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% merel-stein.tex: The field generated by the points of small %
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% prime order on an elliptic curve %
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% 15 April 2001 %
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% Authors: Loic Merel and William A. Stein %
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\documentclass{article}
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\hoffset=-0.075\textwidth
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\textwidth=1.05\textwidth
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\catcode`\@=11
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\begin{document}
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\def\#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi} % trema
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\def\B#1{{\bf #1}} % bold
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\def\lc{{\it loc.\thinspace{}cit.}} % loc. cit.
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\def\mod#1{\ \hbox{{\rm mod}$#1$}} % modulo
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\def\eps{\varepsilon}
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\font\titchap=cmr17 at 20pt % for the titles of chapters.
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\font\pc=cmcsc10 % for the titles of sections, props, etc.
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\def\th#1{\noindent{\pc Theorem}\ #1. --- \ignorespaces} %Theorem 1.
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\def\prop#1{\noindent{\pc Proposition}\ #1. --- \ignorespaces}%Proposition 1.
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\def\Def#1{\noindent{\pc Definition}\ #1. --- \ignorespaces} %Definition 1.
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\def\cor#1{\noindent{\pc Corollary}\ #1. --- \ignorespaces} %Corollary 1.
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\def\conj#1{\noindent{\pc Conjecture}\ #1. --- \ignorespaces} %Conjecture 1.
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\def\lem#1{\noindent{\it Lemma}\ #1. --- \ignorespaces} %Lemma 1.
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\def\rem#1{\noindent{\it Remark}\ #1: \ignorespaces} %Remark 1.
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\def\exe#1{\noindent{\it Example}\ #1: \ignorespaces} %Example 1.
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\def\exr#1{\noindent{\it Exercise}\ #1: \ignorespaces} %Exercise 1.
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\def\rems{\noindent{\it Remarks}: \ignorespaces} %Remarks 1.
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\def\exes{\noindent{\it Examples}: \ignorespaces} %Examples 1.
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\def\exrs{\noindent{\it Exercises}: \ignorespaces} %Exercises 1.
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\def\thp{\noindent{\pc Theorem}. --- \ignorespaces} %Theorem 1.
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\def\propp{\noindent{\pc Proposition}. --- \ignorespaces} %Proposition 1.
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\def\Defp{\noindent{\pc Definition}. --- \ignorespaces} %Definition 1.
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\def\corp{\noindent{\pc Corollary}. --- \ignorespaces} %Corollairy 1.
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\def\conjp{\noindent{\pc Conjecture}. --- \ignorespaces} %Conjecture 1.
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\def\lemp{\noindent{\it Lemma}. --- \ignorespaces} %Lemma 1.
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\def\remp{\noindent{\it Remark}: \ignorespaces} %Remark 1.
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\def\exep{\noindent{\it Example}: \ignorespaces} %Example 1.
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\def\dm{\noindent{\it Proof}. --- \ignorespaces}
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\def\raw{\longrightarrow}
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\def\Hom{{\rm Hom}}
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\def\Gal{{\rm Gal}}
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\def\cP{{\cal P}}
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\def\cO{{\cal O}}
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\def\cI{{\cal I}}
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\def\rp{{\rm Re}}
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\def\ip{{\rm Im}}
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\def\End{{\rm End}}
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\def\Agashe{{$[1]$}}
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\def\Cremona{{$[2]$}}
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\def\Merel{{$[3]$}}
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\def\Mes{{$[4]$}}
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\def\change#1{[[{\bf Change:} #1]]}
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\centerline
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{\titchap The field generated by the points of small}
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\centerline
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{\titchap prime order on an elliptic curve}
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\medskip
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\bigskip\bigskip\bigskip
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\centerline{\pc Lo\"\i c Merel {\rm and} William A.~Stein}
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\bigskip\bigskip\bigskip
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\bigskip\bigskip\noindent
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{\bf Introduction}
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\bigskip
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Let $\bar\B Q$ be an algebraic closure of $\B Q$, and for any prime
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number~$p$, denote by $\B Q(\mu_p)$ the cyclotomic subfield of $\bar\B
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Q$ generated by the $p$th roots of unity.
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\bigskip
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\th{}{\it
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Let~$p$ be a prime. If there exists an elliptic curve~$E$ over $\B
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Q(\mu_p)$ such that the points of order~$p$ of $E(\bar\B Q)$ are all
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$\B Q(\mu_p)$-rational, then $p=2,3,5,13$ or $p>1000$.}
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\bigskip
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The case $p=7$ was treated by Emmanuel Halberstadt. The
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part of the theorem that concerns the case $p\equiv 3\!\!\pmod{4}$ is
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given in~\Merel. In this paper, we give the details that permit our
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treating the more difficult case in which $p\equiv 1 \!\!\pmod{4}$.
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We treat this last case with the aid of Proposition~2 below, which is
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not present in \lc.
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The case $p=13$ is currently under investigation by Marusia Rebolledo,
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as part of her Ph.D.{} thesis.
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\bigskip\noindent
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{\bf 1. Counterexamples define points on $X_0(p)(\B Q(\sqrt{p}))$}
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\bigskip
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First we recall some of the results and notation of \Merel.
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Let $S_2(\Gamma_0(p))$ denote the space of cusp forms of weight~$2$ for
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the congruence subgroup $\Gamma_0(p)$. Denote by $\B T$ the
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subring of ${\rm End}\,S_2(\Gamma_0(p))$ generated by the
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Hecke operators $T_n$ for all integers~$n$.
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Let $f\in S_2(\Gamma_0(p))$ have $q$-expansion
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$\sum_{n=1}^\infty a_nq^n$. When $\chi$ is a Dirichlet character,
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denote by $L(f,\chi,s)$ the entire function which extends the
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Dirichlet series $\sum_{n=1}^\infty a_n\chi(n)/n^s$.
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Let $S$ be the set of isomorphism classes of supersingular elliptic
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curves in characteristic~$p$. Denote by $\Delta_S$ the group formed
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by the divisors of degree~$0$ with support on~$S$. It is equipped with
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a structure of $\B T$-module (induced, for example, from the action
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of the Hecke correspondences on the fiber at~$p$ of the regular minimal
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model of $X_0(p)$ over $\B Z$).
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Let $j\in\bar\B F_p-J_S$, where $J_S$ denotes the set of supersingular modular
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invariants. We denote by $\iota_j$ the homomorphism of
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groups $\Delta_S\raw \bar\B F_p$ that associates to $\sum_E n_E[E]$
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the quantity
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$\sum_E n_E/(j-j(E))$, where $j(E)$ denotes the modular invariant of~$E$.
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One says that an element $j\in\B F_p$ is {\it anomalous}
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if there exists an elliptic curve over $\B F_p$ with modular invariant~$j$
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that possesses an $\B F_p$-rational point of order~$p$
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(then necessarily $j\notin{}J_S$).
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Let~$p$ be a prime that is congruent to~$1$ modulo~$4$.
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In the following proposition we prove, under a hypothesis on~$p$, that
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if~$E$ is an elliptic curve over $\B Q(\mu_p)$ all of whose torsion is
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$\B Q(\mu_p)$-rational, then for each subgroup $C\subset{}E(\bar\B Q)$
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of order~$p$,
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the point $(E,C)$ on $X_0(p)$ is defined over $\B Q(\sqrt{p})$. As we
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will see in Proposition~2, this $\B Q(\sqrt{p})$-rationality
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conclusion is contrary to fact, from which we conclude that such
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elliptic curves~$E$ do not exist when the hypothesis on~$p$
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is satisfied. In Section~3 we verify this hypothesis
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for $p=11$ and $13 < p < 1000$.
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\bigskip
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\prop{1}{\it Suppose that~$p$ is congruent to~$1$ modulo~$4$.
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Suppose that for all anomalous
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$j\in\B F_p$ and all
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non-quadratic Dirichlet characters $\chi \colon (\B Z/p\B Z)^*\raw \B
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C^*$, there exists $t_\chi\in \B T$ and
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$\delta\in\Delta_S$ such that $L(f,\chi,1)\ne0$ for every newform
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$f\in t_\chi S_2(\Gamma_0(p))$ and
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$\iota_j(t_\chi\delta)\ne0$.
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Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the
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points of order~$p$ of
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$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.
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Then for all subgroups~$C$ of order~$p$ of $E(\bar \B Q)$, there exists an
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elliptic curve $E_C$ over $\B Q(\sqrt p)$ equipped with a
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$\B Q(\sqrt p)$-rational subgroup $D_C$ of order~$p$, and
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the pairs $(E,C)$ and $(E_C,D_C)$ are $\bar \B Q$-isomorphic.}
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\dm We prove the proposition using the results of~\Merel.
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The hypothesis
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$\iota_j(t_\chi\delta)\ne0$ forces $t_\chi\notin p\B T$
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and, {\it a fortiori}, $t_\chi\ne0$; in addition,
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the non-vanishing hypothesis on the $L$-series
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forces the hypothesis $H_p(\chi)$ of \lc, introduction.
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By assumption, hypothesis $H_p(\chi)$ is satisfied for all
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non-quadratic Dirichlet characters~$\chi$ of conductor~$p$.
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Thus Corollary~3 of Proposition~6 of \lc{} implies that~$E$ has
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potentially good reduction at the prime ideal
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$\cP$ of $\B Z[\mu_p]$ that lies above~$p$.
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Denote by~$j$ the modular invariant of the fiber at~$\cP$ of the
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N\'eron model of~$E$.
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According to the corollary of Proposition~15 of \lc,
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$j$ is anomalous.
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Let~$C$ be a subgroup of $E(\bar\B Q)$ of order~$p$.
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By assumption~$E$ is an elliptic curve over~$\B Q(\mu_p)$ whose points
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of order~$p$ are all $\B Q(\mu_p)$-rational, so
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the pair $(E,C)$ defines a $\B Q(\mu_p)$-rational point~$P$
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of the modular curve $X_0(p)$.
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Consider the morphism $\phi_{\chi}=\phi_{t_\chi}:X_0(p)\rightarrow J_0(p)$
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obtained by composing the standard embedding of $X_0(p)$ into $J_0(p)$
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with $t_{\chi}$. As in section 1.3 of \lc, $\phi_{\chi}$
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extends to a map from the minimal regular model of $X_0(p)$ to the
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N\'eron model of $J_0(p)$.
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When $\iota_j(t_\chi\delta)\ne0$, this map is a formal
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immersion at the point $P_{/\B F_p}$, according to \lc,
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Proposition~4. The hypothesis that $L(f,\chi,1)\ne0$ for
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every newform
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$f\in t_\chi S_2(\Gamma_0(p))$, translates into $L(t_\chi J_0(p),
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\chi,1)\ne0$, which in turn implies that the $\chi$-isotypical
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component of
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$t_\chi J_0(p)(\B Q(\mu_p))$ is finite (this is Kato's theorem, see the
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discussion in section 1.5 of \lc).
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We can then apply Corollary~1 of Proposition~6 of \lc. This proves
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that~$P$ is
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$\B Q(\sqrt p)$-rational, which translates into the conclusion of
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Proposition~1.
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\bigskip
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\rem{1}
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Proposition~1 is true even under the weaker hypothesis
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that $t_{\chi}$ lies in $\B T\otimes \B Z[\chi]$, which acts $\B
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Z[\chi]$-linearly on modular forms.
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\bigskip\bigskip\noindent
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{\bf 2. Elliptic curves and quadratic fields}
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\bigskip
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\prop{2}{\it Let~$p$ be a prime number $>5$ and congruent to~$1$
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modulo~$4$. Let~$E$ be an elliptic curve over $\bar\B Q$.
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There exists a subgroup~$C\subset{}E(\bar\B Q)$ of order~$p$
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such that $(E,C)$ can not be defined over $\B Q(\sqrt{p})$.}
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\dm
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We procede by contradiction, i.e., we assume that for
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all cyclic subgroups~$C$ of order~$p$ of $E(\bar\B Q)$,
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the pair $(E,C)$ can be defined over ${\B Q(\sqrt{p})}$.
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We choose such a pair $(E_0,C_0)$ over ${\B Q(\sqrt{p})}$.
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Assume first that all twists of $E$ are quadratic, i.e.
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that $j(E)$ is neither~$0$ nor $1728$.
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We show that the group
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$\Gal(\bar\B Q/{\B Q(\sqrt p)})$ acts by scalars
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on the $\B F_p$-vector space $E_0(\bar\B Q)[p]$. For this it
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suffices to show that all subgroups of order~$p$ of
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$E_0(\bar\B Q)[p]$ are stable by $\Gal(\bar\B Q/{\B Q(\sqrt p)})$.
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Suppose $C_1$ is a cyclic subgroup of order~$p$ of $E_0(\bar\B Q)[p]$.
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By assumption, there exists a quadratic twist $E_1$ of $E_0$ and
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a cyclic subgroup $C_1'$ of $E_1(\bar\B Q)[p]$
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that is defined over $\B Q(\sqrt{p})$, such that
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the image of $C_1$ by the isomorphism $E_0\simeq E_1$ is $C'_1$.
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Since $\Gal(\bar\B Q/{\B Q(\sqrt p)})$ leaves $C_1'$ stable and
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the action of $\Gal(\bar\B Q/\B Q(\sqrt p))$
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on $E_0(\bar\B Q)[p]$ is
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a quadratic twist of the action on $E_1(\bar\B Q)[p]$,
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we see that $\Gal(\bar\B Q/\B Q(\sqrt p))$ leaves $C_1$ stable.
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Thus $\Gal(\bar\B Q/\B Q(\sqrt p))$ fixes all lines in
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$E_0(\bar\B Q)[p]$, and hence
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acts by scalars. Denote by~$\alpha$ the corresponding character
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of $\Gal(\bar\B Q/\B Q(\sqrt p))$.
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Because of the Weil pairing, $\alpha^2$ coincides
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with the cyclotomic character modulo~$p$, and it factors through
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$\Gal(\B Q(\mu_p)/\B Q(\sqrt p))$. But, when
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$p\equiv 1\!\!\pmod 4$, the group $\Gal(\B Q(\mu_p)/{\B Q(\sqrt p)})$ is of
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even order, and the characters modulo~$p$ form a group generated by the
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reduction modulo~$p$ of the cyclotomic character, which, therefore,
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can not be a square.
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Next suppose that $j(E)=0$ or $j(E)=1728$. Indeed, in these
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two cases~$E$ has
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complex multiplication by an order of
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$K=\B Q[\sqrt{-3}]$ or $\B Q[\sqrt{-1}]$. Let $d_K=3$ or $d_K=2$ in
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these two cases respectively. Let $C$ be a subgroup of order $p$ of
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$E(\bar\B Q)$. Consider
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the map
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$\rho_0 : \Gal(\bar \B Q/\B Q(\sqrt p))\longrightarrow{\rm
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Aut}\,E_0(\bar\B Q)[p]$. Since $E$ has complex multiplication, the image
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of $\rho_0$ has no element of order~$p$. Therefore, there are at least two
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subgroups, including $C_0$, of order~$p$ of
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$E(\bar\B Q)$ stable under the image of $\rho_0$. Call the other subgroup
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$C_1$. Let $C_2$ be a subgroup of order $p$ of $E(\bar\B Q)$ which is
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distinct from $C_0$ and $C_1$. The pair $(E,C_2)$ can be defined over
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$\B Q(\sqrt p)$.
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Therefore, there exists an extension field $K_2$ of $\B Q(\sqrt p)$,
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whose degree $d_2$ divides $2d_K$, such that
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the image of the restriction of $\rho_0$ to
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$\Gal(\bar \B Q/K_2)$ leaves stable three distinct subgroups of
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order~$p$ of $E_0(\bar\B Q)$, and therefore consists only of scalars.
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If $d_2\le2$, one concludes as in the cases where
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$j(E)\ne0$ and $j(E)\ne 1728$. We suppose now that $d_2>2$.
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The projective image of $\rho_0$ has order $d_K$.
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Since~$E$ is an elliptic curve over $\bar \B Q$ with complex
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multiplication by a field of class number one, there is a model
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for~$E$ that is defined over $\B Q$. Consider the map
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$\rho$ : $\Gal(\bar \B Q/\B Q)\longrightarrow{\rm Aut}\,E(\bar\B Q)[p]$.
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By the theory of complex multiplication, the projective image of
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$\rho$ has order $2(p+1)$ or $2(p-1)$. There exists a field extension~$L$
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of degree dividing
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$d_K$ of $\B Q(\sqrt p)$ such that the restrictions to
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$\Gal(\bar \B Q/L)$ of the projective
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images of $\rho$ and $\rho_0$ coincide.
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Therefore one has $(p-1)|d_K^2$ or $(p+1)|d_K^2$.
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This imposes $p=5$ and $d_K=2$.
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\bigskip\bigskip\noindent
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{\bf 3. Verification of the hypothesis of Proposition~1}
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\bigskip
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Let~$p$ be a prime number. In this section we explain how we used
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a computer to verify that the second hypothesis of Proposition~1 are
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satisfied for $p=11$ and $13 < p < 1000$. (In the present paper,
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this verification is only required for $p$ that are congruent to~$1$
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modulo~$4$.)
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We first list the anomalous $j$-invariants $j\in\B F_p$. Since~$p$ is
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fairly small in the range of our computations, we created this list by
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simply enumerating all of the elliptic curves over $\B F_p$ and
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counting the number of points on each curve. For example, when $p=31$
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the anomalous $j$-invariants are $j=10,14$.
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Let~$\chi: \B Z/p\B Z\raw \B C$ be a non-quadratic
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Dirichlet character, and denote by $\B Z[\chi]$ the
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subring of $\B Q(\zeta_{p-1})$ generated by the image of~$\chi$.
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Denote by $S_2(\Gamma_0(p);\B Z)$ the set of modular forms
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$f\in S_2(\Gamma_0(p))$ whose Fourier expansion at the cusp~$\infty$
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lies in $\B Z[[q]]$.
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We study the $\B T$-modules $\B T$, $\Delta_S$, and $S_2(\Gamma_0(p);\B Z)$.
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After extension of scalars to~$\B Q$, these
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are $\B T\otimes\B Q$-modules that are free of rank~$1$, of which the
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irreducible sub-$\B T\otimes\B Q$ modules are the annihilators of the
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minimal prime ideals of $\B T$. We compute a list of the minimal
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prime ideals of $\B T$ by computing appropriate kernels and
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characteristic polynomials of Hecke operators of small index on
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$\Delta_S$, which we find using the graph method of Mestre and
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Oesterl\'e \Mes{}.
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Having computed the minimal prime ideals of $\B T$, we verify that
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some nontrivial ideal $\cI$ of $\B T$ (always a minimal prime
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ideal in the range of our computations) simultaneously satisfies
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the following three conditions:
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\vskip 2ex
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1)
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For each anomalous $j$-invariant, there exists $x\in\Delta_S$ such that
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$\cI x=0$ and $\iota_j(x)\ne 0$.\vskip 1ex
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2) Each of the newforms~$f\in S_2(\Gamma_0(p))$ with
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$\cI f=0$ satisfies $L(f,\chi,1)\ne 0$.
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\vskip 1ex
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3) The image of~$\cI$ in the $\B T$-module $\B T/p\B T$
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is a direct factor.\vskip 2ex
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Let $\cI$ be an ideal of $\B T$. Here is how we verify these conditions
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for $\cI$.
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\bigskip
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{\it \noindent Verification of condition 1.}
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We verified that $\cI$ satisfies the first condition by
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finding a $\B T$-eigenvector~$v$ of $\Delta_S\otimes \bar\B Z$ that is
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annihilated by $\cI$ and satisfies $\iota_j(v)\neq 0$ for all anomalous $j$-invariants. Because $\iota_j$
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is a homomorphism, this implies the existence of~$x$ as in condition 1.
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\bigskip
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{\it \noindent Verification of condition 2.}
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We verified the second condition using modular symbols.
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Our method is purely algebraic, so we do not perform
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any approximate computation of integrals.
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Using the algorithm described in \Cremona, we compute the action of
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the Hecke algebra $\B T$ on the space
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$\Hom_{\B Q[\chi]}(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$. By intersecting
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the kernels of appropriate elements of $\B T$, we find a basis
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$\varphi_1,\ldots,\varphi_n$ for the subspace of
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$\Hom_{\B Q[\chi]}(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$ that is
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annihilated by~$\cI$. Let~$\Phi_{\cI}=\varphi_1\times \cdots \times
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\varphi_n$ denote the linear map
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$H_1(X_0(p);\B Q[\chi])\raw \B Q[\chi]^n$
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defined by the $\varphi_i$.
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Let $\B T_{\B Q[\chi]} = \B T \otimes \B Q[\chi]$, where $\B Q[\chi]$
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is the number field generated the image of~$\chi$.
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The {\it $\chi$-twisted winding element} (denoted $\theta_\chi$ in
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\Merel)
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$$\B e_\chi = \sum_{a\in (\B Z/p\B Z)^*} \bar\chi(a)
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\Big\{\infty,{a \over p}\Big\}$$
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generates the {\it $\chi$-twisted winding submodule}
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$\B T_{\B Q[\chi]}\cdot \B e_\chi$. To compute this submodule,
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we use that $\B T$ is generated, even as a $\B Z$-module,
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by $T_1,T_2,\ldots, T_b$, for any $b\geq (p+1)/6$
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(see \Agashe).
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\bigskip
387
\lem 3
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{\it Let $\cI$ be a minimal prime ideal of~$\B T$, and
389
let $\chi:(\B Z/N\B Z)^*\raw \B C^*$
390
be a nontrivial Dirichlet character.
391
Then the dimension of the $\B Q[\chi]$-vector space $\Phi_{\cI}(\B T_{\B
392
Q[\chi]}
393
\cdot
394
\B e_\chi)$ is equal to the cardinality of the set of newforms~$f$ such
395
that
396
$\cI f=0$ and $L(f,\chi,1) \neq 0$.
397
}
398
399
\dm
400
We have
401
$$\dim_{\B Q[\chi]} \Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_\chi)
402
= \dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi).$$
403
This dimension is invariant upon changing the basis
404
$\varphi_1,\ldots, \varphi_n$ used to define $\Phi_{\cI}$.
405
In particular, over $\B C$ there is a basis
406
$\varphi_1',\ldots, \varphi_n'$ so that the resulting
407
map $\Phi_{\cI}'$ satisfies
408
$$\Phi_{\cI}'(x) =
409
\Bigl(\rp(\int_x f^{(1)}), \ip(\int_x f^{(1)}),
410
\ldots,
411
\rp(\int_x f^{(d)}),\ip(\int_x f^{(d)})\Bigr),$$
412
where $f^{(1)}, \ldots, f^{(d)}$ are the Galois conjugates
413
of a newform~$f^{(1)}=\sum a_n^{(1)} q^n$ such that $\cI f^{(1)}=0$.
414
Furthermore, $\Phi_{\cI}'$ is a $\B T_{\B C}$-module homomorphism
415
if we declare that $\B T_{\B C}$ acts on $\B R^{2d} = \B C^d$ via
416
$$T_n(x_1,y_1, \ldots, x_d, y_d) =
417
T_n(z_1,\ldots,z_d) = (a_n^{(1)} z_1,\ldots, a_n^{(d)}z_d),$$
418
where $z_j = x_j + i y_j$ and
419
the $a_n^{(j)}$ are Fourier coefficients of the $f^{(j)}$.
420
421
As explained in Section 2.2 of~\Merel,
422
$\int_{\B e_\chi} f = *\cdot L(f,\chi,1)$, where~$*$
423
is some nonzero real or pure-imaginary complex number,
424
according to whether $\chi(-1)$ equals~$1$ or~$-1$,
425
respectively.
426
Combining this observation with the equality
427
$$\dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi)
428
= \dim_{\B C} (\B T_{\B C}\cdot \Phi_{\cI}(\B e_\chi)),$$
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and that the image of $\B T_{\B C}$ in $\End(\B C^d)$ is
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equal to the diagonal matrices, proves the asserted equality.
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\bigskip
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\rem{2} The dimension of $\Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_{\chi})$
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is unchanged if~$\chi$ is
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replaced by a Galois-conjugate character.
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\bigskip
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In practice, computations over the cyclotomic field $\B Q[\chi]$ are
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extremely expensive. Fortunately, for our application it suffices to
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give a lower bound on the dimension appearing in the lemma. Such a
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bound can be efficiently obtained by instead computing the reductions
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of~$\Phi$,~$\chi$, and the $\chi$-twisted winding submodule modulo a
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suitable maximal ideal of the ring of integers of $\B Q[\chi]$ that
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splits completely; this amounts to performing the above linear algebra
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over a relatively small finite field $\B F_\ell$ where~$\ell$ is
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congruent to~$1$ modulo $p-1$.
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\bigskip
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\rem{3} For every newform~$f$ in $S_2(\Gamma_0(p))$, with $p\leq 1000$,
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and every mod~$p$ Dirichlet character~$\chi$, we found that
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$L(f,\chi,1)\neq 0$ if and only if
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$L(f^{\sigma},\chi,1)\neq 0$ for all conjugates $f^{\sigma}$
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of~$f$.
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More generally, for any~$f$ and~$\chi$, this equivalence holds if
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$\B Q[\chi]$ is linearly disjoint from the
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field $K_f=(\B T/\cI)\otimes\B Q$.
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The first few primes
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for which there is a form~$f$ and a mod~$p$ character~$\chi$
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such that the linear disjointness hypothesis fails are
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$p=31, 113, 127$, and $191$.
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The analogue of this nonvanishing observation is false if we instead consider
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newforms on $\Gamma_1(p)$ and allow~$\chi$ to be arbitrary.
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For example, let~$f$
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be one of the two Galois-conjugate newforms in $S_2(\Gamma_1(13))$.
467
Then there is a character $\chi:(\B Z/7\B Z)^*\raw \B C^*$ of
468
order~$3$ such that $L(f,\chi,1) = 0$ and $L(f^{\sigma},\chi,1)\neq 0$.
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\bigskip
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{\it \noindent Verification of condition 3.}
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The third condition is satisfied for all $p<10000$, except possibly
475
$p = 389$,
476
because we have verified that the discriminant of $\B T$ is
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prime to~$p$ for all such $p\neq 389$,
478
so the ring $\B T/p\B T$ is semisimple.
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The discriminant computation was carried out by the second author
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as follows.
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Using the method of \Mes{}, we computed discrimininants of characteristic
482
polynomials mod~$p$ of the Hecke operators $T_2$, $T_3$, $T_5$, and $T_7$.
483
In the few cases when all four of these characteristic polynomials had
484
discriminant equal to~$0$ mod~$p$, we resorted to modular symbols to
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compute several more characteristic polynomials until we found one
486
having nonzero discriminant modulo~$p$.
487
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We consider the remaining case $p=389$ in detail. There are exactly
489
five minimal prime ideals of $\B T$, which we denote $\cP_1$, $\cP_2$,
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$\cP_3$, $\cP_6$, and $\cP_{20}$, where the quotient field of $\B
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T/\cP_i$ has dimension~$i$. The discriminant of the characteristic
492
polynomial of $T_2$ is exactly divisible by $389$. Since the field of
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fractions of $\B T/\cP_{20}$ has discriminant divisible by $389$, we
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see that $389$ is not the residue characteristic of any congruence
495
prime. 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
498
nonquadratic characters $\chi:(\B Z/p\B Z)^*\rightarrow \B C^*$ have
499
orders $1, 4, 97, 193, 388$. We must verify that for each of these
500
degrees, one of the ideals $\cP_i$ satisfies conditions 1--3. We
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check as above that conditions 1--3 for~$\chi$ of order~$4$ are
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satisfied by $\cP_2$ and conditions 1--3 for~$\chi$ of order greater
503
than~$4$ are satisfied by $\cP_1$. When~$\chi$ is the trivial
504
character, conditions~1--3 are satisfied only by $\cP_{20}$.
505
506
\bigskip
507
{\it \noindent Summary.}
508
509
For each prime $p<1000$ different than $2,3,5,7, 13$, we
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verified the existence of an ideal that satisfies the three conditions
511
given above, as follows. We consider each Galois conjugacy class of
512
non-quadratic characters~$\chi$. We find a single newform~$f$ such
513
that $L(f,\chi,1)\ne 0$ for all conjugates of~$f$ and of~$\chi$. Then
514
we let $\cI$ be the annihilator of~$f$, and try to verify condition~1
515
for {\it all} of the anamolous $j$-invariants in $\B F_p$.
516
When the three conditions are satisfied for an ideal~$\cI$ of~$\B T$,
517
there exists $t_\chi\in\B T$ that is annihilated by $\cI$ and is the
518
inverse 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
522
characteristic~$p$, it follows that $\delta$ is annihilated by~$\cI$ and
523
$t_\chi\in 1+p\B T+\cP$).
524
Every newform $f\in t_\chi S_2(\Gamma_0(p))$ satisfies
525
$\cI f=0$, and therefore, by our second condition, $L(f,\chi,1)\ne0$.
526
The pair $(t_\chi,\delta)$ then satisfies the conditions required by
527
Proposition~1.
528
529
\bigskip
530
\noindent
531
{\it Acknowledgment}: We would like to thank Barry Mazur for
532
providing us several useful comments.
533
534
\bigskip\bigskip
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\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},
544
C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), no. 5, 369--374.
545
\vskip 1ex
546
547
\item[]{\Cremona} {\pc J. Cremona},
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{\it Algorithms for modular elliptic curves},
549
second 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
555
elliptiques},
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To appear in Duke Math. Journal.
557
\vskip 1ex
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559
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\item[]{\Mes} {\pc J.-F. Mestre}, {\it La m\'ethode des graphes.
561
Exemples et applications}, Proceedings of the international
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conference on class numbers and fundamental units of algebraic number
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fields (Katata), 217--242, (1986).
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\end{enumerate}
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\end{document}
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