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20\DeclareMathOperator{\Aut}{Aut}
21\DeclareMathOperator{\Gal}{Gal}
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25\DeclareMathOperator{\Tr}{Tr}
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29\title{The Method of Graphs. Examples and Applications}
30\author{J.-F. Mestre. \\{}Tr. Andrei Jorza}
31
32
33\begin{document}
34\maketitle
35
36\section{Introduction}
37
38Let $S_k(N,\e)$ be the space of cusp forms of weight $k$, level
39$N$ and character $\e$, where $k$ and $N$ are integers $\geq 1$,
40and $\e$ is a Dirichlet character $N$. There are several ways
41to construct a basis. For example one can use Selberg's trace
42formula.  Denote by $\Tr(n)$ the trace of $T_n$, the $n$-th Hecke
43operator.  The function $f=\sum_{n=1}^\infty \Tr(n)q^n$ is in
44$S_k(N,\e)$. The set of $f_i=T_if$ generate this space, and this
45theoretically allows us to construct a basis. For example, if $N$
46is prime, $\e=1$ and $k=2$, then the set of the $f_i$ ($1\leq 47i\leq g$ where $g$ is the genus of $X_0(N)$) is a basis of
48$S_2(N,1)$.
49
50But, even in that case, which is the most favorable, the
51computations become hard: on an average computer we can only hope
52to treat $N$ of the size of $5000$ (always with $N$ prime, weight
532 and trivial character); actually, the computation of $\Tr(n)$
54requires the knowledge of many class numbers of imaginary quadratic
55fields of discriminant of order at most~$n$ and
56to obtain a basis
57$\vvn{f}{g}$ one needs to compute $\Tr(n)$ for $n\leq g^2$.
58
59In the following section we describe the method of graphs'', which
60relies on the results of Deuring and Eichler, and developed by J.
61Oestrl\'e and myself, which allows us to obtain a basis for $S_2(N,1)$
62more quickly (at least when $N$ is a prime).
63
64In the second section, we indicate how this method allows us to prove
65that certain elliptic curves defined over $\Q$ are Weil
66curves\footnote{It is now a theorem that every elliptic curve
67is a Weil curve, i.e., the Shimura-Taniyama-Weil conjecture
68is true. -- William Stein}
69(which, by providing an adequate Weil curve, yields all the
70imaginary quadratic fields of class number at most 3, due to a
71result of Goldfeld and recent works of Gross and Zagier).
72
73The third section is dedicated to the verification of a conjecture
74of Serre in certain particular cases; this is
75possible because of the method described in the first section. It is
76known that this conjecture, if it is true, has numerous
77consequences (e.g., the Shimura-Taniyama-Weil conjecture, and thus
78Fermat's Last Theorem).
79
80\section{The method of graphs}
81
82\subsection{Definitions and notations}
83
84In the following $p$ is a prime number and $N_1$ is a positive
85integer coprime to~$p$. Set $N=pN_1$.
86
87Let $M_N=\oplus_S\Z[S]$where $S$ is taken over all supersingular
88points of $X_0(N_1)$ in characteristic $p$, i.e., over the set of
89isomorphism classes of pairs $(E,C)$ consisting of an elliptic
90curve $E$ defined over $\overline{\F}_p$ and a cyclic group $C$ of $E$
91of order $N_1$. Two such pairs are identified if they are, in the
92obvious sense, $\overline{\F}_p$-isomorphic.
93
94Let $\a_S=\frac{|\Aut(S)|}{2},$where $\Aut(S)$ is the group of
95$\overline{\F}_p$-automorphisms of $S$. We always have $\a_S\leq 9612$, and if $p$ does not divide 6 then $\a_S\leq 3$.
97
98Therefore we can define a scalar product on $M_N$ by
99$\s{S,S}=\a_S$ and $\s{S,S'}=0$ if $S\neq S'$. Let
100$\Eis=\sum\a_S^{-1}[S]$ and let
101$102 M_N^0=\left\{\sum x_S[S] : \sum x_S=0\right\} 103$
104be the subspace orthogonal to $\Eis$.
105
106For all integers $n\geq 1$ coprime with $p$ we define an operator
107$T_n$ on $M$ by $T_n(E,C)=\sum_{C_n}(E/C_n,(C+C_n)/C_n)$
108where $C_n$ runs over all the cyclic subgroups of
109order~$n$ such that $C\i C_n=0$.
110
111For all $q\mid{}N_1$ and coprime with $q'=N_1/q$, we define the same
112way the Atkin-Lehner involutions $W_q$ by
113$W_q(E,C)=(E/q'C,(E_q+C)/q'C),$where $E_q$ is the group of
114points of order dividing $q$ of $E$.
115
116Finally we define an involution $W_p$ by $W_p=-\Frob_p$, where
117$\Frob_p$ is the endomorphism of $M_N$ that transforms $(E,C)$ to
118$(E^p,C^p)$. (The fact that it is an involution reflects that the
119supersingular points are defined over $\F_{p^2}$.)
120
121These operators have the following properties: the set of $W_q$
122and $T_n$ ($n$ coprime with $N$) generate an abelian semigroup of
123hermitian operators with respect to the scalar product $\s{\cdot, 124\cdot}$. The $T_n$ commute with each other for all $n$ coprime to $p$. If
125$q=q_1q_2$ ($q_1,q_2$ coprime) and if $n=n_1n_2$ ($n_1,n_2$
126coprime with each other and with $p$) then $W_q=W_{q_1}W_{q_2}$
127and $T_n=T_{n_1}T_{n_2}$.
128
129For all $d\mid N_1$ we have a homomorphism $\phi_d:M_N\to M_{N/d}$
130that transforms $(E,C)$ to $(E,dC)$. This
131homomorphism commutes with $T_n$ ($n$ coprime with $N$), and with
132$W_q$ (for $q\mid{}N/d$). For $d\mid{}N_1$ and coprime with $N_1/d$ we have
133$T_d\phi_d=\phi_d(T_d+W_d)$
134
135\subsection{An isomorphism with $S_2(N)$}
136
137We consider here the space $S_2(N)$ of cusp forms of weight 2 over
138$\Gamma_0(N)$, with its natural structure of $\T$-module, where~$\T$
139is the Hecke algebra \cite{1}.
140
141%% NEWFORMS
142
143\begin{theorem} There exists an isomorphism compatible with the
144action of the Hecke operators, between $M_N^0\otimes \CC$ and the
145subspace of $S_2(N)$ generated by the newforms of level $N$ and
146oldforms coming from the cusp forms of weight 2 and level $pd$,
147$d\mid{}N_1$.
148\end{theorem}
149\begin{remark}Assume $N$ (or without loss $N_1$) is
150square-free. We can determine efficiently the subspace $M_N^0$
151corresponding to newforms in $S_2(N)$; it is the subspace
152formed by all~$x$ so that for all divisors~$d$ of $N_1$ we have
153$154 \phi_d(x)=\phi_d(W_d(x))=0. 155$
156In particular if $N=pq$, $q$ prime, it is the subspace of $M_{pq}$
157intersection of the kernel of $\phi_q$ and of $\phi_qW_q$.
158\end{remark}
159
160\subsection{Relation to the quaternion algebra}
161
162The matrices of the operators $T_n$ acting on $M_N$ are the same
163as the classical Brandt matrices \cite{15}, constructed using
164quaternion algebras.
165
166Let $B_{p,\infty}$ be the quaternion algebra over~$\Q$ ramified exactly
167at~$p$ and infinity, and let~$\O$ is an Eichler order of level $N_1$
168(defined by Eichler \cite{6} in the case when $N_1$ is
169square-free, and defined in general by Pizer \cite{14}), and let
170$\vvn{I}{h}$ be representatives of the left ideal classes of
171$\O$.
172
173Let $\O_i$ be the right order (i.e., right normalizers) of the
174ideals $I_i$, and $e_i$ be the number of units of $\O_i$.
175The Brandt matrix $B(n)=(b_{i,j}^{(n)})$ has $i,j$ entry
176$177b_{i,j}^{(n)} = 178 e_j^{-1}\cdot |\{\a : \a\in 179 I_j^{-1}I_i,\,\Nor(\a)\Nor(I_j)/\Nor(I_i)=n\}| 180$
181where $\Nor$ is
182the norm over $B_{p,\infty}$ (the norm of an ideal being the
183$\gcd$ of the norms of its nonzero elements).
184
185In the language of supersingular curves of characteristic $p$, we
186may give these matrices (actually their transposes) the following
187interpretation:
188
189Let $S$ be a supersingular point as in $I.1$, i.e., a
190supersingular elliptic curve $E$ defined over $\overline{\F}_p$
191together with a cyclic group $C$ of order $N_1$. The ring of
192endomorphisms $\O_1$ of $S$ is an Eichler order of level $N_1$. To
193all the other supersingular points $S'=(E',C')$ we associate the
194set $I_{S,S'}$ of homomorphisms from $S$ to $S'$, i.e. the set of
195all homomorphisms $\a$ from $E$ to $E'$ that send $C$ to $C'$.
196This is obviously a left ideal over $\O_1$, and its inverse ideal
197is $I_{S',S}$. We can prove that all the right ideals of $\O_1$
198are obtained in this way, and the whole Eichler order of level
199$N_1$ if the rign of endomorphisms of a supersingular point $S$.
200It is clear that the general term $B_{i,j}^{(n)}$ of the $n$-th
201Brandt matrix is the number of isogenies of $S_i$ to $S_j$ (the
202supersingular points being conveniently indexed,) two such
203isogenies being identified is different by an automorphism of
204$S_j$. We can retrieve the matrix of the operator $T_n$ acting
205over $M_n$.
206
207On the other hand if for all pairs of supersingular points
208$(S,S')$ we associate the function $\t_{S,S'}(q)=\sum_\a 209q^{\deg\a}$where $\a$ goes through all the homomorphisms of $S$
210to $S'$, we retrieve the functions $\t$ classically associated
211with the ideals of the orders of the quaternions, or, if one
212prefers, associated with the positive integer quadratic forms in 4
213variables.
214
215It is therefore easy to prove that if $\sum{x_S}[S]$ is an
216elements of $M_N\otimes \CC$ eigenvector of all the Hecke
217operators and if $f(q)$ is the corresponding modular form, we
218have, for all $S'$
219$x_{S'}f(q)=\sum_Sx_S\t_{S,S'}$which allows, in theory, to find
220the coefficients $a_n$ of $f$, using the $x_S$. In practice,
221unfortunately, the computation of $a_n$ demands the knowledge of
222all the isogenies of degree $n$ to $S'$, and there doesn't seem to
223be a simple algorithm for that.
224
225Nevertheless, in certain cases, there exists a different method to
226calculate the coefficients of $f$, which is easy as far as
227computation is concerned. Suppose that $N$ is a prime (thus equal
228to $p$), or $N$ is a product of primes $pq$ and $X_0(q)$ is of
229genus $0$ (thus $q=2,3,5,7$ or $13$).
230
231In the appendix, we give for each such case an equation of
232$X_0(q)$ of the form $xy=p^k$, thus the action of the Hecke
233operators $T_2$ and $T_3$ over $X_0(q)$, which is given by an
234equation much simpler than the equation of modular polynomials
235$\Phi_2(j,j'),\Phi_3(j,j')$ (which give the action of $T_2,T_3$ on
236$X_0(1)$, parametrized by the modular invariant $j$; cf. section
2372.4).
238
239Let $u=x$ if $N=pq$ and $u=j$ if $N=p$. The Fourier expansion of
240$u$ at infinity is $1/q+\cdots$. Let $f(q)=\sum a_nq^n$ a normalized
241newform of level $N$ and weight 2 corresponding to a vector
242$\sum x_S[S]$ of $M_N^0\otimes K$, where $K$ is the extension of
243$\Q$ generated by the $a_n$. Therefore there exists a prime ideal
244$\wp$ of $K$ over $p$ so that \bean\left(\sum
245x_S \cdot u(S)\right)f(q)\frac{dq}{q}\equiv \sum x_S\frac{du}{u-u(S)} \pmod{\wp}
246.\eean (it is about the congruence between Laurent series in
247$q$).
248
249Suppose for example that $f$ corresponds to a Weil curve of
250conductor $N$, so that $a_n$ are in $\Z$. The $x_S$ are in $\Z$
251and one can prove that $\sum x_Su(S)\neq 0$. Thus we know $a_n \m 252p$ for all $n$. Hasse's inequality $|a_l|<2\sqrt{l}$ for $l$ prime
253proves that we know the $a_n$ for $n<p^2/16$.
254
255%% NETWORK
256
257
258\subsection{Explicit construction of the net $M_N$}
259
260In this section we suppose that $N$ is odd. Suppose that given an
261explicit model of the curve $X_0(N_1)$, and so the action of the
262Hecke operator $T_2$ on that model (cf. Appedix).
263
264%%INERT
265
266First we need to find a supersingular points. Note that they are
267defined over $\F_{p^2}$. For example suppose that $N=p$. First we
268check to see if $p$ is inert in one of the 9 imaginary quadratic
269fields of class number 1. If yes, then one can take for the
270initial value of $j$ the modular invariant of the curve of complex
271multiplication by the ring of integers of corresponding fields. If
272not, one can know a list of minimal polynomials of modular
273invariants of elliptic curves of complex multiplication by
274imaginary quadratic fields of small class numbers, and apply the
275same method. One needs here to solve over $\F_{p^2}$ a polynomial
276equation, which can be done in $\log p$ operations -- at least
277probabilistically. Finally suppose that all these attempts fail.
278There remains the possibility to enumerate all the values of
279$\F_p$ until finding a supersingular value. We know there must
280exist a supersingular $j$-invariant in $\F_p$,
281but unfortunately only a very small number---on the
282order of the size of the class group of $\Q(\sqrt{-p})$, or
283approximately $\sqrt{p}$.
284
285So assume we know a supersingular point $S_1$. Knowing the action
286of $T_2$ on the model given by $X_0(N)$ allows us to obtain the
287three supersingular points $S_2,S_3,S_4$ (not necessarily
288distinct) related to $S_1$ by a 2-isogeny. It comes down to
289solving a degree 3 polynomial over $\F_{p^2}$, which needs
290extracting cubic and square roots, operations that need $O(\log 291p)$ operations. Sometimes we may as well exlude this computation.
292Suppose that $n=p$ and that we have, say $p\equiv 2 (\m 3)$. Thus
293$p$ is inert in $\Q(\sqrt{-3})$, so $j=0$ is a supersingular
294value, and we know that the three isogenies of degree 2 send the
295curve of the invariant to the curve of complex multiplication by
296$\Z[\sqrt{-3}]$, for which the invariant is $j=54000$.
297
298In any case, we have at most one time when we need to solve a 3rd
299degree equation: once $S_2$ is known, we search from $S_i$ ($i\geq 3002$) the three supersingular points which are related, but we
301already know one, so we only need to solve a second degree
302equation, which comes down to square roots over $\F_{p^2}$ which
303is fast (probabilistic methods require $O(\log p)$ operations
304using an algorithm that is very simple to implement).
305
306To prove that we can find, step by step, all the supersingular
307points of $M_N$ it is enough to prove that the graph of $T_2$ (and
308more generally of $T_n$) is connected. But, as Serre remarked, the
309eigenvalue $a_2=3$ of $T_2$ over $M_N$ has multiplicity equal to
310the number of connected components of the graph of $T_2$. But in
311$M_N$, the space $M_N^0$ corresponding to the cusp forms of
312codimension 1, so 3 is a simple eigenvalue in $M_N$ (because for a
313cusp form we have $|a_2|<2\sqrt{2}$), so the graph of $T_2$ is
314connected.
315
316In conclusion, an algorithm in $O(N\log N)$ operations gives all
317the supersingular points and the Brandt matrix $B_2$ associated to
318them. One of the advantages of this matrix is that it is very
319sparse; on each line and column there are at most 3 nonzero terms,
320which are integers whose sum is 3. This allows, given an
321eigenvalue, to find very quickly, if $N$ is large, the
322corresponding eigenvectors.
323
324\subsection{Examples}
325
326\be
327
328\item Take for example $N=p=37$. Since $37$ is inert in
329$\Q(\sqrt{-2})$, one can take as the first vertex of our graph the
330curve $E_1$ of complex multiplications by $\Z[\sqrt{-2}]$, for
331which the modular invariant is $j_1=8000\equiv 8 \m 37$. We need
332to find now all the invariants of curves 2-isogenous to this,
333i.e., to solve the equation $\Phi_2(x,8000)\equiv 0 (\m 37)$. But
334$\sqrt{-2}$ is an endomorphism of degree 2 of $E_1$, so $j_1$ is a
335root (over $\Q$) of the polynomial $\Phi_2(x,8000)$. Dividing this
336polynomial by $x-8000$ we get a second degree polynomials with
337roots $j_2,j_3$, the invariants of the other two curves, $E_2,E_3$
338related to $E_1$ by a degree 2 isogeny. Let $\w\in\F_{p^2}$ so
339that $\w^2=-2$. One gets that then $j_2=3+14\w,j_3=3-14\w$.
340
341Another method to find $j_2,j_3$ consists in remarking that 37 is
342equally inert in the field $K=\Q(\sqrt{-15})$, for which the class
343number is 2. The second degree polynomial giving the values of the
344modular invariants of 2 curves of complex multiplication by the
345ring of integers of $K$ is $x^2+191025x-121287375$, whose roots
346generate $\Q(\sqrt{5})$,  so modulo 37 are conjugate in
347$\F_{37^2}$. We can thus find $j_2,j_3$.
348
349For $N$ prime congruent to 1 mod 12, the number of supersingular
350curves mod $N$ is $(N-1)/12$. For $N=37$ we get 3 supersingular
351curves. It remains to show that the action of $T_2$ on $E_2$ (by
352conjugations we get the action on $E_3$). It is not possible to
353have 2 isogenies of $E_2$ on $E_1$, because then we would have 5
354isogenies of degree 2 starting in $E_1$. Therefore there is one
3552-isogeny of $E_2$ over $E_2$.
356
357Actually, if there is a 2-isogeny of an elliptic curve of
358invariant $j$ on itself, this invariant is the root of the
359equation $\Phi_2(x,x)=0$, a fourth degree equation that can be
360written as $(x-1728)(x-8000)(x+3375)^2$. (To see this, one can
361make the computation of the equation of $\Phi_2(j,j')$ above. One
362can also search which are the curves of complex multiplication
363that admit a degree 2 endomorphism, i.e., which are the imaginary
364quadratic fields that contains an element of norm 2. One finds, by
365multiplication by the units of the (corps pres?'') the elements
366$1+i,\sqrt{-2}, \frac{1+\sqrt{-7}}{2}$ and $\frac{1-\sqrt{-7}}{2}$
367that are the endomorphisms of degree 2 of the curves of invariant
368$j=1728,j=8000$ and for the last two, $j=-3375$.)
369
370By order, mod $p$, the graph of $T_2$ cannot contain a loop of a
371supersingular curve on itself -- although this curve is defined
372over $\F_p$ (and, more precisely, it is one of 3 curves described
373above). Therefore, there are 2 isogenies relating $E_2$ to $E_3$
374and the graph of $T_2$ acting on $M_{37}$ is completely
375determined.
376
377To compute the corresponding eigenvectors, one can evidently
378diagonalize the matrix $(3,3)$ of $T_2$ but there is a simpler
379method:
380
381the involution $W_{37}=-\Frob_{37}$ separates $M_{37}$ in an
382obvious way into two orthogonal proper subspaces, one generated by
383$u_1=[E_2]-[E_3]$, associated with the eigenvalue 1, and the other
384associated with the eigenvalue -1, generates by
385$\Eis=[E_1]+[E_2]+[E_3]$ and the vector product of $u_1$ and $\Eis$,
386let it be $u_2=2[E_1]-[E_2]-[E_3]$. One can deduce, without
387recourse to $T_2$, that there exist 2 newforms for which the
388$q$-expansion has rational coefficient, and thus that $J_0(37)$,
389the jacobian of $X_0(37)$ is isogenous to the product of 2
390elliptic curves (which is well-known, see for example \cite{9}).
391Formula (1) above allows us to obtain the first 83 terms of their
392function $L$.
393
394\item $p-37,N=2\cdot 37$.
395
396To study $X_0(74)$ one uses the homomorphism $\phi_2$ of $M_{74}$
397to $M_{37}$ defined previously. The fibres of reach of the three
398supersingular points $[E_1],[E_2]$ and $[E_3]$ of $X_0(1) \m 37$
399are formed by three distinct supersingular points of $X_0(2) \m 4002$. In a general way, write that if $\vvn{S}{k}$ are the
401supersingular points of $X_0(qM) \m p$ above a supersingular point
402$S$ of $X_0(M) \m p$ ($p,q$ coprime and coprime with $M$), one has
403the formula
404$\frac{q+1}{\Aut S}=\sum_1^k\frac{1}{\Aut S_i}.$
405
406The equation of $X_0(2)$ used here is that described in the
407appendix: $uv=2^{12}$, the involution $W_2$ switching $u$ and $v$.
408Recall that $W_{37}=-\Frob_{37}$ and that $j=(u+16)^3/u$ (where $j$
409is 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
412three supersingular points of $E_1$, of coordinates
413$u_1=(-1+\w)/2,u_2=(-1-\w)/2=W_2(u_1)$ and $u_3=27=W_2(u_2)$.
414(Here again, it is possible to guess the value of $u_3$, because
415it is clear by the action of $T(2)$ on $X_0(1) \m 37$ done
416previously that one of the above $E_1$ must be invariant relative
417to $W_2$; or the two solutions of $u^2=2^{12}$ are $u_1,-u_1$.
418Replacing them in the equation that gives $j$ one can see that it
419is about $u_1$. To get $u_2,u_3$ it is enough to solve a second
420degree equation.)
421
422One can compute that $u_4=W_2(u_1)=2^{12}/u_1=-5-5\w$, and one
423finds that the corresponding invariant $j(u_4)$ is $j_2=3+14\w$.
424One solves the second degree equation given 2 other points above
425by $j_2$ 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
428supersingular points over $E_3$ ($x\to \bar{x}$ being the
429nontrivial automorphism of $\F_{p^2}$.) We get the list of all
430supersingular points of $X_0(2) \m 37$.
431
432As said above, the space $M_{74}^{new}$ corresponding to the
433newforms is the intersection of the kernel of $\phi_2$ and the
434kernel of $\phi_2W_2$. If we write $[u_i]$, $i=1,\ldots, 9$) the
435generators of $M_{74}$ corresponding to the supersingular points
436of $x$-coordinate $u_i$, an examination of the action of $W_{37}$
437and $W_2$ prove that $M_{74}^{new}$ is the direct sum of two
4382-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
441other, $G_2$, 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
445Using the equation of $T_3$ acting on $X_0(2)$ (cf. appendix), one
446can 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
459One deduces that $J_0^{new}(74)$ is isogenous to the product of
460two abelian simple varieties, $A_1$ (resp. $A_2$), of real
461multiplication by the ring of integers of $\Q(\sqrt{5}),$
462(respectively $\Q(\sqrt{13})$.)
463
464If $\l=\frac{-1+\sqrt{5}}{2},\mu=\frac{3+\sqrt{13}}{2}$, then the
465vectors $v_1=e_1+(\l+1)e_2,v_2=e_1-\l e_2,v_3=\mu 466e_3+e_4,v_4=(3-\mu)e_3+e_4$ corresponding to the 4 newforms
467$f_1,f_2,f_3,f_4$ of weights 2 and level $74$. Using (1) one gets
468the first 83 values of the coefficients of these newforms. For
469example 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
475and 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
485Let $f=\sum a_nq^n$ be a newform of weight 2 and level $N$ with
486integer coefficients. They correspond to a strong Weil curve $\E$
487of conductor $N$. Unfortunately the coefficients $a_n$ don't give
488too much information on $\E$ and do not allow us to obtain a
489simple equation for $\E$. (In \cite{10} there is a method due to
490Serre that sometimes allows us to get such an equation, but that
491method is not systematic.) Here we give a method that at least
492when $N=p$ is a prime, one can solve the problem.
493
494From now on let $N$ be a prime. According to the last section, for
495a 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.
497Theorem 1 doesn't describe the isomorphism (which is not
498canonical) between $S_2(N)$ and $M_N^0\otimes\CC$. But suppose
499known the terms $a_n$ of $f$ ($a_2$ is sufficient in general). The
500construction 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
502the eigenspace $V_2$ associated with the eigenvalue $a_2$. If it
503is of dimension 1, we also have $v_f$, or at least the space it
504generates. Otherwise, we apply $T_3$ to $V_2$ (which is of
505tentatively small dimension---for conductors $< 80000$, $\dim V_2$
506doesn't goes beyond 6), until finding a 1-dimensional space,
507corresponding 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
510just mean that the gcd of coefficients is $1$?};
511then $r_f$ is determined up to sign.
512
513To go further, we need a geometric interpretation of the $x_E$.
514Let $\d=\pm N^\delta$, the discriminant of the minimal Weierstrass
515model of $\E$, let $\phi:X_0(N)\to\E$ be a minimal cover of $\E$
516 of degree $n=\deg \phi$.
517
518According to Deligne-Rapoport \cite{5}, there exists a model
519$X_0(N)_{/\Z}$ of $X_0(N)$ defined over $\Z$ for which reduction
520mod $N$ is the union of two projective lines, one $C_\infty$
521classifying the elliptic curves of characteristic $N$ provided
522with the group scheme kernel of the Frobenius (this
523corresponding to inseparable isogenies), the other one, $C_0$,
524classifying the curves provided with Verschiebung. These two
525lines intersect at supersingular points. As far as the curve $\E$
526is concerned, reduction mod $N$ of its Neron model
527has identity component  $\E^0_{/\F_N}$ isomorphic to $\F_{N^2}$ of the
528multiplicative group $G_m$. One can prove that the cover extends
529to $X_0(N)_{/\Z}-\S$ where $\S$ is the set of all supersingular
530points of characteristic $N$, and define by restriction a regular
531application ?'' of $C_\infty$ on $\E^0_{/\overline{\F}_N}$, of a
532rational function $\phi$ over $C_\infty$, for which the poles and
533zeros are in $\E$. The divisor $\sum \l_E[E]$ of $\phi$, $E$ going
534through all the supersingular curves $\m N$, and thus an element
535of $M_N^0$, defined up to sign (depending on the choice of
536isomorphism 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
542It is not difficult to see that $(\Phi)$ is proportional to $r_f$.
543By contradiction, the fact that the $l_E$ are coprime with one
544another 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
546weight 2 and level $Np$ (where $Np$ is square-free) for which the
547associated representation mod $l$ is irreducible and not ramified
548at $p$, 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
550that the Taniyama-Weil conjecture implies the Fermat theorem.)
551
552To prove the previous theorem, one proves first that $\delta$ is
553related 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
555number $l$ divides the gcd is $\l_E$. It also divides $\delta$,
556and one deduces from here that $p$ is not ramified in the field of
557points of order $l$ or $\E$. If $l$ is coprime with 6 Ribet's
558\footnote{K.Ribet, {\it Lectures on Serre's conjectures}, MSRI,
559Fall 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
5611, which cannot be but the Eisenstein series. The curve $\E$ is
562semi-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 $l=2,3$
564we get the same result due to \cite{4}, Appendix. Now, we know
565explicitly the curves of prime conductor with torsion \cite{11}
566namely the curves 11A and 11B of \cite{19}, which have a point of
567order 5, curves 17A,17B,17C (point of order 4), 17D (point of
568order 2), 19A and 19B (point of order 3), 37B, 37C (point of order
5693) and the curves of Setzer-Neumann \cite{18}, which have a point
570of order 2. In each of these cases, we know $\delta$, which is
571equal to the number of finite points rational over $\Q$ of the
572considered curves, and one can verify that the $\l_E$ are coprime
573with one another. This proves the proposition. Note that along the
574proof we showed that Ribet's theorem implies the following
575
576\begin{theorem}
577Let $E$ be a strong Weil curve of prime conductor $N$. The
578valuation of its discriminant in $N$ is equal to the number of
579torsion points of $E(\Q)$.
580\end{theorem}
581
582We state without proof the theorem that allows us to get an
583explicit equation for $\E$ once we know the $\l_E$.
584
585\begin{theorem}
586Let $\E$ 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
588constructions above. There exists an equation of $\E$
589$y^2=x^3-\frac{c_4}{48}x-\frac{c_6}{864}$
590with $c_4,c_6\in\Z$ 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
597supersingular 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
608If the $\l_E$ are known then 5 allows us to get $n$ and 1 allows
609us 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
611congruences 3 and 4 allow us to reduce the number of computations
612significantly. Thus we have found an equation of a strong Weil
613curve corresponding to the initial newform $f$.
614
615This method also allows us to prove that an elliptic curve of
616small prime conductor is a Weil curve. Suppose that we are given
617such a curve by its equation. Then we may compute the number of
618its points $N_l$ mod $l$ for $l=2,3,\ldots$. Next we search, by
619the 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
621false. If yes, then continue with $T_3$ acting on the found
622eigenspace, if it is not of dimension 1, until we get an
623eigenspace of dimension 1 for the Hecke operators, with integers
624eigenvalues. If there is no such thing, then we get a
625counterexample to the Taniyama-Weil conjecture. If there is one,
626we compute the equation of a corresponding Weil curve. If this
627curve is isogenous to the initial curve, we are done. Otherwise,
628the initial curve is not a Weil curve.
629
630In 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
633This curve seems to be the smallest curve (ordering the curves by
634their conductors) having a Mordell-Weil rank $\geq 3$ \cite{3}.
635The interest in it is the following:
636
637Let $f(z)=\sum a_nq^n$ ($q=e^{2\pi iz}$), a newform of weight 2
638and 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
640proved 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
643discriminant $-p$. We have other formulas, but more complicated,
644in the case of imaginary quadratic fields of non-prime
645discriminant (see \cite{13} for example).
646
647If the Birch and Swinnerton-Dyer conjecture is true, all the Weil
648curves for which the Mordell-Weil group over $\Q$ is of rank $\geq 6493$ have to be given by such modular forms, but until the work of
650Gross and Zagier \cite{8}, there was no way to verify that the
651derivative at 1 of the $L$ function of a Weil curve is indeed 0.
652The results of Gross and Zagier allow to write $L'(1)$ as the
653product of a non-zero factor easily computable and the
654N\'eron-Tate height of a Heegner point (cf. \cite{8} for more
655details.) It is therefore possible, by decreasing the height of
656rational points on the curve and increasing $L'(1)$ by a careful
657computation, to prove that $L$ is of order $\geq 3$ at $s=1$. (In
658all the previous, we considered odd Weil curves, i.e., for which
659the $L$ function has an odd order at 1 -- or if one prefers for
660which the sign of the functional equation is -1.)
661
662One has several method to construct Weil curves for which the
663Mordell-Weil group is of rank $\geq 3$ (and which are good
664candidates for the preceding question: by the method of
665Gross-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
667value of the discriminant of imaginary quadratic fields of given
668class numbers; if it is non-zero, the conjecture of Birch and
669Swinnerton-Dyer is false.) One can, for example, search for curves
670of complex multiplication of rank 3 (we know that they are Weil
671curves), but the constant $C_f$ is very large. One can deform\footnote{Twist?} a
672Weil curve (for example the curve 37C of \cite{19} until getting a
673rank 3 curve (for the curve 37C, one can deform by
674$\Q(\sqrt{-139})$, as shown by Gross and Zagier \cite{8}.) This
675leads to a constant $C_f$ of order of 7000
676
677One may choose some elliptic curve defined over $\Q$, or rank 3,
678and 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
680trace formula. But the computation is very long. The method of
681graphs allows us to do it in about 5 seconds an a computer that
682needed 5 hours with the mentioned method.
683
684For this curve, one has $C_f<50$: all imaginary quadratic curves
685of discriminant $d$ with $|d|>e^{150}$ therefore has a class
686number $\geq 4$. On the other hand, there is no imaginary
687quadratic field of discriminant $d$ and class number 3 for
688$907<|d|<10^{2500}$ \cite {12}. Therefore (after an examination of
689a table of class numbers of the first quadratic fields):
690
691\begin{theorem}
692The imaginary quadratic fields of class number 3 are the 16 fields
693of 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
699Let $\r$ be a continuous representation of $Gal(\overline{\Q}/\Q)$
700in $GL_2(V)$ where $V$ is a dimension 2 vector space over a finite
701field $\F_q$ of characteristic $p$. Assume this is an odd
702representation, i.e., that $\r(c)$ the image of the complex
703conjugation, seen as an element of $Gal(\overline{\Q}/\Q)$ has
704eigenvalues 1 and -1. In that case put $G=Im\r$.
705
706In \cite{17} Serre defines the level, the character and the weight
707of such a representation:
708
709\be
710
711\item The level.
712
713Let $l$ 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
718The conductor of the representation $\r$ is defined as
719$N=\prod_{l\neq p}l^{n(l)}.$
720
721\item The character.
722
723The determinant of $\r$ yields a character of
724$Gal(\overline{\Q}/\Q)$ in $\F_q^*$, for which the conductor
725divides $pN$. Therefore, one can write
726$\det\r=\e\chi^{k-1},$where $\chi$ is the cyclotomic character
727of conductor $p$ and $\e$ is the character $(\Z/N\Z)^*\to\F_q^*.$
728The integer $k$ is defined mod $(p-1)$, and the fact that the
729representation is odd implies that $\e(-1)=(-1)^k$.
730
731By definition, $\e$ is the character of the representation $\r$.
732
733\item The weight.
734
735The integer $k$ above is defined mod $(p-1)$. Read Serre's article
736for 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
738places coprime with $p$, the definition of weight only uses the
739local properties at $p$ of the representation $\r$.
740
741\ee
742
743Then Serre's conjecture is:
744
745\begin{conjecture}
746Let $\r$ be a representation as above, of weight $k$, level $N$
747and character $\e$. Assume this representation is irreducible.
748Then it comes from a cusp form $\m p$ of weight $k$, level $N$ and
749character $\e$.
750\end{conjecture}
751
752This conjectures, if true, has numerous consequences: it implies
753the Taniyama-Weil conjecture and Fermat's theorem.
754
755Many such representations $\r$ are modular, either by
756construction, or because they are part of classical conjectures
757(Langlands, Artin, $\ldots$) that carry on the conjecture (but
758sometimes in a weak form, i.e., with a weight or conductor bigger
759than those defined in \cite{17}.)
760
761In order to verify (or contradict) Serre's conjecture, we need to
762find 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
764not difficult to calculate, for $l$ prime and not too large, the
765trace $a_l$ of $\Frob_l$ in $GL_2(\F_q)$: if $P(x)$ is a polynomial
766whose roots generate $K$ the decomposition of $P \m l$ usually
767will suffice.
768
769It is however, much harder to find modular forms $\m p$, if they
770exist, that correspond to the representation $\r$ given by the
771field $K$: the discriminant of $K$ is usually large, thus so is
772the conductor of $\r$, which is related to it, so it is not easy
773to make the computations.
774
775\subsection{The case $SL_2(\mathbb{F}_4)$}
776
777A troubling case is that of $p=2$, because, since $-1\equiv 1 (\m 7782)$ all representations are odd.
779
780The representations of $Gal(\overline{\Q}/\Q)$ in $GL_2(\F_2)=S_3$
781(although altogether real, cf. \cite{17}) come from weight 1
782modular forms; the group $S_3$ can be realized as a subgroup of
783$GL_2(\CC)$. One can hope that by multiplication with convenient
784Eisenstein series, one can obtain a modular form of weight and
785level predicted by the Serre conjecture (cf. \cite{17} for
786examples.)
787
788In order to obtain the most interesting case for characteristic 2,
789one considers the representations with values in $GL_2(\F_4)$. The
790isomorphism $A_5\simeq SL_2(\F_4)$ allows us to obtain several
791examples. Let $K$ be an extension of $\Q$ of Galois group $A_5$.
792Since $A_5$ immerses ?'' into $PGL_2(\CC)$, if the field is not
793completely real, the associated representation $\r$ comes from a
794weight 1 modular form (module Artin's conjecture, cf. \cite{2}).
795Suppose now that $K$ is real. None of the classical conjectures
796allow us to suspect that $\r$ comes from a modular form, even if
797of higher weight or level. It is this case that we will study in
798what follows. The method of graphs here is indispensable, the
799modular forms that we look at having a conductor too large to be
800studied with the Eichler-Selberg trace formula.
801
802Let $P(x)=x^5+a_1x^4+a_2x^3+a_3x^2+a_4x+a_5$ be a rational
803polynomial of discriminant $D$. In order that the field of roots
804of $P$ be $A_5$ it is sufficient and necessary that $P$ be
805irreducible, that $D$ be square-free, and that there exist a prime
806number $l$ not dividing $D$ so that $P \m l$ having exactly two
807roots in $\F_l$ (this last condition assuring that the group is
808all of $A_5$).
809
810It is clear that $\e=1$. If $p\mid{}D$, $p$ coprime with 30, $n(p)=1$
811if it seulment si l'inertie en p ==?'' is of order 2, and thus the
812polynomial $P$ has at most double roots mod $p$. As far as the
813weight $k$ is concerned, it is either 2 or 4 according to the
814ramification of $K$ at 2. To simplify the computation, we have
815limited to searching examples among the representations of prime
816level and weight 2.
817
818On 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
820it exists, cannot be in $\F_4$, but in $\F_{16}$. This comes from
821the fact that the coefficient $a_l$ of a modular form $\m l$ is
822equal to an eigenvalue of $\Frob_l$, and not to its trace. Now, if
823a matrix in $SL_2(\F_4)$ is of order 5, its eigenvalues are in
824$\F_{16}$ not in $\F_4$.
825
826The examples treated above were obtained by making a systematic
827search on a computer of convenient polynomials (totally real, of
828type $A_5$, for which the conductor of the associated
829representation is a prime $N$, and for which the weight is 2).
830
831Thereafter, for each such polynomial $P$, one computes the
832corresponding eigenvalue $a_2$ (in $\F_{16}$), and one tries to
833find whether there exists a modular form mod 2 of level $N$ and
834weight 2 so that $T_2$ has $a_2$ as an eigenvalue. In all the
835cases considered, we have thereafter found an eigenspace of
836dimension 1 or 2. Using the operators $T_3,T_5$, one calculates
837the coefficients $a_3,a_5$, and verifies that they correspond to
838the values predicted by the decomposition of $P$ in 3 and 5.
839
840Clearly, this doesn't really prove that the representation $\r$
841associated to $P$ is modular: we have only exhibited a modular
842form 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
844the truthfulness of the conjecture of Serre in the considered
845cases: an exhaustive search over numerous primes $N$ of the
846coefficients $a_2$ of modular forms of weight 2 and level $N$
847proves that it is rare that there are fields of small degree.
848(Actually, is seems that 2, and in general the small primes, are
849the most inert'' possible in the fields that appear in the Hecke
850algebra of modular forms, fields which themselves in general
851appear to have the largest degree possible, taking into account
852constraints such as the Atkin-Lehner involutions, primes of
853Eisenstein, etc. One gets that one has small factors, --
854corresponding 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
863The discriminant is $(2^3887)^2$. This polynomial is irreducible
864mod 5, thus irreducible over $\Q$. Its roots are all real (apply
865Sturm's algorithm). One has that $P(x)\equiv x(x-1)(x^3+x^2-1) \m 8663,$ which gives a cycle of order 3; the Galois group of $K$, the
867field 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
878From $P(x)\equiv (x-462)(x-755)^2(x-788)^2 \m 887$ one gets that
879the conductor $N$ of the associated representation is $N=887$. One
880can also prove that 2 is little ramified'' in the sense of
881\cite{17}, thus $\r$ has weight 2. Examining the reduction mod 2
882of $P$ proves that the coefficients $a_2,a_3,a_5$ of the modular
883form mod 2 of level 887 (which must correspond to $\r$ via the
884Serre conjecture) are 1, 1, j (where $j\in\F_4$ has the property
885that $j^2+j+1=0$).
886
887One therefore applies the method of graphs: the space of modular
888forms mod 2 of weight 2 and level 887 has dimension 73, and
889computation shows that the eigenspace $G_1$ of $T_2$ corresponding
890to 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 $G_1$,
892from 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
895operators. These corroborate the conjecture.
896
897\item $P(x)=x^5-23x^3+55x^2-33x-1$.
898
899Then $D=13613^2,P(x)\equiv (x-6308)(x-2211)^2(x-8248)^2 \m 13613$,
900$N=13613$; $P$ being irreducible mod 2, $\Frob_2$ is a cycle of
901order 5, and $a_2=\zeta_5$ is a fifth root of unity, viewed as an
902element of $\F_{16}$. Computation also shows that in the space of
903modular forms mod 2 of level 13613 and weight 2, which has
904dimension 1134, $\zeta_5$ is a simple eigenvalue of $T_2$. The
905coefficients $a_3,a_5$ are respectively equal to
906$1+\zeta_5^2+\zeta_5^3=j$ and $\zeta_5^2+\zeta_5^3=j^2$, which are
907the traces of $\Frob_3,\Frob_5$ in $SL_2(\F_4)$.
908
909\item We write the other found polynomials; in each case there
910exists a modular form of weight 2 and appropriate level, for which
911the first terms $a_n$ correspond to those values predicted by the
912Serre 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
922In \cite{5}, it is proven that if $p$ is a prime number then the
923curve $X_0(p)$ over $\Z_p$ is formally isomorphic to the curve of
924equation $xy=p^k$, in the neighborhood of each point reducing mod
925$p$ to a supersingular point $S$, $k$ being one half the number of
926automorphisms of $S$.
927
928If $X_0(p)$ has genus 0 (i.e., $p=2,3,5,7,13$) one has such a
929model over $\Z$, given by the function \bean
930x=\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
933This results from Fricke \cite{7}, who gives for each of the above
934$p$'s an expression of the oubli ?'' homomorphism $j:X_0(p)\to 935X_0(1)$, which associates to each point $(E,C)$ of $X_0(p)$ the
936point $(E)$ of $X_0(1)$, parametrized by the modular invariant
937$j$.
938
939In the following we recall these equations and give the
940expressions of the correspondences $T_2,T_3$ over these curves.
941The 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
948model 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$x^4+y^4-x^3y^3-2^33^2x^2y^2(x+y)-2^23^25^2xy(x^2+y^2)+2\cdot 9583^21579x^2y^2-2^{15}3^2xy(x+y)-2^{24}xy=0$
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$T_3:x^4+y^4-x^3y^3-2\cdot 3^2x^2y^2(x+y)-3^4xy(x^2+y^2)-2\cdot 9723^223x^2y^2-2250xy(x+y)-5^6xy=0$
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$T_3:x^4+y^4-x^3y^3-2^23x^2y^2(x+y)-2\cdot 3\cdot 7xy(x^2+y^2)-3\cdot 97953x^2y^2-2^23\cdot 7^2xy(x+y)-7^4xy=0$
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$T_3:x^4+y^4-x^3y^3-2\cdot 3x^2y^2(x+y)- 3\cdot 5xy(x^2+y^2)-3\cdot 98611x^2y^2-2\cdot 3\cdot 13xy(x+y)-13^2xy=0$
987
988\ee
989
990The polynomials above that give $T_2,T_3$ are of simpler form than
991the 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 $X_0(1)$).
993For comparison, we recall their expressions:
994
995\bea \Phi_2(j,j') &=& j^3+j'^3-j^2j'^2+2^43\cdot
99631jj'(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') &=&
1000j^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
10023^413\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
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1096