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Author: William A. Stein
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2%                                                                 %
3%  merel-stein.tex:  The field generated by the points of small   %
4%                    prime order on an elliptic curve             %
5%                                                                 %
6%  15 April 2001                                                  %
7%                                                                 %
8%  Authors: Loic Merel and William A. Stein                       %
9%                                                                 %
10%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
11\documentclass{article}
12\hoffset=-0.075\textwidth
13\textwidth=1.05\textwidth
14\catcode\@=11
15\begin{document}
16
17\def\�#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi}   % trema
18\def\B#1{{\bf #1}}                                      % bold
19\def\lc{{\it loc.\thinspace{}cit.}}                     % loc. cit.
20\def\mod#1{\ \hbox{{\rm mod}$#1$}}                      % modulo
21\def\eps{\varepsilon}
22
23\font\titchap=cmr17 at 20pt  % for the titles of chapters.
24\font\pc=cmcsc10             % for the titles of sections, props, etc.
25
26\def\th#1{\noindent{\pc Theorem}\ #1. --- \ignorespaces}      %Theorem 1.
27\def\prop#1{\noindent{\pc Proposition}\ #1. --- \ignorespaces}%Proposition 1.
28\def\Def#1{\noindent{\pc Definition}\ #1. --- \ignorespaces}  %Definition 1.
29\def\cor#1{\noindent{\pc Corollary}\ #1. --- \ignorespaces}   %Corollary 1.
30\def\conj#1{\noindent{\pc Conjecture}\ #1. --- \ignorespaces} %Conjecture 1.
31\def\lem#1{\noindent{\it Lemma}\ #1. --- \ignorespaces}       %Lemma 1.
32\def\rem#1{\noindent{\it Remark}\ #1: \ignorespaces}          %Remark 1.
33\def\exe#1{\noindent{\it Example}\ #1: \ignorespaces}         %Example 1.
34\def\exr#1{\noindent{\it Exercise}\ #1: \ignorespaces}        %Exercise 1.
35\def\rems{\noindent{\it Remarks}: \ignorespaces}              %Remarks 1.
36\def\exes{\noindent{\it Examples}: \ignorespaces}             %Examples 1.
37\def\exrs{\noindent{\it Exercises}: \ignorespaces}            %Exercises 1.
38\def\thp{\noindent{\pc Theorem}. --- \ignorespaces}           %Theorem 1.
39\def\propp{\noindent{\pc Proposition}. --- \ignorespaces}     %Proposition 1.
40\def\Defp{\noindent{\pc Definition}. --- \ignorespaces}       %Definition 1.
41\def\corp{\noindent{\pc Corollary}. --- \ignorespaces}        %Corollairy 1.
42\def\conjp{\noindent{\pc Conjecture}. --- \ignorespaces}      %Conjecture 1.
43\def\lemp{\noindent{\it Lemma}. --- \ignorespaces}            %Lemma 1.
44\def\remp{\noindent{\it Remark}: \ignorespaces}               %Remark 1.
45\def\exep{\noindent{\it Example}: \ignorespaces}              %Example 1.
46\def\dm{\noindent{\it Proof}. --- \ignorespaces}
47\def\raw{\longrightarrow}
48\def\Hom{{\rm Hom}}
49\def\Gal{{\rm Gal}}
50\def\cP{{\cal P}}
51\def\cO{{\cal O}}
52\def\cI{{\cal I}}
53\def\rp{{\rm Re}}
54\def\ip{{\rm Im}}
55\def\End{{\rm End}}
56
57\def\Agashe{{$$}}
58\def\Cremona{{$$}}
59\def\Merel{{$$}}
60\def\Mes{{$$}}
61
62\def\change#1{[[{\bf Change:} #1]]}
63
64\centerline
65{\titchap The field generated by the points of small}
66\centerline
67{\titchap prime order on an elliptic curve}
68\medskip
69
70\bigskip\bigskip\bigskip
71\centerline{\pc Lo\"\i c Merel {\rm and} William A.~Stein}
72\bigskip\bigskip\bigskip
73
74
75\bigskip\bigskip\noindent
76{\bf Introduction}
77\bigskip
78
79Let $\bar\B Q$ be an algebraic closure of $\B Q$, and for any prime
80number~$p$, denote by $\B Q(\mu_p)$ the cyclotomic subfield of $\bar\B 81Q$ generated by the $p$th roots of unity.
82
83\bigskip
84\th{}{\it
85Let~$p$ be a prime.  If there exists an elliptic curve~$E$ over $\B 86Q(\mu_p)$ such that the points of order~$p$ of $E(\bar\B Q)$ are all
87$\B Q(\mu_p)$-rational, then $p=2,3,5,13$ or $p>1000$.}
88\bigskip
89
90The case $p=7$ was treated by Emmanuel Halberstadt.  The
91part of the theorem that concerns the case $p\equiv 3\!\!\pmod{4}$ is
92given in~\Merel.  In this paper, we give the details that permit our
93treating the more difficult case in which $p\equiv 1 \!\!\pmod{4}$.
94We treat this last case with the aid of Proposition~2 below, which is
95not present in \lc.
96The case $p=13$ is currently under investigation by Marusia Rebolledo,
97as part of her Ph.D.{} thesis.
98
99\bigskip\noindent
100{\bf 1. Counterexamples define points on $X_0(p)(\B Q(\sqrt{p}))$}
101\bigskip
102
103First we recall some of the results and notation of \Merel.
104Let $S_2(\Gamma_0(p))$ denote the space of cusp forms of weight~$2$ for
105the congruence subgroup $\Gamma_0(p)$. Denote by $\B T$ the
106subring of ${\rm End}\,S_2(\Gamma_0(p))$ generated by the
107Hecke operators $T_n$ for all integers~$n$.
108Let $f\in S_2(\Gamma_0(p))$ have $q$-expansion
109$\sum_{n=1}^\infty a_nq^n$. When $\chi$ is a Dirichlet character,
110denote by $L(f,\chi,s)$ the entire function which extends the
111Dirichlet series $\sum_{n=1}^\infty a_n\chi(n)/n^s$.
112
113Let $S$ be the set of isomorphism classes of supersingular elliptic
114curves in characteristic~$p$.  Denote by $\Delta_S$ the group formed
115by the divisors of degree~$0$ with support on~$S$.  It is equipped with
116a structure of $\B T$-module (induced, for example, from the action
117of the Hecke correspondences on the fiber at~$p$ of the regular minimal
118model of $X_0(p)$ over $\B Z$).
119
120Let $j\in\bar\B F_p-J_S$, where $J_S$ denotes the set of supersingular modular
121invariants. We denote by  $\iota_j$ the homomorphism of
122groups $\Delta_S\raw \bar\B F_p$ that associates to $\sum_E n_E[E]$
123the quantity
124$\sum_E n_E/(j-j(E))$, where $j(E)$ denotes the modular invariant of~$E$.
125
126
127One says that an element  $j\in\B F_p$ is {\it anomalous}
128if there exists an elliptic curve over $\B F_p$ with modular invariant~$j$
129that possesses an $\B F_p$-rational point  of order~$p$
130(then necessarily $j\notin{}J_S$).
131
132Let~$p$ be a prime that is congruent to~$1$ modulo~$4$.
133In the following proposition we prove, under a hypothesis on~$p$, that
134if~$E$ is an elliptic curve over $\B Q(\mu_p)$ all of whose torsion is
135$\B Q(\mu_p)$-rational, then for each subgroup $C\subset{}E(\bar\B Q)$
136of order~$p$,
137the point $(E,C)$ on $X_0(p)$ is defined over $\B Q(\sqrt{p})$.  As we
138will see in Proposition~2, this $\B Q(\sqrt{p})$-rationality
139conclusion is contrary to fact, from which we conclude that such
140elliptic curves~$E$ do not exist when the hypothesis on~$p$
141is satisfied.  In Section~3 we verify this hypothesis
142for $p=11$ and $13 < p < 1000$.
143
144\bigskip
145\prop{1}{\it Suppose that~$p$ is congruent to~$1$ modulo~$4$.
146Suppose that for all anomalous
147$j\in\B F_p$ and all
148non-quadratic Dirichlet characters $\chi \colon (\B Z/p\B Z)^*\raw \B 149C^*$, there exists $t_\chi\in \B T$ and
150$\delta\in\Delta_S$ such that $L(f,\chi,1)\ne0$ for every newform
151$f\in t_\chi S_2(\Gamma_0(p))$ and
152 $\iota_j(t_\chi\delta)\ne0$.
153
154Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the
155points of order~$p$ of
156$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.
157Then for all subgroups~$C$ of order~$p$ of $E(\bar \B Q)$, there exists an
158elliptic curve $E_C$ over $\B Q(\sqrt p)$ equipped with a
159$\B Q(\sqrt p)$-rational subgroup $D_C$ of order~$p$, and
160the pairs $(E,C)$ and $(E_C,D_C)$ are $\bar \B Q$-isomorphic.}
161
162\dm We prove the proposition using the results of~\Merel.
163The hypothesis
164$\iota_j(t_\chi\delta)\ne0$ forces $t_\chi\notin p\B T$
165and, {\it a fortiori}, $t_\chi\ne0$; in addition,
166the non-vanishing hypothesis on the $L$-series
167forces the hypothesis $H_p(\chi)$ of \lc, introduction.
168
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
175Denote by~$j$ the modular invariant of the fiber at~$\cP$ of the
176N\'eron model of~$E$.
177According to the corollary of Proposition~15 of \lc,
178$j$ is anomalous.
179
180Let~$C$ be a subgroup of $E(\bar\B Q)$ of order~$p$.
181By assumption~$E$ is an elliptic curve over~$\B Q(\mu_p)$ whose points
182of order~$p$ are all $\B Q(\mu_p)$-rational, so
183the pair $(E,C)$ defines a $\B Q(\mu_p)$-rational point~$P$
184of the modular curve $X_0(p)$.
185
186Consider the morphism $\phi_{\chi}=\phi_{t_\chi}:X_0(p)\rightarrow J_0(p)$
187obtained by composing the standard embedding of $X_0(p)$ into $J_0(p)$
188with $t_{\chi}$.  As in section 1.3 of \lc, $\phi_{\chi}$
189extends to a map from the minimal regular model of $X_0(p)$ to the
190N\'eron model of $J_0(p)$.
191When $\iota_j(t_\chi\delta)\ne0$, this map is a formal
192immersion at the point $P_{/\B F_p}$, according to \lc,
193Proposition~4.  The hypothesis that $L(f,\chi,1)\ne0$ for
194every newform
195$f\in t_\chi S_2(\Gamma_0(p))$, translates into $L(t_\chi J_0(p), 196\chi,1)\ne0$, which in turn implies that the $\chi$-isotypical
197component of
198$t_\chi J_0(p)(\B Q(\mu_p))$ is finite (this is Kato's theorem, see the
199discussion in section 1.5 of \lc).
200We can then apply Corollary~1 of Proposition~6 of \lc.  This proves
201that~$P$ is
202$\B Q(\sqrt p)$-rational, which translates into the conclusion of
203Proposition~1.
204
205\bigskip
206\rem{1}
207Proposition~1 is true even under the weaker hypothesis
208that $t_{\chi}$ lies in $\B T\otimes \B Z[\chi]$, which acts $\B 209Z[\chi]$-linearly on modular forms.
210
211
212\bigskip\bigskip\noindent
213{\bf 2. Elliptic curves and quadratic fields}
214
215\bigskip
216\prop{2}{\it Let~$p$ be a prime number $>5$ and congruent to~$1$
217modulo~$4$. Let~$E$ be an elliptic curve over $\bar\B Q$.
218There exists a subgroup~$C\subset{}E(\bar\B Q)$ of order~$p$
219such that $(E,C)$ can not be defined over $\B Q(\sqrt{p})$.}
220
221\dm
222We procede by contradiction, i.e., we assume that for
223all cyclic subgroups~$C$ of order~$p$ of $E(\bar\B Q)$,
224the pair $(E,C)$ can be defined over ${\B Q(\sqrt{p})}$.
225We choose such a pair $(E_0,C_0)$ over ${\B Q(\sqrt{p})}$.
226
227Assume first that all twists of $E$ are quadratic, i.e.
228that $j(E)$ is neither~$0$ nor $1728$.
229We show that the group
230$\Gal(\bar\B Q/{\B Q(\sqrt p)})$ acts by scalars
231on the $\B F_p$-vector space $E_0(\bar\B Q)[p]$. For this it
232suffices to show that all subgroups of order~$p$ of
233$E_0(\bar\B Q)[p]$ are stable by $\Gal(\bar\B Q/{\B Q(\sqrt p)})$.
234
235Suppose $C_1$ is a cyclic subgroup of order~$p$ of $E_0(\bar\B Q)[p]$.
236By assumption, there exists a quadratic twist $E_1$ of $E_0$ and
237a cyclic subgroup $C_1'$ of $E_1(\bar\B Q)[p]$
238that is defined over $\B Q(\sqrt{p})$, such that
239the image of $C_1$ by the isomorphism $E_0\simeq E_1$ is $C'_1$.
240Since $\Gal(\bar\B Q/{\B Q(\sqrt p)})$ leaves $C_1'$ stable and
241the action of $\Gal(\bar\B Q/\B Q(\sqrt p))$
242on $E_0(\bar\B Q)[p]$ is
243a quadratic twist of the action on $E_1(\bar\B Q)[p]$,
244we see that $\Gal(\bar\B Q/\B Q(\sqrt p))$ leaves $C_1$ stable.
245Thus $\Gal(\bar\B Q/\B Q(\sqrt p))$ fixes all lines in
246$E_0(\bar\B Q)[p]$, and hence
247acts by scalars.  Denote by~$\alpha$ the corresponding character
248of $\Gal(\bar\B Q/\B Q(\sqrt p))$.
249
250Because of the Weil pairing, $\alpha^2$ coincides
251with the cyclotomic character modulo~$p$, and it factors through
252$\Gal(\B Q(\mu_p)/\B Q(\sqrt p))$. But, when
253$p\equiv 1\!\!\pmod 4$, the group $\Gal(\B Q(\mu_p)/{\B Q(\sqrt p)})$ is of
254even order, and the characters modulo~$p$ form a group generated by the
255reduction modulo~$p$ of the cyclotomic character, which, therefore,
256can not be a square.
257
258Next suppose that $j(E)=0$ or $j(E)=1728$. Indeed, in these
259two cases~$E$ has
260complex multiplication by an order of
261$K=\B Q[\sqrt{-3}]$ or $\B Q[\sqrt{-1}]$. Let $d_K=3$ or $d_K=2$ in
262these two cases respectively. Let $C$ be a subgroup of order $p$ of
263$E(\bar\B Q)$. Consider
264the map
265$\rho_0 : \Gal(\bar \B Q/\B Q(\sqrt p))\longrightarrow{\rm 266Aut}\,E_0(\bar\B Q)[p]$. Since $E$ has complex multiplication, the image
267of $\rho_0$ has no element of order~$p$. Therefore, there are at least two
268subgroups, including $C_0$, of order~$p$ of
269$E(\bar\B Q)$ stable under the image of $\rho_0$. Call the other subgroup
270$C_1$. Let $C_2$ be a subgroup of order $p$ of $E(\bar\B Q)$ which is
271distinct from $C_0$ and $C_1$. The pair $(E,C_2)$ can be defined over
272$\B Q(\sqrt p)$.
273Therefore, there exists an extension field $K_2$ of $\B Q(\sqrt p)$,
274whose degree $d_2$ divides $2d_K$, such that
275the image of the restriction of $\rho_0$ to
276$\Gal(\bar \B Q/K_2)$ leaves stable three distinct subgroups of
277order~$p$ of $E_0(\bar\B Q)$, and therefore consists only of scalars.
278If $d_2\le2$, one concludes as in the cases where
279$j(E)\ne0$ and $j(E)\ne 1728$. We suppose now that $d_2>2$.
280The projective image of $\rho_0$ has order $d_K$.
281
282Since~$E$ is an elliptic curve over $\bar \B Q$ with complex
283multiplication by a field of class number one, there is a model
284for~$E$ that is defined over $\B Q$. Consider the map
285$\rho$ :  $\Gal(\bar \B Q/\B Q)\longrightarrow{\rm Aut}\,E(\bar\B Q)[p]$.
286By the theory of complex multiplication, the projective image of
287$\rho$ has order $2(p+1)$ or $2(p-1)$. There exists a field extension~$L$
288of degree dividing
289$d_K$ of $\B Q(\sqrt p)$ such that the restrictions to
290$\Gal(\bar \B Q/L)$ of the projective
291images of $\rho$ and $\rho_0$ coincide.
292Therefore one has $(p-1)|d_K^2$ or $(p+1)|d_K^2$.
293This imposes $p=5$ and $d_K=2$.
294
295
296
297\bigskip\bigskip\noindent
298{\bf 3. Verification of the hypothesis of Proposition~1}
299\bigskip
300Let~$p$ be a prime number.  In this section we explain how we used
301a computer to verify that the second hypothesis of Proposition~1 are
302satisfied for $p=11$ and $13 < p < 1000$.  (In the present paper,
303this verification is only required for $p$ that are congruent to~$1$
304modulo~$4$.)
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 $S_2(\Gamma_0(p);\B Z)$ 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$\Hom_{\B Q[\chi]}(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$.   By intersecting
366the kernels of appropriate elements of $\B T$, we find a basis
367$\varphi_1,\ldots,\varphi_n$ for the subspace of
368$\Hom_{\B Q[\chi]}(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$ that is
369annihilated by~$\cI$.  Let~$\Phi_{\cI}=\varphi_1\times \cdots \times 370\varphi_n$ denote the linear map
371 $H_1(X_0(p);\B Q[\chi])\raw \B Q[\chi]^n$
372defined by the $\varphi_i$.
373
374Let $\B T_{\B Q[\chi]} = \B T \otimes \B Q[\chi]$, where $\B Q[\chi]$
375is the number field generated the image of~$\chi$.
376The {\it $\chi$-twisted winding element} (denoted $\theta_\chi$ in
377\Merel)
378 $$\B e_\chi = \sum_{a\in (\B Z/p\B Z)^*} \bar\chi(a) 379 \Big\{\infty,{a \over p}\Big\}$$
380generates the {\it $\chi$-twisted winding submodule}
381$\B T_{\B Q[\chi]}\cdot \B e_\chi$.  To compute this submodule,
382we use that $\B T$ is generated, even as a $\B Z$-module,
383by $T_1,T_2,\ldots, T_b$, for any $b\geq (p+1)/6$
384(see \Agashe).
385
386\bigskip
387\lem  3
388{\it Let $\cI$ be a minimal prime ideal of~$\B T$, and
389let $\chi:(\B Z/N\B Z)^*\raw \B C^*$
390be a nontrivial Dirichlet character.
391Then the dimension of the $\B Q[\chi]$-vector space $\Phi_{\cI}(\B T_{\B 392Q[\chi]} 393\cdot 394\B e_\chi)$ is  equal to the cardinality of the set of newforms~$f$ such
395that
396$\cI f=0$ and $L(f,\chi,1) \neq 0$.
397}
398
399\dm
400We have
401$$\dim_{\B Q[\chi]} \Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_\chi) 402 = \dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi).$$
403This dimension is invariant upon changing the basis
404$\varphi_1,\ldots, \varphi_n$ used to define $\Phi_{\cI}$.
405In particular, over $\B C$ there is a basis
406$\varphi_1',\ldots, \varphi_n'$ so that the resulting
407map $\Phi_{\cI}'$ satisfies
408$$\Phi_{\cI}'(x) = 409\Bigl(\rp(\int_x f^{(1)}), \ip(\int_x f^{(1)}), 410\ldots, 411 \rp(\int_x f^{(d)}),\ip(\int_x f^{(d)})\Bigr),$$
412where $f^{(1)}, \ldots, f^{(d)}$ are the Galois conjugates
413of a newform~$f^{(1)}=\sum a_n^{(1)} q^n$ such that $\cI f^{(1)}=0$.
414Furthermore, $\Phi_{\cI}'$ is a $\B T_{\B C}$-module homomorphism
415if we declare that $\B T_{\B C}$ acts on $\B R^{2d} = \B C^d$ via
416$$T_n(x_1,y_1, \ldots, x_d, y_d) = 417 T_n(z_1,\ldots,z_d) = (a_n^{(1)} z_1,\ldots, a_n^{(d)}z_d),$$
418where $z_j = x_j + i y_j$ and
419the $a_n^{(j)}$ are Fourier coefficients of the $f^{(j)}$.
420
421As explained in Section 2.2 of~\Merel,
422$\int_{\B e_\chi} f = *\cdot L(f,\chi,1)$, where~$*$
423is some nonzero real or pure-imaginary complex number,
424according to whether $\chi(-1)$ equals~$1$ or~$-1$,
425respectively.
426Combining this observation with the equality
427 $$\dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi) 428 = \dim_{\B C} (\B T_{\B C}\cdot \Phi_{\cI}(\B e_\chi)),$$
429and that the image of $\B T_{\B C}$ in $\End(\B C^d)$ is
430equal to the diagonal matrices, proves the asserted equality.
431
432\bigskip
433\rem{2} The dimension of $\Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_{\chi})$
434is unchanged if~$\chi$ is
435replaced by a Galois-conjugate character.
436
437\bigskip
438
439In practice, computations over the cyclotomic field $\B Q[\chi]$ are
440extremely expensive.  Fortunately, for our application it suffices to
441give a lower bound on the dimension appearing in the lemma.  Such a
442bound can be efficiently obtained by instead computing the reductions
443of~$\Phi$,~$\chi$, and the $\chi$-twisted winding submodule modulo a
444suitable maximal ideal of the ring of integers of $\B Q[\chi]$ that
445splits completely; this amounts to performing the above linear algebra
446over a relatively small finite field $\B F_\ell$ where~$\ell$ is
447congruent to~$1$ modulo $p-1$.
448
449\bigskip
450
451\rem{3} For every newform~$f$ in $S_2(\Gamma_0(p))$, with $p\leq 1000$,
452and every mod~$p$ Dirichlet character~$\chi$, we found that
453$L(f,\chi,1)\neq 0$ if and only if
454$L(f^{\sigma},\chi,1)\neq 0$ for all conjugates $f^{\sigma}$
455of~$f$.
456More generally, for any~$f$ and~$\chi$, this equivalence holds if
457$\B Q[\chi]$ is linearly disjoint from the
458field $K_f=(\B T/\cI)\otimes\B Q$.
459The first few primes
460for which there is a form~$f$ and a mod~$p$ character~$\chi$
461such that the linear disjointness hypothesis fails are
462$p=31, 113, 127$, and $191$.
463The analogue of this nonvanishing observation is false if we instead consider
464newforms on $\Gamma_1(p)$ and allow~$\chi$ to be arbitrary.
465For example, let~$f$
466be one of the two Galois-conjugate newforms in $S_2(\Gamma_1(13))$.
467Then there is a character $\chi:(\B Z/7\B Z)^*\raw \B C^*$ of
468order~$3$ such that $L(f,\chi,1) = 0$ and $L(f^{\sigma},\chi,1)\neq 0$.
469
470\bigskip
471
472{\it \noindent Verification of condition 3.}
473
474The third condition is satisfied for all $p<10000$, except possibly
475$p = 389$,
476because we have verified that the discriminant of $\B T$ is
477prime to~$p$ for all such $p\neq 389$,
478so the ring $\B T/p\B T$ is semisimple.
479The discriminant computation was carried out by the second author
480as follows.
481Using the method of \Mes{}, we computed discrimininants of characteristic
482polynomials mod~$p$ of the Hecke operators $T_2$, $T_3$, $T_5$, and $T_7$.
483In the few cases when all four of these characteristic polynomials had
484discriminant equal to~$0$ mod~$p$, we resorted to modular symbols to
485compute several more characteristic polynomials until we found one
486having nonzero discriminant modulo~$p$.
487
488We consider the remaining case $p=389$ in detail.  There are exactly
489five minimal prime ideals of $\B T$, which we denote $\cP_1$, $\cP_2$,
490$\cP_3$, $\cP_6$, and $\cP_{20}$, where the quotient field of $\B 491T/\cP_i$ has dimension~$i$.  The discriminant of the characteristic
492polynomial of $T_2$ is exactly divisible by $389$.  Since the field of
493fractions of $\B T/\cP_{20}$ has discriminant divisible by $389$, we
494see that $389$ is not the residue characteristic of any congruence
495prime.  Let $\cO_i = \B T/\cP_{i}$.  The natural map $\B T \rightarrow 496\prod \cO_i$ has finite kernel and cokernel each of order coprime to
497$389$, so $\B T / 389 \B T \cong \prod \cO_i/389 \cO_i$.  The
498nonquadratic characters $\chi:(\B Z/p\B Z)^*\rightarrow \B C^*$ have
499orders $1, 4, 97, 193, 388$.  We must verify that for each of these
500degrees, one of the ideals $\cP_i$ satisfies conditions 1--3.  We
501check as above that conditions 1--3 for~$\chi$ of order~$4$ are
502satisfied by $\cP_2$ and conditions 1--3 for~$\chi$ of order greater
503than~$4$ are satisfied by $\cP_1$.  When~$\chi$ is the trivial
504character, conditions~1--3 are satisfied only by $\cP_{20}$.
505
506\bigskip
507{\it \noindent Summary.}
508
509For each prime $p<1000$ different than $2,3,5,7, 13$, we
510verified the existence of an ideal that satisfies the three conditions
511given above, as follows.  We consider each Galois conjugacy class of
512non-quadratic characters~$\chi$.  We find a single newform~$f$ such
513that $L(f,\chi,1)\ne 0$ for all conjugates of~$f$ and of~$\chi$.  Then
514we let $\cI$ be the annihilator of~$f$, and try to verify condition~1
515for {\it all} of the anamolous $j$-invariants in $\B F_p$.
516When the three conditions are satisfied for an ideal~$\cI$ of~$\B T$,
517there exists $t_\chi\in\B T$ that is annihilated by $\cI$ and is the
518inverse image of a projector of $\B T/p\B T$ on the complement of
519$\cI+p\B T$.  Putting $\delta=x$, one has
520$\iota_j(t_\chi \delta)=\iota_j(\delta)\ne0$
521(because $\iota_j$ takes its values in
522characteristic~$p$, it follows that $\delta$ is annihilated by~$\cI$ and
523$t_\chi\in 1+p\B T+\cP$).
524Every newform $f\in t_\chi S_2(\Gamma_0(p))$ satisfies
525$\cI f=0$, and therefore, by our second condition, $L(f,\chi,1)\ne0$.
526The pair $(t_\chi,\delta)$ then satisfies the conditions required by
527Proposition~1.
528
529\bigskip
530\noindent
531{\it Acknowledgment}: We would like to thank Barry Mazur for
533
534\bigskip\bigskip
535
536\vskip 1in
537
538\centerline{\pc Bibliography}
539\bigskip
540
541\begin{enumerate}
542\item[]{\Agashe} {\pc A. Agashe},
543{\it On invisible elements of the Tate-Shafarevich group},
544C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), no. 5, 369--374.
545\vskip 1ex
546
547\item[]{\Cremona} {\pc J. Cremona},
548{\it Algorithms for modular elliptic curves},
549second ed., Cambridge University Press, Cambridge,
550(1997).
551\vskip 1ex
552
553\item[]{\Merel} {\pc L. Merel},
554{\it Sur la nature non cyclotomique des points d'ordre fini des courbes
555elliptiques},
556To appear in Duke Math. Journal.
557\vskip 1ex
558
559
560\item[]{\Mes} {\pc J.-F. Mestre}, {\it La m\'ethode des graphes.
561Exemples et  applications}, Proceedings of the international
562conference on class numbers and fundamental units of algebraic number
563fields (Katata), 217--242, (1986).
564\end{enumerate}
565\end{document}
566
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