CoCalc Public Fileswww / papers / merel2 / newproof
Author: William A. Stein
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78\def\sh{\mathop{\rm sh}\nolimits}              %sinus hyperbolique
79\def\ch{\mathop{\rm ch}\nolimits}              %cosinus hyperbolique
80\def\tgh{\mathop{\rm th}\nolimits}             %tangente hyperbolique
81\def\tg{\mathop{\rm tg}\nolimits}              %tangente
82\def\Arcsin{\mathop{\rm Arc\,sin}\nolimits}    %Arc sinus
83\def\Arccos{\mathop{\rm Arc\,cos}\nolimits}    %Arc cosinus
84\def\Arctg{\mathop{\rm Arc\,tg}\nolimits}      %Arc tangente
85\def\Argsh{\mathop{\rm Arg\,sh}\nolimits}      %Arg sinus hyperbolique
86\def\Argch{\mathop{\rm Arg\,ch}\nolimits}      %Arg cosinus hyperbolique
87\def\Argth{\mathop{\rm Arg\,th}\nolimits}      %Arg tangente hyperbolique
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99\def\th#1{\noindent{\pc Theorem}\ #1. --- \ignorespaces}     %Th=C8or=CBme 1=
100=2E-
101\def\prop#1{\noindent{\pc Proposition}\ #1. --- \ignorespaces}
102%Proposition 1.-
103\def\Def#1{\noindent{\pc Definition}\ #1. --- \ignorespaces}
104%D=C8finition 1.-
105\def\cor#1{\noindent{\pc Corollary}\ #1. --- \ignorespaces}
106%Corollaire 1.-
107\def\conj#1{\noindent{\pc Conjecture}\ #1. --- \ignorespaces}
108%Conjecture 1.-
109\def\lem#1{\noindent{\it Lemma}\ #1. --- \ignorespaces}           %Lemme 1.-
110\def\rem#1{\noindent{\it Remark}\ #1: \ignorespaces}        %Remarque 1.-
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113\def\rems{\noindent{\it Remarks}: \ignorespaces}            %Remarques .-
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115\def\exrs{\noindent{\it Exercises}: \ignorespaces}            %Exercices .-
116
117\def\thp{\noindent{\pc Theorem}. --- \ignorespaces}          %Th=C8or=CBme .=
118-
119\def\propp{\noindent{\pc Proposition}. --- \ignorespaces}
120%Proposition .-
121\def\Defp{\noindent{\pc Definition}. --- \ignorespaces}
122%D=C8finition .-
123\def\corp{\noindent{\pc Corollary}. --- \ignorespaces}
124%Corollaire .-
125\def\conjp{\noindent{\pc Conjecture}. --- \ignorespaces}
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127\def\lemp{\noindent{\it Lemma}. --- \ignorespaces}                %Lemme .-
128\def\scholie{\noindent{\it Scholie}. --- \ignorespaces}           %Scholie .=
129-
130\def\remp{\noindent{\it Remark} : \ignorespaces}             %Remarque .-
131\def\exep{\noindent{\it Example} : \ignorespaces}              %Exemple .-
132
133\def\dm{\noindent{\it Proof}. --- \ignorespaces}
134%Exemple .-
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153
154%\def\Mazur{{$[1]$}}
155\def\Agashe{{$[1]$}}
156\def\Cremona{{$[2]$}}
157\def\Merel{{$[3]$}}
158\def\Mes{{$[4]$}}
159
160\centerline
161{\titchap The field generated by the points of small}
162\centerline
163{\titchap prime order on an elliptic curve}
164\medskip
165
166\bigskip\bigskip\bigskip
167\centerline{\pc Lo\"\i c Merel {\rm and} William A.~Stein}
168\bigskip\bigskip\bigskip
169
170
171\bigskip\bigskip\noindent
172{\bf Introduction}
173\bigskip
174
175Let~$p$ be a prime number.
176Let $\bar\B Q$ be an algebraic closure of $\B Q$, and
177denote by $\B Q(\mu_p)$ the cyclotomic subfield of $\bar\B Q$
178generated by the $p$th roots of unity.
179Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the
180points of order~$p$ of
181$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.
182\bigskip
183\th{}{\it One has $p=3D2,3,5, 13$ or $p>1000$.}
184\bigskip
185
186We note that the case $p=3D7$ was treated by Emmanuel Halberstadt.
187The part of the theorem that concerns the case
188$p\equiv 3\!\!\pmod{4}$ is given in~\Merel.
189We propose to give the details that permit our treating the
190more difficult case in which $p\equiv 1 \!\!\pmod{4}$.
191We treat this last case with the aid of Proposition~2 below, which is
192not present in {\it loc. cit.}
193
194We did not thoroughly study the case $p=3D13$.
195
196\bigskip\noindent
197{\bf 1. We recall the results of \Merel}
198\bigskip
199Let $S_2(\Gamma_0(p))$ denote the space of cusp forms of weight $2$ for
200the congruence subgroup $\Gamma_0(p)$. Denote by $\B T$ the
201subring of ${\rm End}\,S_2(\Gamma_0(p))$ generated by the Hecke operators.
202Let $f\in S_2(\Gamma_0(p))$ have $q$-expansion
203$\sum_{n=3D1}^\infty a_nq^n$. When $\chi$ is a Dirichlet character,
204denote by $L(f,\chi,s)$ the entire function which extends the
205Dirichlet series $\sum_{n=3D1}^\infty a_n\chi(n)/n^s$.
206
207Let $S$ be the set of isomorphism classes of supersingular elliptic
208curves in characteristic~$p$.  Denote by $\Delta_S$ the group formed
209by the divisors of degree~$0$ with support on~$S$.  It is equipped with
210a structure of $\B T$-module (induced, for example, from the action
211of the Hecke correspondences on the fiber at~$p$ of the regular minimal
212model of $X_0(p)$ over $\B Z$).
213
214Let $j\in\bar\B F_p-J_S$, where $J_S$ denotes the set of supersingular modul=
215ar
216invariants. We denote by  $\iota_j$ the homomorphism of
217groups $\Delta_S\raw \bar\B F_p$ that associates to $\sum_E n_E[E]$
218the quantity
219$\sum_E n_E/(j-j(E))$, where $j(E)$ denotes the modular invariant of~$E$.
220
221
222One says that an element  $j\in\B F_p$ is {\it anomalous}
223if there exists an elliptic curve over $\B F_p$ with modular invariant~$j$
224that possesses an $\B F_p$-rational point  of order~$p$
225(then necessarily $j\notin{}J_S$).
226
227\bigskip
228\prop{1}{\it Suppose that~$p$ is congruent to~$1$ modulo~$4$.
229Suppose that for all anomalous
230$j\in\B F_p$ and all
231non-quadratic Dirichlet characters $\chi$ : $(\B Z/p\B Z)^*\raw \B C$,
232there exists $t_\chi\in \B T$ and
233$\delta\in\Delta_S$ such that $L(f,\chi,1)\ne0$ for every newform
234$f\in t_\chi S_2(\Gamma_0(p))$ and
235=20 $\iota_j(t_\chi\delta)\ne0$.
236
237Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the
238points of order~$p$ of
239$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.
240Then for all subgroups~$C$ of order~$p$ of $E(\bar \B Q)$, there exists an
241elliptic curve $E_C$ over $\B Q(\sqrt p)$ equipped with a
242$\B Q(\sqrt p)$-rational subgroup $D_C$ of order~$p$, and
243the pairs $(E,C)$ and $(E_C,D_C)$ are $\bar \B Q$-isomorphic.}
244
245\dm We indicate how this is deduced from~\Merel.
246The hypothesis
247$\iota_j(t_\chi\delta)\ne0$ forces $t_\chi\notin p\B T$
248and, {\it a fortiori}, $t_\chi\ne0$; in addition,
249the non-vanishing hypothesis on the $L$-series
250forces the hypothesis $H_p(\chi)$ of {\it loc. cit}, introduction.
251
252According to Corollary~3 of Proposition~6 of {\it loc. cit},~$E$ has
253potentially good reduction at the prime ideal
254$\cP$ of $\B Z[\mu_p]$ that lies above~$p$
255once we know that hypothesis $H_p(\chi)$ is satisfied for all
256non-quadratic Dirichlet characters~$\chi$ of conductor~$p$
257(this is the case by hypothesis).
258
259Denote by~$j$ the modular invariant of the fiber at~$\cP$ of the
260N\'eron model of~$E$.
261According to the corollary of Proposition~15 of {\it loc. cit.},
262$j$ is anomalous.
263
264Let~$C$ be a subgroup of $E(\bar\B Q)$ of order~$p$.
265By assumption~$E$ is an elliptic curve over~$\B Q(\mu_p)$ whose points
266of order~$p$ are all $\B Q(\mu_p)$-rational, so
267the pair $(E,C)$ defines a $\B Q(\mu_p)$-rational point~$P$
268of the modular curve $X_0(p)$.
269
270Consider the morphism $\phi_{\chi}=3D\phi_{t_\chi}$ (see {\it loc. cit.}
271section 1.3). When $\iota_j(t_\chi\delta)\ne0$, this is a formal
272immersion at the point $P_{/\B F_p}$, according to {\it loc. cit.},
273Proposition~4.  The hypothesis that $L(f,\chi,1)\ne0$ for
274every newform
275$f\in t_\chi S_2(\Gamma_0(p))$, translates into $L(t_\chi J_0(p), 276\chi,1)\ne0$, which in turn implies that the $\chi$-isotypical
277component of
278$t_\chi J_0(p)(\B Q(\mu_p))$ is finite (this is Kato's theorem, see the
279discussion in {\it loc. cit.} section 1.5).  We can then apply
280Corollary~1 of Proposition~6 of {\it loc. cit}.  This proves that $P$ is
281$\B Q(\sqrt p)$-rational, which translates into the conclusion of
282Proposition~1.
283
284\bigskip
285
286\rem{} In this proposition we content ourself with a stronger hypothesis
287than the one generally used in {\it loc. cit}: the Hecke operator
288$t_\chi$ is required to belong to $\B T$ and not to $\B T\otimes\B 289Z[\chi]$.
290
291\bigskip\bigskip\noindent
292{\bf 2. Elliptic curves and quadratic fields}
293
294\bigskip
295\prop{2}{\it Let~$p$ be a prime number that is congruent to~$1$
296modulo~$4$. Let~$E$ be an elliptic curve over $\bar\B Q$.
297There exists a cyclic subgroup~$C$ of order~$p$ of
298$E(\bar\B Q)[p]$, such that for all elliptic curves~$E'$ over
299${\B Q(\sqrt{p})}$
300equipped with a ${\B Q(\sqrt p)}$-rational subgroup~$C'$, the
301pairs $(E,C)$ and $(E', C')$ are not $\bar\B Q$-isomorphic.
302}
303
304\dm
305We procede by contradiction, {\it i.e.} we assume that for
306all cyclic subgroup $C$ of order $p$ of $E(\bar\B Q)$,
307the pair $(E,C)$ can be defined over ${\B Q(\sqrt{p})}$.
308We choose such a pair $(E_0,C_0)$ over ${\B Q(\sqrt{p})}$.
309
310We show first that such a situation prevents that $j(E)=3D0$ or
311$j(E)=3D1728$. Indeed in those cases
312$E$ has complex multiplication by an order $R_K$ of $K=3D\B Q[\sqrt{-1}]$ or
313$\B Q[\sqrt{-3}]$.
314Consider the map $\rho$ :
315$\Gal(\bar \B Q/\B Q(\sqrt p))\longrightarrow{\rm Aut}\,E_0(\bar\B Q)[p]$.
316If its image had an element of order $p$, so would have
317$\rho(\Gal(\bar \B Q/K(\sqrt p)))$. Let $L_p$ be the class field of ray $p$
318of $K$. It contains $\B Q(\sqrt p)$ since $p\equiv 1\pmod 4$.
319By the theory
320of complex multiplication,
321$\rho(\Gal(\bar \B Q/L_p))$ is trivial. By class field theory, $\Gal(L_p/K)$
322has no element of order $p$, since $K$ has class number $1$. Therefore
323$\rho(\Gal(\bar \B Q/\B Q(\sqrt p)))$ has no element of order $p$.
324Since it is contained in the Borel subgroup of ${\rm Aut}\,E_0(\bar\B Q)[p]$
325which stabilizes $C_0$, it is an abelian group. By the theory of complex
326multiplication, it is the semi-direct product of $\Gal(L_p/K(\sqrt p))$ and
327$\Gal(K(\sqrt p)/\B Q(\sqrt p))\simeq\Gal(\B C/\B R)$. Such a group
328is not abelian since
329$\Gal(L_p/K(\sqrt p))$ is not a $2$-group, hence the contradiction.
330
331We can now assume that all twists of $E$ are quadratic.
332We now show that the group
333$\Gal(\bar\B Q/{\B Q(\sqrt p)})$ acts by scalars
334on the $\B F_p$-vector space $E_0(\bar\B Q)[p]$. For this it
335suffices to show that all subgroups of order $p$ of
336$E_0(\bar\B Q)[p]$ are stable by $\Gal(\bar\B Q/{\B Q(\sqrt p)})$.
337
338Let $C_1$ be a cyclic subgroup of order $p$ of $E_0(\bar\B Q)[p]$.
339By assumption, there exists a quadratic twist $E_1$ of $E_0$ and
340a cyclic subgroup $C_1'$ of $E_1(\bar\B Q)[p]$ of order $p$, such that
341the image of $C_1$ by the isomorphism $E_0\simeq E_1$ is $C'_1$.
342Since the Galois action on $E_1(\bar\B Q)[p]$ is the one obtained
343by twisting by a character the action on $E_0(\bar\B Q)[p]$.
344The group $\Gal(\bar\B Q/{\B Q(\sqrt p)})$ leaves $C_1$ stable.
345That implies that $\Gal(\bar\B Q/\B Q(\sqrt p))$ acts
346by scalars on $E_0(\bar\B Q)[p]$. Denote by ~$\alpha$ the corresponding
347character of $\Gal(\bar\B Q/\B Q(\sqrt p))$.
348
349Because of the Weil pairing, $\alpha^2$ coincides
350with the cyclotomic character modulo~$p$, and it factors through
351$\Gal(\B Q(\mu_p)/\B Q(\sqrt p))$. But, when
352$p\equiv 1\!\!\pmod 4$, the group $\Gal(\B Q(\mu_p)/{\B Q(\sqrt p)})$ is of
353even order, and the characters modulo~$p$ form a group generated by the
354reduction modulo~$p$ of the cyclotomic character, which, therefore,
355can not be a square.
356
357\bigskip\bigskip\noindent
358{\bf 3. Verification of the hypothesis of Proposition~1}
359\bigskip
360Let $p$ be a prime number.  In this section we explain how to use a
361computer to verify that the hypothesis of Proposition~1 are satisfied.
362We have carried out this verification for $p=3D11$ and $13 < p < 1000$.
363
364We first list the anomalous $j$-invariants $j\in\B F_p$.  Since~$p$ is
365fairly small in the range of our computations, we created this list by
366simply enumerating all of the elliptic curves over $\B F_p$ and
367counting the number of points on each curve.  For example, when $p=3D31$
368the anomalous $j$-invariants are $j=3D10,14$.
369
370Let~$\chi: (\B Z/p\B Z)^*\raw \B C^*$ denote a non-quadratic
371Dirichlet character  and
372suppose $j\in\B P^1(\B F_p)$ is anomalous.
373
374We study the following three $\B T$-modules: $\B T$, $\Delta_S$, and
375$S_2(\Gamma_0(p);\B Z)$. After extension of scalars to~$\B Q$, these
376are $\B T\otimes\B Q$-modules that are free of rank~$1$, of which the
377irreducible sub-$\B T\otimes\B Q$ modules are the annihilators of the
378minimal prime ideals of $\B T$.  We compute a list of the minimal
379prime ideals of $\B T$ by computing appropriate kernels and
380characteristic polynomials of Hecke operators of small index on
381$\Delta_S$, which we find using the graph method of Mestre and
382Oesterl\'e \Mes{}.
383
384Having computed the minimal prime ideals of $\B T$, we verify that
385some nontrivial ideal $\cI$ of $\B T$ (usually a minimal prime
386ideal) simultaneously satisfies the following three conditions:
387\vskip 2ex
388
3891)
390There exists $x\in\Delta_S$ such that
391=20 $\cI x=3D0$ and  $\iota_j(x)\ne 0$.\vskip 1ex
392
3932) Each of the newforms~$f\in S_2(\Gamma_0(p))$ with
394$\cI f=3D0$ satisfies $L(f,\chi,1)\ne 0$.
395\vskip 1ex
396
3973) The image of~$\cI$ in the $\B T$-module $\B T/p\B T$
398is a direct factor.\vskip 2ex
399
400Let $\cI$ be an ideal of $\B T$. Here is how we verify these conditions
401for $\cI$.
402
403\bigskip
404{\it \noindent Condition 1.}
405
406We verified that $\cI$ satisfies the first condition by
407finding a $\B T$-eigenvector~$v$ of $\Delta_S\otimes \bar\B Z$ that is
408annihilated by $\cI$ and satisfies $\iota_j(v)\neq 0$.  Because $\iota_j$
409is a homomorphism, this implies the existence of an~$x$ as in condition 1.
410
411\bigskip
412{\it \noindent Condition 2.}
413
414We verified the second condition using modular symbols.
415Our method is purely algebraic, avoiding the need to perform
416any approximate computation of integrals.
417Using the algorithm described in \Cremona, we compute the action of
418the Hecke algebra $\B T$ on the space
419$\Hom(H_1(X_0(p);\B Q(\chi)),\B Q(\chi))$.   By intersecting the kernels
420of appropriate elements of $\B T$, we find a basis
421$\varphi_1,\ldots,\varphi_n$ for the subspace of
422$\Hom(H_1(X_0(p);\B Q(\chi)),\B Q(\chi))$ that is annihilated by~$\cI$.
423Let~$\Phi_{\cI}=3D(\varphi_1,\ldots,\varphi_n)$ denote the linear map
424=20 $H_1(X_0(p);\B Q(\chi))\raw \B Q(\chi)^n$
425defined by the $\varphi_i$.
426
427Let $\B T_{\B Q(\chi)} =3D \B T \otimes \B Q(\chi)$, where $\B Q(\chi)$
428is the number field generated the image of~$\chi$.
429The {\it $\chi$-twisted winding element} (denoted by $\theta_\chi$ in
430\Merel)
431=20 $$\B e_\chi =3D \sum_{a\in (\B Z/p\B Z)^*} \bar\chi(a) 432=20 \Big\{\infty,{a \over p}\Big\},$$
433generates the {\it $\chi$-twisted winding submodule}
434$\B T_{\B Q(\chi)}\cdot \B e_\chi$.  To compute this submodule,
435we use that $\B T$ is generated, even as a $\B Z$-module,
436by $T_1,T_2,\ldots, T_b$, for any $b\geq (p+1)/6$
437(see \Agashe).
438
439\bigskip
440\lem  3
441{\it Let $\cI$ be a minimal prime ideal of~$\B T$, and
442let $\chi:(\B Z/p\B Z)^*\raw \B C^*$
443be a nontrivial Dirichlet character.
444Then the dimension of $\Phi_{\cI}(\B T_{\B Q(\chi)} \cdot \B e_\chi)$ is
445equal to the cardinality of the set of newforms~$f$ such that
446$\cI f=3D0$ and $L(f,\chi,1) \neq 0$.
447}
448
449\dm
450Since $\B C$ is faithfully flat over $\B Q(\chi)$,
451we have
452$$\dim_{\B Q(\chi)} \Phi_{\cI}(\B T_{\B Q(\chi)}\cdot \B e_\chi) 453=20 =3D \dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi).$$
454This dimension is invariant upon changing the basis
455$\varphi_1,\ldots, \varphi_n$ used to define $\Phi_{\cI}$.
456In particular, over $\B C$ there is a basis
457$\varphi_1',\ldots, \varphi_n'$ so that the resulting
458map $\Phi_{\cI}'$ satisfies
459$$\Phi_{\cI}'(x) =3D 460\Bigl(\rp(\int_x f^{(1)}), \ip(\int_x f^{(1)}), 461\ldots, 462=20 \rp(\int_x f^{(d)}),\ip(\int_x f^{(d)})\Bigr),$$
463where $f^{(1)}, \ldots, f^{(d)}$ are the Galois conjugates
464of a newform~$f^{(1)}=3D\sum a_n^{(1)} q^n$ such that $\cI f^{(1)}=3D0$.
465=46urthermore, $\Phi_{\cI}'$ is a $\B T_{\B C}$-module homomorphism
466if we declare that $\B T_{\B C}$ as acts on $\B R^{2d} =3D \B C^d$ via
467$$T_n(x_1,y_1, \ldots, x_d, y_d) =3D 468=20 T_n(z_1,\ldots,z_d) =3D (a_n^{(1)} z_1,\ldots, a_n^{(d)}z_d),$$
469where $z_j =3D x_j + i y_j$ and
470the $a_n^{(j)}$ are Fourier coefficients of the $f^{(j)}$.
471
472As explained in Section 2.2 of~\Merel,
473$\int_{\B e_\chi} f =3D *\cdot L(f,\chi,1)$, where~$*$
474is some nonzero real or pure-imaginary complex number,
475according to whether $\chi(-1)$ equals~$1$ or~$-1$,
476respectively.
477Combining this observation with the equality
478=20 $$\dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi) 479=20 =3D \dim_{\B C} (\B T_{\B C}\cdot \Phi_{\cI}(\B e_\chi)),$$
480and that the image of $\B T_{\B C}$ in $\End(\B C^d)$ is
481equal to the diagonal matrices, gives the equality asserted
482in the lemma.
483
484\bigskip
485\rem {} The dimension of $\Phi_{\cI}(\B T_{\B Q(\chi)}\cdot \B e_{\chi})$
486is unchanged if~$\chi$ is
487replaced by a Galois-conjugate character.  Also, the
488conductor of~$\chi$ can be arbitrary.
489
490\bigskip
491
492In practice, computations over the cyclotomic field $\B Q(\chi)$ are
493extremely expensive.  Fortunately, for our application it suffices to
494give a lower bound on the dimension appearing in the lemma.  Such a
495bound can be efficiently obtained by instead computing the reductions
496of~$\Phi$,~$\chi$, and the $\chi$-twisted winding submodule modulo a
497suitable maximal ideal of the ring of integers of $\B Q(\chi)$ that
498splits completely; this amounts to performing the above linear algebra
499over a relatively small prime finite field $\B F_\ell$ such
500that~$\ell$ is congruent to~$1$ modulo $p-1$.
501
502\bigskip
503
504\rem {} For every newform~$f$ in $S_2(\Gamma_0(p))$, with $p\leq 1000$,
505and every mod~$p$ Dirichlet character~$\chi$, we found that
506$L(f,\chi,1)\neq 0$ if and only if
507$L(f^{\sigma},\chi,1)\neq 0$ for all conjugates $f^{\sigma}$
508of~$f$.
509More generally, for any~$f$ and~$\chi$, this equivalence holds if
510$\B Q(\chi)$ is linearly disjoint from the
511field $K_f=3D(\B T/\cI)\otimes\B Q$.
512The first few primes
513for which there is a form~$f$ and a mod-$p$ character~$\chi$
514such that the linear disjointness hypothesis fails are
515$p=3D31, 113, 127$, and $191$.
516
517In general, the equivalence need not hold.  For example,
518let~$f$ be one of the two Galois-conjugate weight-$2$ newforms on
519$X_1(13)$ and  $\chi:(\B Z/7\B Z)^*\raw \B C^*$ be one of the
520two conjugate characters of order~$3$.   Then for one choice
521of~$\chi$, we have $L(f,\chi,1) =3D 0$ but
522$L(f^{\sigma},\chi,1)\neq 0$.
523
524\bigskip
525
526{\it \noindent Condition 3.}
527
528The third condition is satisfied for all $p<10000$, except $p =3D 389$,
529because one of us has verified that the discriminant of $\B T$ is
530prime to~$p$, so the ring $\B T/p\B T$ is semi-simple.  This was
531accomplished by computing discrimininants of characteristic
532polynomials mod~$p$ of the Hecke operators $T_2$, $T_3$, $T_5$, $T_7$.
533In the few cases when all four of these characteristic
534polynomials had discriminant equal to~$0$ mod~$p$, we resorted to
535modular symbols to compute several more characteristic polynomials
536until we found one with nonzero discriminant mod~$p$.
537
538In the case when $p=3D389$, we find $\cI$ in the following way.  There
539exists at least two minimal prime ideals $\cP_1$ and $\cP_2$ that satisfy
540the conditions 1 and 2. Therefore the ideal $\cP_1\cap \cP_2$ satisfy
541also the conditions 1 and 2. Because the discriminant of~$\B T$ has
542$p$-adic valuation~$1$, the image of at least one of the ideals $\cP_1$,
543$\cP_2$ and $\cP_1\cap\cP_2$ is a direct factor of $\B T/p\B T$.  We
544choose~$\cI$ to be one of these that is appropriate.
545
546
547\bigskip
548{\it \noindent Conclusion.}
549
550=46or all prime numbers~$p<1000$ different than $2,3,5,7, 13$, we
551verified the existence of an ideal that satisfies the three conditions
552given above.  For each~$p$, we consider each Galois conjugacy class of
553non-quadratic characters~$\chi$.  We find a single newform~$f$ such
554that $L(f,\chi,1)\ne 0$ for all conjugates of~$f$ and of $\chi$.  Then
555we let $\cI$ be the annihilator of~$f$, and try to verify condition~1
556for {\it all} of the anamolous $j$-invariants in $\B F_p$.
557
558When the three conditions are satisfied for an ideal~$\cI$ of~$\B T$,
559there exists $t_\chi\in\B T$ which is annihilated by $\cI$ and is the
560inverse image of a projector of $\B T/p\B T$ on the supplement of
561$\cI+p\B T$.  Putting $\delta=3Dx$, one has
562$\iota_j(t_\chi \delta)=3D\iota_j(\delta)\ne0$
563(because $\iota_j$ takes its values in
564characteristic~$p$, $\delta$ is annihilated by~$\cI$ and
565$t_\chi\in 1+p\B T+\cP$).
566Every newform $f\in t_\chi S_2(\Gamma_0(p))$ satisfies
567$\cI f=3D0$, and therefore, by our second condition, $L(f,\chi,1)\ne0$.
568The pair $(t_\chi,\delta)$ thus satisfies the conditions required by
569Proposition~1.
570
571
572
573\bigskip\bigskip
574
575
576
577\centerline{\pc Bibliography}
578\bigskip
579
580\item{\Agashe}{\pc A. Agashe},
581{\it On invisible elements of the Tate-Shafarevich group},
582C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), no. 5, 369--374.
583
584\item{\Cremona}{\pc J. Cremona},
585{\it Algorithms for modular elliptic curves},
586second ed., Cambridge University Press, Cambridge,
587{\oldstyle 1997}.
588\vskip 1ex
589
590\item{\Merel}{\pc L. Merel},
591{\it Sur la nature non cyclotomique des points d'ordre fini des courbes
592elliptiques},
593To appear in Duke Math. Journal.
594\vskip 1ex
595
596\item{\Mes}{\pc J.-F. Mestre}, {\it La m\'ethode des graphes.
597Exemples et  applications}, Proceedings of the international
598conference on class numbers and fundamental units of algebraic number
599fields (Katata), 217--242, {\oldstyle 1986}.
600\end
601--============_-1236242168==_============
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