^2)+2^33^231j^2j'^2(j+j')\\1000&&-2^{16}5^33^517\cdot 263jj'(j+j')+2^{15}3^25^3(j^3+j'^3)+2\cdot10013^413\cdot 193\cdot 6367j^2j'^2\\1002&& - 2^{31}5^622973jj'+2^{30}3^35^6(j^2+j'^2)+2^{45}3^35^9(j+j')1003\eea100410051006\begin{thebibliography}{99}10071008\bibitem{1}1009A.O.L. 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