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146\def\Merel{{$[2]$}}
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148
149\centerline
150{\titchap The field generated by the points of small}
151\centerline
152{\titchap prime order on an elliptic curve}
153\medskip
154
155\bigskip\bigskip\bigskip
156\centerline{\pc Lo\"\i c Merel {\rm and} William A.~Stein}
157\bigskip\bigskip\bigskip
158
159\bigskip
160
161\bigskip\bigskip\noindent
162{\bf Introduction}
163\bigskip\bigskip
164
165Let~$p$ be a prime number.
166Let $\bar\B Q$ be an algebraic closure of $\B Q$.
167Denote by $\B Q(\mu_p)$ the cyclotomic subfield of $\bar\B Q$
168generated by the $p$th roots of unity.
169Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the
170points of order~$p$ of
171$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.
172\bigskip
173\th{}{\it One has $p>1000$, $p<6$ or $p=13$.}
174\bigskip
175
176We note that the case $p=7$ was treated by Emmanuel Halberstadt.
177The part of the theorem that concerns the case
178$p\equiv 3\pmod 4$ is given in~\Merel.
179We propose to give the details that permit our treating the
180more difficult case in which $p\equiv 1 \pmod 4$.
181We treat this last case with the aid of Proposition~2 below, which is
182not present in {\it loc. cit.}
183
184It may be possible to exclude the case when $p=13$
185by studying the modular curve $X_1(13)$ and its Jacobian $J_1(13)$.
186
187\bigskip\noindent
188{\bf 1. We recall the results of \Merel}
189\bigskip
190Denote by
191 $J_0(p)$ the Jacobian of the modular curve $X_0(p)$ (whose points are
192isomorphism classes of
193generalized elliptic curves equipped with a cyclic subgroup of order~$p$).
194Consider the subring $\B T$ of ${\rm End}\,J_0(p)$
195generated by the Hecke operators. We refer the reader to~\Mazur{}
196for an in-depth study of these objects.
197
198Let~$S$ be the set of isomorphism classes of supersingular elliptic
199curves in characteristic~$p$.  Denote by $\Delta_S$ the group formed
200by the divisors of degree~$0$ with support on~$S$.  It is equipped with
201a structure of $\B T$-module (deduced, for example, from the action
202of the Hecke correspondences on the fiber at~$p$ of the regular minimal
203model of $X_0(p)$ over $\B Z$).
204
205Let $j\in\bar\B F_p-J_S$, where $J_S$ denotes the set of supersingular modular
206invariants. We denote by  $\iota_j$ the homomorphism of
207groups $\Delta_S\raw \bar\B F_p$ that associates to $\sum_E n_E[E]$
208the quantity
209$\sum_E n_E/(j-j(E))$, where $j(E)$ denotes the modular invariant of~$E$.
210
211
212One says that an element  $j\in\B F_p$ {\it presents an anomaly}
213if there exists an elliptic curve over $\B F_p$ with modular invariant~$j$
214that possesses an $\B F_p$-rational point  of order~$p$
215(then necessarily $j\notin{}J_S$).
216
217\bigskip
218\prop{1}{\it Suppose that~$p$ is congruent to~$1$ modulo~$4$.
219Suppose that for all
220$j\in\B F_p$ that present an anomaly and all
221non-quadratic Dirichlet characters $\chi$ : $(\B Z/p\B Z)^*\raw \B C$,
222there exists $t_\chi\in \B T$ and
223$\delta\in\Delta_S$ such that $L(t_\chi J_0(p),\chi,1)\ne0$ and
224 $\iota_j(t_\chi\delta)\ne0$.
225
226Then for all subgroups~$C$ of order~$p$ of $E(\bar \B Q)$, there exists an
227elliptic curve $E_C$ over $\B Q(\sqrt p)$ equipped with a
228$\B Q(\sqrt p)$-rational subgroup $D_C$ or order~$p$, and
229the pairs $(E,C)$ and $(E_C,D_C)$ are $\bar \B Q$-isomorphic.}
230
231\dm We indicate how this is deduced from~\Merel.
232The hypothesis
233$\iota_j(t_\chi\delta)\ne0$ forces $t_\chi\notin p\B T$
234and, {\it a fortiori}, $t_\chi\ne0$; in addition,
235the non-vanishing hypothesis on the $L$-series
236forces the hypothesis $H_p(\chi)$ of {\it loc. cit}, introduction.
237
238According to Corollary~3 of Proposition~6 of {\it loc. cit},~$E$ has
239potentially good reduction at the prime ideal
240$\cP$ of $\B Z[\mu_p]$ that lies above~$p$
241once we know that hypothesis $H_p(\chi)$ is satisfied for all
242non-quadratic Dirichlet characters~$\chi$ of conductor~$p$
243(this is the case by hypothesis).
244
245Denote by~$j$ the modular invariant of the fiber at~$\cP$ of the
246N\'eron model of~$E$.
247According to the corollary of Proposition~15 of {\it loc. cit.},
248$j$~presents an anomaly.
249
250Let~$C$ be a subgroup of $E(\bar\B Q)$ of order~$p$.
251By assumption~$E$ is an elliptic curve over~$\B Q(\mu_p)$ whose points
252of order~$p$ are all $\B Q(\mu_p)$-rational, so
253the pair $(E,C)$ defines a $\B Q(\mu_p)$-rational point~$P$
254of the modular curve $X_0(p)$.
255
256Consider the morphism $\phi_{\chi}=\phi_{t_\chi}$ (see {\it loc. cit.}
257section 1.3). When $\iota_j(t_\chi\delta)\ne0$, this is a formal
258immersion at the point $P_{/\B F_p}$, according to {\it loc. cit.},
259Proposition~4.  The hypothesis that $L(t_\chi J_0(p),\chi,1)\ne0$
260implies that the $\chi$-isotypical component of $t_\chi J_0(p)(\B 261Q(\mu_p))$ is finite (this is Kato's theorem, see the discussion in
262{\it loc. cit.} section 1.5).  We can then apply Corollary~1 of
263Proposition~6 of {\it loc. cit}.  This proves that $P$ is $\B Q(\sqrt 264p)$-rational; this translates into the conclusion of Proposition~1.
265
266
267
268\bigskip\bigskip\noindent
269{\bf 2. A lemma about elliptic curves}
270\bigskip
271\prop{2}{\it Let~$p$ be a prime number that is congruent to~$1$
272modulo~$4$. Let~$E$ be an elliptic curve over $\bar\B Q$.
273There exists a cyclic subgroup~$C$ of order~$p$ of
274$E(\bar\B Q)[p]$, such that for all elliptic curves~$E'$ over
275${\B Q(\sqrt{p})}$
276equipped with a ${\B Q(\sqrt p)}$-rational subgroup~$C'$, the
277pairs $(E,C)$ and $(E', C')$ are not $\bar\B Q$-isomorphic.
278}
279
280\dm
281We procede by contradiction.
282Let $E_0$ be an elliptic curve over $\B Q(\sqrt p)$
283that is $\bar\B Q$ isomorphic to $E$ (it exists by hypothesis).
284We first show that the subgroup
285$\Gal(\bar\B Q/{\B Q(\sqrt p)})$ acts by scalars
286on the $\B F_p$-vector space $E_0(\bar\B Q)[p]$.
287
288Denote by $X(p)$ the algebraic curve over $\B Q$ that parametrizes classes
289(fine because $p>2$) of generalized elliptic curves equipped with an
290embedding
291$\pi$ : $(\B Z/p\B Z)^2\raw E[p]$.
292Consider the morphism (of algebraic varieties
293over $\B Q$) $\phi$ :
294$X(p)\raw X_0(p)^{\B P^1(\B F_p)}$ that to $(E,\pi)$ associates
295$\prod_{t\in\B P^1(\B F_p)}(E,\pi(t))$.
296Denote by $X_\Delta(p)$ the image of~$\phi$.
297The covering (of algebraic curves over $\B Q$)
298$\phi'$ :
299$X(p)\raw X_\Delta(p)$ is Galois with Galois group isomorphic to
300$\B F_p^*$ (the action being deduced from the scalar action of
301$\B F_p^*$ on $E[p]$).
302
303Let $\pi_0$ be an embedding $(\B Z/p\B Z)^2\raw E_0[p]$.
304Denote by~$P$ the $\bar\B Q$-rational point of $X(p)$ deduced from
305$(E_0,\pi_0)$.
306Its image by~$\phi$ is $\B Q(\sqrt p)$-rational by hypothesis.
307We have then a character  $\alpha$ : $\Gal(\bar\B Q/\B Q(\sqrt p))\raw\B 308F_p^*$ such that $\sigma(P)=\alpha(\sigma).P$ ($\sigma\in \Gal(\bar\B Q/\B 309Q(\sqrt p))$).  In other words, $\Gal(\bar\B Q/\B Q(\sqrt p))$ acts
310by scalars on $E_0(\bar\B Q)[p]$ via the character~$\alpha$.
311
312Because of the Weil pairing, $\alpha^2$ coincides
313with the cyclotomic character modulo~$p$, and it factors through
314$\Gal(\B Q(\mu_p)/\B Q(\sqrt p))$. But, when
315$p\equiv 1\pmod 4$, the group $\Gal(\B Q(\mu_p)/{\B Q(\sqrt p)})$ is of
316even order, and the characters modulo~$p$ form a group generated by the
317reduction modulo~$p$ of the cyclotomic character;
318it thus can not be a square.
319
320\bigskip\bigskip\noindent
321{\bf 3. Verification of the hypothesis of Proposition~1}
322\bigskip
323Let $p$ be a prime number.
324We now indicate how to computationally
325verify that the hypothesis of Proposition~1 are
326satisfied.  We denote by $S_2(\Gamma_0(p))$ the group formed by the cuspidal
327modular forms of weight~$2$ for the congruence subgroup
328$\Gamma_0(p)$ which have integer Fourier coefficients.
329By a newform we mean a newform in the
330$\bar\B Q$-vector space generated by $S_2(\Gamma_0(p))$.
331
332We first list the modular $j$-invariants that present an anomaly;
333one is only interested in these.
334Let~$\chi$ be a Dirichlet character $(\B Z/p\B Z)^*\raw \B C^*$ and
335suppose $j\in\B P^1(\B F_p)$ presents an anomaly.
336
337We study the following three $\B T$-modules: $\B T$, $\Delta_S$, and
338$S_2(\Gamma_0(p))$. After extension of scalars to~$\B Q$,
339these are $\B T\otimes\B Q$-modules that are free of rank~$1$, of which
340the irreducible sub-$\B T$ modules are the annihilators of the
341minimal prime ideals of  $\B T$.
342We establish a list of the minimal prime ideals of $\B T$.
343From an algorithmic point of view,
344these are generated by the
345minimal polynomials of small degree of Hecke operators of small index;
346we find these polynomials by utilizing the
347graph method of Mestre and Oesterl\'e \MO.
348
349For each ideal $\cP$ of $\B T$, we can determine if
350the following three conditions are satisfied:
351
352\noindent i)
353There exists $x\in\Delta_S$ such that
354 $\cP x=0$ and  $\iota_j(x)\ne 0$
355for all anomalous~$j$.
356
357\noindent
358ii) All newforms~$f$ such that $\cP f=0$ satisfy $L(f,\chi,1)\ne 0$.
359
360\noindent
361iii) The image of~$\cP$ in the $\B T$-module $\B T/p\B T$
362is a direct factor.
363
364We study the case when~$\cP$ is a minimal prime.
365The first condition is studied by the graph method of Mestre and Oesterl\'e.
366The second condition is verified using the theory of modular symbols
367(without recourse to the calculation of integrals).
368The third condition  is verified for $p<1000$ and $p\ne 389$,
369because one of us has verified that the discriminant of
370$\B T$ is prime to~$p$ and so the ring  $\B T/p\B T$ is semi-simple.
371In fact, we verified this for $p<10000$; this was accomplished by
372computing discrimininants of
373characteristic polynomials mod~$p$ of $T_p$, for $p\leq 7$,
374using the method of graphs.  If all four characteristic polynomials
375had discriminant equal to~$0$ mod~$p$, we resorted to modular symbols
376to compute several more characteristic polynomials until one is
377found with nonzero discriminant mod~$p$.
378
379For all prime numbers~$p$ different than
380$2,3,5,7, 13$ and $389$, we thus
381verified the existence of a minimal prime ideal
382that satisfies the three conditions given above.
383
384In the case when $p=389$,
385we find $\cP$ in the following way.
386There exists two minimal prime ideals $\cP_1$ and $\cP_2$
387that satisfy the first two conditions.
388Because the discriminant of~$\B T$ has $p$-adic valuation~$1$,
389the image of at least one of the ideals
390$\cP_1$, $\cP_2$ and $\cP_1\cap\cP_2$ is a direct factor of
391$\B T/p\B T$.  We choose~$\cP$ to be one of these that is
392appropriate.
393
394When our three conditions are satisfied for an ideal~$\cP$
395of~$\B T$, there exists
396$t_\chi\in\B T$ which is annihilated by $\cP$ and
397is the inverse image of a projector of
398$\B T/p\B T$ on the complement of $\cP+p\B T$.
399Putting $\delta=x$, one has $\iota_j(t_\chi \delta)=\iota_j(\delta)\ne0$
400(because $\iota_j$ takes its values in characteristic~$p$,
401$\delta$ is annihilated by~$\cP$ and $t_\chi\in 1+p\B T+\cP$).
402The assertion $L(t_\chi J_0(p),\chi,1)\ne0$ follows
403easily from the second condition,
404since $L(t_\chi J_0(p),\chi,s)$ is the product of
405$L(f,\chi,s)$, where~$f$ runs over the
406newforms that are not annihilated by~$t_\chi$.
407
408The pair $(t_\chi,\delta)$ thus satisfy the required conditions.
409
410\bigskip\bigskip
411
412
413
414\centerline{\pc Bibliography}
415\bigskip
416
417\item{\Mazur}{\pc B. Mazur},
418{\it Modular curves and the Eisenstein ideal},
419Pub. math. de l'IHES {\bf 47}, 33--186, {\oldstyle 1977}.
420
421\item{\Merel}{\pc L. Merel},
422{\it Sur la nature non cyclotomique des points d'ordre fini des courbes
423elliptiques},
424Pr\'epublication , {\oldstyle
4251999}.
426
427\item{\MO}{\pc J.-F. Mestre} and {\pc J. Oesterl\'e},
428{\it Courbes elliptiques de conducteur premier}, Manuscrit non publi\'e.
429\end