CoCalc Public Fileswww / merel-stein_v6.tex
Author: William A. Stein
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1% merel-stein_v6.tex
2% 12 February 2001
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12\def\�#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi}   % trema
13\def\B#1{{\bf #1}}                                      % bold
14\def\lc{{\it loc.\thinspace{}cit.}}                     % loc. cit.
15\def\mod#1{\ \hbox{{\rm mod}$#1$}}                      % modulo
16\def\eps{\varepsilon}
17
18\font\titchap=cmr17 at 20pt  % for the titles of chapters.
19\font\pc=cmcsc10             % for the titles of sections, props, etc.
20
21\def\th#1{\noindent{\pc Theorem}\ #1. --- \ignorespaces}      %Theorem 1.
22\def\prop#1{\noindent{\pc Proposition}\ #1. --- \ignorespaces}%Proposition 1.
23\def\Def#1{\noindent{\pc Definition}\ #1. --- \ignorespaces}  %Definition 1.
24\def\cor#1{\noindent{\pc Corollary}\ #1. --- \ignorespaces}   %Corollary 1.
25\def\conj#1{\noindent{\pc Conjecture}\ #1. --- \ignorespaces} %Conjecture 1.
26\def\lem#1{\noindent{\it Lemma}\ #1. --- \ignorespaces}       %Lemma 1.
27\def\rem#1{\noindent{\it Remark}\ #1: \ignorespaces}          %Remark 1.
28\def\exe#1{\noindent{\it Example}\ #1: \ignorespaces}         %Example 1.
29\def\exr#1{\noindent{\it Exercise}\ #1: \ignorespaces}        %Exercise 1.
30\def\rems{\noindent{\it Remarks}: \ignorespaces}              %Remarks 1.
31\def\exes{\noindent{\it Examples}: \ignorespaces}             %Examples 1.
32\def\exrs{\noindent{\it Exercises}: \ignorespaces}            %Exercises 1.
33\def\thp{\noindent{\pc Theorem}. --- \ignorespaces}           %Theorem 1.
34\def\propp{\noindent{\pc Proposition}. --- \ignorespaces}     %Proposition 1.
35\def\Defp{\noindent{\pc Definition}. --- \ignorespaces}       %Definition 1.
36\def\corp{\noindent{\pc Corollary}. --- \ignorespaces}        %Corollairy 1.
37\def\conjp{\noindent{\pc Conjecture}. --- \ignorespaces}      %Conjecture 1.
38\def\lemp{\noindent{\it Lemma}. --- \ignorespaces}            %Lemma 1.
39\def\remp{\noindent{\it Remark}: \ignorespaces}               %Remark 1.
40\def\exep{\noindent{\it Example}: \ignorespaces}              %Example 1.
41\def\dm{\noindent{\it Proof}. --- \ignorespaces}
42\def\raw{\longrightarrow}
43\def\Hom{{\rm Hom}}
44\def\Gal{{\rm Gal}}
45\def\cP{{\cal P}}
46\def\cO{{\cal O}}
47\def\cI{{\cal I}}
48\def\rp{{\rm Re}}
49\def\ip{{\rm Im}}
50\def\End{{\rm End}}
51
52\def\Agashe{{$[1]$}}
53\def\Cremona{{$[2]$}}
54\def\Merel{{$[3]$}}
55\def\Mes{{$[4]$}}
56
57\def\change#1{[[{\bf Change:} #1]]}
58
59\centerline
60{\titchap The field generated by the points of small}
61\centerline
62{\titchap prime order on an elliptic curve}
63\medskip
64
65\bigskip\bigskip\bigskip
66\centerline{\pc Lo\"\i c Merel {\rm and} William A.~Stein}
67\bigskip\bigskip\bigskip
68
69
70\bigskip\bigskip\noindent
71{\bf Introduction}
72\bigskip
73
74Let~$p$ be a prime number.
75Let $\bar\B Q$ be an algebraic closure of $\B Q$, and
76denote by $\B Q(\mu_p)$ the cyclotomic subfield of $\bar\B Q$
77generated by the $p$th roots of unity.
78Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the
79points of order~$p$ of
80$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.
81\bigskip
82\th{}{\it One has $p=2,3,5,13$ or $p>1000$.}
83\bigskip
84
85The case $p=7$ was treated by Emmanuel Halberstadt.  The
86part of the theorem that concerns the case $p\equiv 3\!\!\pmod{4}$ is
87given in~\Merel.  In this paper, we give the details that permit our
88treating the more difficult case in which $p\equiv 1 \!\!\pmod{4}$.
89We treat this last case with the aid of Proposition~2 below, which is
90not present in \lc.
91The case $p=13$ is currently under investigation by Marusia Rebolledo,
92as part of her Ph.D.{} thesis.
93
94\bigskip\noindent
95{\bf 1. We recall the results of \Merel}
96\bigskip
97\change{Change the title to something like
98Counterexamples define points on $X_0(p)(\B Q(\sqrt{p}))$''.}
99
100Let $S_2(\Gamma_0(p))$ denote the space of cusp forms of weight~$2$ for
101the congruence subgroup $\Gamma_0(p)$. Denote by $\B T$ the
102subring of ${\rm End}\,S_2(\Gamma_0(p))$ generated by the
103Hecke operators $T_n$ for all integers~$n$.
104Let $f\in S_2(\Gamma_0(p))$ have $q$-expansion
105$\sum_{n=1}^\infty a_nq^n$. When $\chi$ is a Dirichlet character,
106denote by $L(f,\chi,s)$ the entire function which extends the
107Dirichlet series $\sum_{n=1}^\infty a_n\chi(n)/n^s$.
108
109Let $S$ be the set of isomorphism classes of supersingular elliptic
110curves in characteristic~$p$.  Denote by $\Delta_S$ the group formed
111by the divisors of degree~$0$ with support on~$S$.  It is equipped with
112a structure of $\B T$-module (induced, for example, from the action
113of the Hecke correspondences on the fiber at~$p$ of the regular minimal
114model of $X_0(p)$ over $\B Z$).
115
116Let $j\in\bar\B F_p-J_S$, where $J_S$ denotes the set of supersingular modular
117invariants. We denote by  $\iota_j$ the homomorphism of
118groups $\Delta_S\raw \bar\B F_p$ that associates to $\sum_E n_E[E]$
119the quantity
120$\sum_E n_E/(j-j(E))$, where $j(E)$ denotes the modular invariant of~$E$.
121
122
123One says that an element  $j\in\B F_p$ is {\it anomalous}
124if there exists an elliptic curve over $\B F_p$ with modular invariant~$j$
125that possesses an $\B F_p$-rational point  of order~$p$
126(then necessarily $j\notin{}J_S$).
127
128\change{
129Let~$p$ be a prime that is congruent to~$1$ modulo~$4$.
130In the following proposition we prove, under a hypothesis on~$p$, that
131if~$E$ is an elliptic curve over $\B Q(\mu_p)$ all of whose torsion is
132$\B Q(\mu_p)$-rational, then for each subgroup $C\subset{}E(\B Q)$
133of order~$p$,
134the point $(E,C)$ on $X_0(p)$ is defined over $\B Q(\sqrt{p})$.  As we
135will see in Proposition~2, this $\B Q(\sqrt{p})$-rationality
136conclusion is contrary to fact, from which we conclude that such
137elliptic curves~$E$ do not exist when the hypothesis on~$p$
138is satisfied.  In Section~3 we verify this hypothesis
139for $p=11$ and $13 < p < 1000$.
140}
141
142\bigskip
143\prop{1}{\it Suppose that~$p$ is congruent to~$1$ modulo~$4$.
144Suppose that for all anomalous
145$j\in\B F_p$ and all
146non-quadratic Dirichlet characters $\chi \colon \B Z/p\B Z\raw \B C$,
147there exists $t_\chi\in \B T$ and
148$\delta\in\Delta_S$ such that $L(f,\chi,1)\ne0$ for every newform
149$f\in t_\chi S_2(\Gamma_0(p))$ and
150 $\iota_j(t_\chi\delta)\ne0$.
151
152Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the
153points of order~$p$ of
154$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.
155Then for all subgroups~$C$ of order~$p$ of $E(\bar \B Q)$, there exists an
156elliptic curve $E_C$ over $\B Q(\sqrt p)$ equipped with a
157$\B Q(\sqrt p)$-rational subgroup $D_C$ of order~$p$, and
158the pairs $(E,C)$ and $(E_C,D_C)$ are $\bar \B Q$-isomorphic.}
159
160\dm We prove the proposition using the results of~\Merel.
161The hypothesis
162$\iota_j(t_\chi\delta)\ne0$ forces $t_\chi\notin p\B T$
163and, {\it a fortiori}, $t_\chi\ne0$; in addition,
164the non-vanishing hypothesis on the $L$-series
165forces the hypothesis $H_p(\chi)$ of \lc, introduction.
166
167\change{
168{\bf (the following paragraph)}
169By assumption, hypothesis $H_p(\chi)$ is satisfied for all
170non-quadratic Dirichlet characters~$\chi$ of conductor~$p$.
171Thus Corollary~3 of Proposition~6 of \lc{} implies that~$E$ has
172potentially good reduction at the prime ideal
173$\cP$ of $\B Z[\mu_p]$ that lies above~$p$.}
174
175According to Corollary~3 of Proposition~6 of \lc,~$E$ has
176potentially good reduction at the prime ideal
177$\cP$ of $\B Z[\mu_p]$ that lies above~$p$
178once we know that
179hypothesis $H_p(\chi)$ is satisfied for all
180non-quadratic Dirichlet characters~$\chi$ of conductor~$p$
181(this is the case by hypothesis).
182
183Denote by~$j$ the modular invariant of the fiber at~$\cP$ of the
184N\'eron model of~$E$.
185According to the corollary of Proposition~15 of \lc,
186$j$ is anomalous.
187
188Let~$C$ be a subgroup of $E(\bar\B Q)$ of order~$p$.
189By assumption~$E$ is an elliptic curve over~$\B Q(\mu_p)$ whose points
190of order~$p$ are all $\B Q(\mu_p)$-rational, so
191the pair $(E,C)$ defines a $\B Q(\mu_p)$-rational point~$P$
192of the modular curve $X_0(p)$.
193
194Consider the morphism $\phi_{\chi}=\phi_{t_\chi}:X_0(p)\rightarrow J_0(p)$
195obtained by composing the standard embedding of $X_0(p)$ into $J_0(p)$
196with $t_{\chi}$.  As in section 1.3 of \lc, $\phi_{\chi}$
197extends to a map from the minimal regular model of $X_0(p)$ to the
198N\'eron model of $J_0(p)$.
199When $\iota_j(t_\chi\delta)\ne0$, this map is a formal
200immersion at the point $P_{/\B F_p}$, according to \lc,
201Proposition~4.  The hypothesis that $L(f,\chi,1)\ne0$ for
202every newform
203$f\in t_\chi S_2(\Gamma_0(p))$, translates into $L(t_\chi J_0(p), 204\chi,1)\ne0$, which in turn implies that the $\chi$-isotypical
205component of
206$t_\chi J_0(p)(\B Q(\mu_p))$ is finite (this is Kato's theorem, see the
207discussion in section 1.5 of \lc).
208\change{Mazur asked us to verify that section 1.5 of Merel's other
209paper properly refers to Scholl's paper.}
210We can then apply Corollary~1 of Proposition~6 of \lc.  This proves
211that~$P$ is
212$\B Q(\sqrt p)$-rational, which translates into the conclusion of
213Proposition~1.
214
215\bigskip
216
217\change{Delete this remark.}
218\rem{1} In this proposition we content ourself with a stronger hypothesis
219than the one generally used in \lc: the Hecke operator
220$t_\chi$ is required to belong to $\B T$ and not to $\B T\otimes\B 221Z[\chi]$.
222
223\bigskip\bigskip\noindent
224{\bf 2. Elliptic curves and quadratic fields}
225
226\bigskip
227\prop{2}{\it Let~$p$ be a prime number that is congruent to~$1$
228modulo~$4$. Let~$E$ be an elliptic curve over $\bar\B Q$.
229\change{There exists a subgroup~$C\subset{}E(\B Q)$ of order~$p$
230such that $(E,C)$ can not be defined over $\B Q(\sqrt{p})$.}
231There exists a cyclic subgroup~$C$ of order~$p$ of
232$E(\bar\B Q)[p]$, such that for all elliptic curves~$E'$ over
233${\B Q(\sqrt{p})}$
234equipped with a ${\B Q(\sqrt p)}$-rational subgroup~$C'$, the
235pairs $(E,C)$ and $(E', C')$ are not $\bar\B Q$-isomorphic.
236}
237
238\dm
239We procede by contradiction, i.e., we assume that for
240all cyclic subgroups~$C$ of order~$p$ of $E(\bar\B Q)$,
241the pair $(E,C)$ can be defined over ${\B Q(\sqrt{p})}$.
242We choose such a pair $(E_0,C_0)$ over ${\B Q(\sqrt{p})}$.
243
244Assume first that all twists of $E$ are quadratic, i.e.,
245that $j(E)$ is neither~$0$ nor $1728$.
246We show that the group
247$\Gal(\bar\B Q/{\B Q(\sqrt p)})$ acts by scalars
248on the $\B F_p$-vector space $E_0(\bar\B Q)[p]$. For this it
249suffices to show that all subgroups of order~$p$ of
250$E_0(\bar\B Q)[p]$ are stable by $\Gal(\bar\B Q/{\B Q(\sqrt p)})$.
251
252\change{I reworded the following paragraph.}
253Suppose $C_1$ is a cyclic subgroup of order~$p$ of $E_0(\bar\B Q)[p]$.
254By assumption, there exists a quadratic twist $E_1$ of $E_0$ and
255a cyclic subgroup $C_1'$ of $E_1(\bar\B Q)[p]$
256that is defined over $\B Q(\sqrt{p})$, such that
257the image of $C_1$ by the isomorphism $E_0\simeq E_1$ is $C'_1$.
258Since $\Gal(\bar\B Q/{\B Q(\sqrt p)})$ leaves $C_1'$ stable and
259the action of $\Gal(\bar\B Q/\B Q(\sqrt p))$
260on $E_0(\bar\B Q)[p]$ is
261a quadratic twist of the action on $E_1(\bar\B Q)[p]$,
262we see that $\Gal(\bar\B Q/\B Q(\sqrt p))$ leaves $C_1$ stable.
263Thus $\Gal(\bar\B Q/\B Q(\sqrt p))$ fixes all lines in
264$E_0(\bar\B Q)[p]$, and hence
265acts by scalars.  Denote by~$\alpha$ the corresponding character
266of $\Gal(\bar\B Q/\B Q(\sqrt p))$.
267
268Because of the Weil pairing, $\alpha^2$ coincides
269with the cyclotomic character modulo~$p$, and it factors through
270$\Gal(\B Q(\mu_p)/\B Q(\sqrt p))$. But, when
271$p\equiv 1\!\!\pmod 4$, the group $\Gal(\B Q(\mu_p)/{\B Q(\sqrt p)})$ is of
272even order, and the characters modulo~$p$ form a group generated by the
273reduction modulo~$p$ of the cyclotomic character, which, therefore,
274can not be a square.
275
276\change{I don't understand this argument.}
277Next suppose that $j(E)=0$ or $j(E)=1728$. Indeed, in these
278two cases~$E$ has
279complex multiplication by an order $R_K$ of $K=\B Q[\sqrt{-1}]$ or
280$\B Q[\sqrt{-3}]$.
281Consider the map
282$\rho : \Gal(\bar \B Q/\B Q(\sqrt p))\longrightarrow{\rm Aut}\,E_0(\bar\B Q)[p]$.
283%Suppose $\rho(\Gal(\bar \B Q/K(\sqrt p)))$ contains an element of order~$p$.
284Let $L_p$ be the ray class field of conductor~$p$ of~$K$.
285It contains $\B Q(\sqrt p)$ since $p\equiv 1\!\!\!\pmod 4$.
286By the theory
287of complex multiplication,
288$\rho(\Gal(\bar \B Q/L_p))$ is trivial. By class field theory, $\Gal(L_p/K)$
289has no element of order~$p$, since~$K$ has class number~$1$. Therefore
290$\rho(\Gal(\bar \B Q/\B Q(\sqrt p)))$ has no elements of order~$p$.
291Since it is contained in the Borel subgroup of ${\rm Aut}\,E_0(\bar\B Q)[p]$
292which stabilizes $C_0$, it is an abelian group. By the theory of complex
293multiplication, it is the semi-direct product of $\Gal(L_p/K(\sqrt p))$ and
294$\Gal(K(\sqrt p)/\B Q(\sqrt p))\simeq\Gal(\B C/\B R)$. Such a group
295is not abelian since
296$\Gal(L_p/K(\sqrt p))$ is not a $2$-group, hence the contradiction.
297
298
299\bigskip\bigskip\noindent
300{\bf 3. Verification of the hypothesis of Proposition~1}
301\bigskip
302Let $p$ be a prime number.  In this section we explain how we used
303a computer to verify that the hypothesis of Proposition~1 are satisfied
304for $p=11$ and $13 < p < 1000$.
305
306We first list the anomalous $j$-invariants $j\in\B F_p$.  Since~$p$ is
307fairly small in the range of our computations, we created this list by
308simply enumerating all of the elliptic curves over $\B F_p$ and
309counting the number of points on each curve.  For example, when $p=31$
310the anomalous $j$-invariants are $j=10,14$.
311
312Let~$\chi: \B Z/p\B Z\raw \B C$ be a non-quadratic
313Dirichlet character, and denote by $\B Z[\chi]$ the
314subring of $\B Q(\zeta_{p-1})$ generated by the image of~$\chi$.
315Denote by $S_2(\Gamma_0(p);\B Z)$ be the set of modular forms
316$f\in S_2(\Gamma_0(p))$ whose Fourier expansion at the cusp~$\infty$
317lies in $\B Z[[q]]$.
318
319We study the $\B T$-modules $\B T$, $\Delta_S$, and $S_2(\Gamma_0(p);\B Z)$.
320After extension of scalars to~$\B Q$, these
321are $\B T\otimes\B Q$-modules that are free of rank~$1$, of which the
322irreducible sub-$\B T\otimes\B Q$ modules are the annihilators of the
323minimal prime ideals of $\B T$.  We compute a list of the minimal
324prime ideals of $\B T$ by computing appropriate kernels and
325characteristic polynomials of Hecke operators of small index on
326$\Delta_S$, which we find using the graph method of Mestre and
327Oesterl\'e \Mes{}.
328
329Having computed the minimal prime ideals of $\B T$, we verify that
330some nontrivial ideal $\cI$ of $\B T$ (always a minimal prime
331ideal in the range of our computations) simultaneously satisfies
332the following three conditions:
333\vskip 2ex
334
3351)
336For each anomalous $j$-invariant, there exists $x\in\Delta_S$ such that
337 $\cI x=0$ and  $\iota_j(x)\ne 0$.\vskip 1ex
338
3392) Each of the newforms~$f\in S_2(\Gamma_0(p))$ with
340$\cI f=0$ satisfies $L(f,\chi,1)\ne 0$.
341\vskip 1ex
342
3433) The image of~$\cI$ in the $\B T$-module $\B T/p\B T$
344is a direct factor.\vskip 2ex
345
346Let $\cI$ be an ideal of $\B T$. Here is how we verify these conditions
347for $\cI$.
348
349\bigskip
350{\it \noindent Verification of condition 1.}
351
352We verified that $\cI$ satisfies the first condition by
353finding a $\B T$-eigenvector~$v$ of $\Delta_S\otimes \bar\B Z$ that is
354annihilated by $\cI$ and satisfies $\iota_j(v)\neq 0$ for all anomalous $j$-invariants.  Because $\iota_j$
355is a homomorphism, this implies the existence of~$x$ as in condition 1.
356
357\bigskip
358{\it \noindent Verification of condition 2.}
359
360We verified the second condition using modular symbols.
361Our method is purely algebraic, so we do not perform
362any approximate computation of integrals.
363Using the algorithm described in \Cremona, we compute the action of
364the Hecke algebra $\B T$ on the space
365$\Hom(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$.   By intersecting the kernels
366of appropriate elements of $\B T$, we find a basis
367$\varphi_1,\ldots,\varphi_n$ for the subspace of
368$\Hom(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$ that is annihilated by~$\cI$.
369Let~$\Phi_{\cI}=\varphi_1\times \cdots \times \varphi_n$ denote the linear map
370 $H_1(X_0(p);\B Q[\chi])\raw \B Q[\chi]^n$
371defined by the $\varphi_i$.
372
373Let $\B T_{\B Q[\chi]} = \B T \otimes \B Q[\chi]$, where $\B Q[\chi]$
374is the number field generated the image of~$\chi$.
375The {\it $\chi$-twisted winding element} (denoted $\theta_\chi$ in
376\Merel)
377 $$\B e_\chi = \sum_{a\in (\B Z/p\B Z)^*} \bar\chi(a) 378 \Big\{\infty,{a \over p}\Big\}$$
379generates the {\it $\chi$-twisted winding submodule}
380$\B T_{\B Q[\chi]}\cdot \B e_\chi$.  To compute this submodule,
381we use that $\B T$ is generated, even as a $\B Z$-module,
382by $T_1,T_2,\ldots, T_b$, for any $b\geq (p+1)/6$
383(see \Agashe).
384
385\bigskip
386\lem  3
387{\it Let $\cI$ be a minimal prime ideal of~$\B T$, and
388let $\chi:(\B Z/N\B Z)^*\raw \B C^*$
389be a nontrivial Dirichlet character.
390Then the dimension of $\Phi_{\cI}(\B T_{\B Q[\chi]} \cdot \B e_\chi)$ is
391equal to the cardinality of the set of newforms~$f$ such that
392$\cI f=0$ and $L(f,\chi,1) \neq 0$.
393}
394
395\dm
396We have
397$$\dim_{\B Q[\chi]} \Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_\chi) 398 = \dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi).$$
399This dimension is invariant upon changing the basis
400$\varphi_1,\ldots, \varphi_n$ used to define $\Phi_{\cI}$.
401In particular, over $\B C$ there is a basis
402$\varphi_1',\ldots, \varphi_n'$ so that the resulting
403map $\Phi_{\cI}'$ satisfies
404$$\Phi_{\cI}'(x) = 405\Bigl(\rp(\int_x f^{(1)}), \ip(\int_x f^{(1)}), 406\ldots, 407 \rp(\int_x f^{(d)}),\ip(\int_x f^{(d)})\Bigr),$$
408where $f^{(1)}, \ldots, f^{(d)}$ are the Galois conjugates
409of a newform~$f^{(1)}=\sum a_n^{(1)} q^n$ such that $\cI f^{(1)}=0$.
410Furthermore, $\Phi_{\cI}'$ is a $\B T_{\B C}$-module homomorphism
411if we declare that $\B T_{\B C}$ as acts on $\B R^{2d} = \B C^d$ via
412$$T_n(x_1,y_1, \ldots, x_d, y_d) = 413 T_n(z_1,\ldots,z_d) = (a_n^{(1)} z_1,\ldots, a_n^{(d)}z_d),$$
414where $z_j = x_j + i y_j$ and
415the $a_n^{(j)}$ are Fourier coefficients of the $f^{(j)}$.
416
417As explained in Section 2.2 of~\Merel,
418$\int_{\B e_\chi} f = *\cdot L(f,\chi,1)$, where~$*$
419is some nonzero real or pure-imaginary complex number,
420according to whether $\chi(-1)$ equals~$1$ or~$-1$,
421respectively.
422Combining this observation with the equality
423 $$\dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi) 424 = \dim_{\B C} (\B T_{\B C}\cdot \Phi_{\cI}(\B e_\chi)),$$
425and that the image of $\B T_{\B C}$ in $\End(\B C^d)$ is
426equal to the diagonal matrices, proves the asserted equality.
427
428\bigskip
429\rem{2} The dimension of $\Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_{\chi})$
430is unchanged if~$\chi$ is
431replaced by a Galois-conjugate character.
432
433\bigskip
434
435In practice, computations over the cyclotomic field $\B Q[\chi]$ are
436extremely expensive.  Fortunately, for our application it suffices to
437give a lower bound on the dimension appearing in the lemma.  Such a
438bound can be efficiently obtained by instead computing the reductions
439of~$\Phi$,~$\chi$, and the $\chi$-twisted winding submodule modulo a
440suitable maximal ideal of the ring of integers of $\B Q[\chi]$ that
441splits completely; this amounts to performing the above linear algebra
442over a relatively small prime finite field $\B F_\ell$ such
443that~$\ell$ is congruent to~$1$ modulo $p-1$.
444
445\bigskip
446
447\rem{3} For every newform~$f$ in $S_2(\Gamma_0(p))$, with $p\leq 1000$,
448and every mod~$p$ Dirichlet character~$\chi$, we found that
449$L(f,\chi,1)\neq 0$ if and only if
450$L(f^{\sigma},\chi,1)\neq 0$ for all conjugates $f^{\sigma}$
451of~$f$.
452More generally, for any~$f$ and~$\chi$, this equivalence holds if
453$\B Q[\chi]$ is linearly disjoint from the
454field $K_f=(\B T/\cI)\otimes\B Q$.
455The first few primes
456for which there is a form~$f$ and a mod~$p$ character~$\chi$
457such that the linear disjointness hypothesis fails are
458$p=31, 113, 127$, and $191$.
459The analogue of this nonvanishing observation is false if we instead consider
460newforms on $\Gamma_1(p)$ and allow~$\chi$ to be arbitrary.
461For example, let~$f$
462be one of the two Galois-conjugate newforms in $S_2(\Gamma_1(13))$.
463Then there is a character $\chi:(\B Z/7\B Z)^*\raw \B C^*$ of
464order~$3$ such that $L(f,\chi,1) = 0$ and $L(f^{\sigma},\chi,1)\neq 0$.
465
466\bigskip
467
468{\it \noindent Verification of condition 3.}
469
470The third condition is satisfied for all $p<10000$, except possibly
471$p = 389$,
472because we have verified that the discriminant of $\B T$ is
473prime to~$p$ for all such $p\neq 389$,
474so the ring $\B T/p\B T$ is semisimple.
475The discriminant computation was carried out by the second author
476as follows.
477Using the method of \Mes{}, we computed discrimininants of characteristic
478polynomials mod~$p$ of the Hecke operators $T_2$, $T_3$, $T_5$, and $T_7$.
479In the few cases when all four of these characteristic polynomials had
480discriminant equal to~$0$ mod~$p$, we resorted to modular symbols to
481compute several more characteristic polynomials until we found one
482having nonzero discriminant modulo~$p$.
483
484We consider the remaining case $p=389$ in detail.  There are exactly
485five minimal prime ideals of $\B T$, which we denote $\cP_1$, $\cP_2$,
486$\cP_3$, $\cP_6$, and $\cP_{20}$, where the quotient field of $\B 487T/\cP_i$ has dimension~$i$.  The discriminant of the characteristic
488polynomial of $T_2$ is exactly divisible by $389$.  Since the field of
489fractions of $\B T/\cP_{20}$ has discriminant divisible by $389$, we
490see that $389$ is not the residue characteristic of any congruence
491prime.  Let $\cO_i = \B T/\cP_{i}$.  The natural map $\B T \rightarrow 492\prod \cO_i$ has finite kernel and cokernel each of order coprime to
493$389$, so $\B T / 389 \B T \cong \prod \cO_i/389 \cO_i$.  The
494nonquadratic characters $\chi:(\B Z/p\B Z)^*\rightarrow \B C^*$ have
495orders $1, 4, 97, 193, 388$.  We must verify that for each of these
496degrees, one of the ideals $\cP_i$ satisfies conditions 1--3.  We
497check as above that conditions 1--3 for~$\chi$ of order~$4$ are
498satisfied by $\cP_2$ and conditions 1--3 for~$\chi$ of order greater
499than~$4$ are satisfied by $\cP_1$.  When~$\chi$ is the trivial
500character, conditions~1--3 are satisfied only by $\cP_{20}$.
501
502\bigskip
503{\it \noindent Summary.}
504
505For each prime $p<1000$ different than $2,3,5,7, 13$, we
506verified the existence of an ideal that satisfies the three conditions
507given above, as follows.  For each~$p$, we consider each Galois conjugacy class of
508non-quadratic characters~$\chi$.  We find a single newform~$f$ such
509that $L(f,\chi,1)\ne 0$ for all conjugates of~$f$ and of~$\chi$.  Then
510we let $\cI$ be the annihilator of~$f$, and try to verify condition~1
511for {\it all} of the anamolous $j$-invariants in $\B F_p$.
512When the three conditions are satisfied for an ideal~$\cI$ of~$\B T$,
513there exists $t_\chi\in\B T$ that is annihilated by $\cI$ and is the
514inverse image of a projector of $\B T/p\B T$ on the complement of
515$\cI+p\B T$.  Putting $\delta=x$, one has
516$\iota_j(t_\chi \delta)=\iota_j(\delta)\ne0$
517(because $\iota_j$ takes its values in
518characteristic~$p$, it follows that $\delta$ is annihilated by~$\cI$ and
519$t_\chi\in 1+p\B T+\cP$).
520Every newform $f\in t_\chi S_2(\Gamma_0(p))$ satisfies
521$\cI f=0$, and therefore, by our second condition, $L(f,\chi,1)\ne0$.
522The pair $(t_\chi,\delta)$ then satisfies the conditions required by
523Proposition~1.
524
525
526
527\bigskip\bigskip
528
529\vskip .5in
530
531\centerline{\pc Bibliography}
532\bigskip
533
534\item{\Agashe}{\pc A. Agashe},
535{\it On invisible elements of the Tate-Shafarevich group},
536C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), no. 5, 369--374.
537\vskip 1ex
538
539\item{\Cremona}{\pc J. Cremona},
540{\it Algorithms for modular elliptic curves},
541second ed., Cambridge University Press, Cambridge,
542{\oldstyle 1997}.
543\vskip 1ex
544
545\item{\Merel}{\pc L. Merel},
546{\it Sur la nature non cyclotomique des points d'ordre fini des courbes
547elliptiques},
548To appear in Duke Math. Journal.
549\vskip 1ex
550
551\item{\Mes}{\pc J.-F. Mestre}, {\it La m\'ethode des graphes.
552Exemples et  applications}, Proceedings of the international
553conference on class numbers and fundamental units of algebraic number
554fields (Katata), 217--242, {\oldstyle 1986}.
555\end
556
557
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