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Author: William A. Stein
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\def\ie{{\it i.e.~}} %id est
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\def\lc{{\it loc.cit.}} %loc. cit.
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\def\af{{\it a fortiori\ }} %a fortiori
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\def\mod#1{\ \hbox{{\rm mod}$#1$}} %modulo
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\def\eps{\varepsilon} %nom court pour varepsilon
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\def\sh{\mathop{\rm sh}\nolimits} %sinus hyperbolique
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\def\ch{\mathop{\rm ch}\nolimits} %cosinus hyperbolique
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\def\tgh{\mathop{\rm th}\nolimits} %tangente hyperbolique
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\def\tg{\mathop{\rm tg}\nolimits} %tangente
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\def\Arcsin{\mathop{\rm Arc\,sin}\nolimits} %Arc sinus
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\def\Arccos{\mathop{\rm Arc\,cos}\nolimits} %Arc cosinus
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\def\Arctg{\mathop{\rm Arc\,tg}\nolimits} %Arc tangente
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\def\Argsh{\mathop{\rm Arg\,sh}\nolimits} %Arg sinus hyperbolique
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\def\Argch{\mathop{\rm Arg\,ch}\nolimits} %Arg cosinus hyperbolique
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\def\Argth{\mathop{\rm Arg\,th}\nolimits} %Arg tangente hyperbolique
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\def\RE{\mathop{\rm Re}\nolimits} %Partie r�elle
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\def\IM{\mathop{\rm Im}\nolimits} %Partie imaginaire
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\def\H{\mathop{\bf H}\nolimits} %1/2 plan
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\font\titchap=cmr17 at 20pt % pour les titres de chapitres
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\font\pc=cmcsc10 % pour les titres de paragraphe, propositions,etc.
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\def\th#1{\noindent{\pc Theorem}\ #1. --- \ignorespaces} %Th�or�me 1.-
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\def\prop#1{\noindent{\pc Proposition}\ #1. --- \ignorespaces}
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%Proposition 1.-
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\def\Def#1{\noindent{\pc Definition}\ #1. --- \ignorespaces}
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%D�finition 1.-
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\def\cor#1{\noindent{\pc Corollary}\ #1. --- \ignorespaces}
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%Corollaire 1.-
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\def\conj#1{\noindent{\pc Conjecture}\ #1. --- \ignorespaces}
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%Conjecture 1.-
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\def\lem#1{\noindent{\it Lemma}\ #1. --- \ignorespaces} %Lemme 1.-
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\def\rem#1{\noindent{\it Remark}\ #1 : \ignorespaces} %Remarque 1.-
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\def\exe#1{\noindent{\it Example}\ #1 : \ignorespaces} %Exemple 1.-
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\def\exr#1{\noindent{\it Exercise}\ #1 : \ignorespaces} %Exercice 1.-
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\def\rems{\noindent{\it Remark} : \ignorespaces} %Remarques .-
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\def\exes{\noindent{\it Examples} : \ignorespaces} %Exemples .-
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\def\exrs{\noindent{\it Exercises} : \ignorespaces} %Exercices .-
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\def\thp{\noindent{\pc Theorem}. --- \ignorespaces} %Th�or�me .-
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\def\propp{\noindent{\pc Proposition}. --- \ignorespaces}
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%Proposition .-
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\def\Defp{\noindent{\pc Definition}. --- \ignorespaces}
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%D�finition .-
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\def\corp{\noindent{\pc Corollary}. --- \ignorespaces}
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%Corollaire .-
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\def\conjp{\noindent{\pc Conjecture}. --- \ignorespaces}
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%Conjecture .-
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\def\lemp{\noindent{\it Lemma}. --- \ignorespaces} %Lemme .-
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\def\scholie{\noindent{\it Scholie}. --- \ignorespaces} %Scholie .-
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\def\remp{\noindent{\it Remark} : \ignorespaces} %Remarque .-
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\def\exep{\noindent{\it Example} : \ignorespaces} %Exemple .-
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\def\dm{\noindent{\it Proof}. --- \ignorespaces}
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%Exemple .-
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\def\spec{{\rm Spec\, }}
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\def\cP{{\cal P}}
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\def\Mazur{{$[1]$}}
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\def\Merel{{$[2]$}}
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\def\MO{{$[3]$}}
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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
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\bigskip\bigskip\noindent
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{\bf Introduction}
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\bigskip\bigskip
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Let~$p$ be a prime number.
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Let $\bar\B Q$ be an algebraic closure of $\B Q$.
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Denote by $\B Q(\mu_p)$ the cyclotomic subfield of $\bar\B Q$
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generated by the $p$th roots of unity.
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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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\bigskip
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\th{}{\it One has $p>1000$, $p<6$ or $p=13$.}
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\bigskip
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We note that the case $p=7$ was treated by Emmanuel Halberstadt.
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The part of the theorem that concerns the case
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$p\equiv 3\pmod 4$ is given in~\Merel.
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We propose to give the details that permit our treating the
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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 {\it loc. cit.}
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It may be possible to exclude the case when $p=13$
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by studying the modular curve $X_1(13)$ and its Jacobian $J_1(13)$.
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\bigskip\noindent
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{\bf 1. We recall the results of \Merel}
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\bigskip
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Denote by
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$J_0(p)$ the Jacobian of the modular curve $X_0(p)$ (whose points are
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isomorphism classes of
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generalized elliptic curves equipped with a cyclic subgroup of order~$p$).
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Consider the subring $\B T$ of ${\rm End}\,J_0(p)$
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generated by the Hecke operators. We refer the reader to~\Mazur{}
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for an in-depth study of these objects.
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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 (deduced, 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 $j\in\B F_p$ {\it presents an anomaly}
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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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\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
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$j\in\B F_p$ that present an anomaly and all
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non-quadratic Dirichlet characters $\chi$ : $(\B Z/p\B Z)^*\raw \B C$,
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there exists $t_\chi\in \B T$ and
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$\delta\in\Delta_S$ such that $L(t_\chi J_0(p),\chi,1)\ne0$ and
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$\iota_j(t_\chi\delta)\ne0$.
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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 $D_C$ or 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 indicate how this is deduced from~\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 {\it loc. cit}, introduction.
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According to Corollary~3 of Proposition~6 of {\it loc. cit},~$E$ 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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once we know that hypothesis $H_p(\chi)$ is satisfied for all
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non-quadratic Dirichlet characters~$\chi$ of conductor~$p$
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(this is the case by hypothesis).
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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 {\it loc. cit.},
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$j$~presents an anomaly.
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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 $\phi_{\chi}=\phi_{t_\chi}$ (see {\it loc. cit.}
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section 1.3). When $\iota_j(t_\chi\delta)\ne0$, this is a formal
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immersion at the point $P_{/\B F_p}$, according to {\it loc. cit.},
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Proposition~4. The hypothesis that $L(t_\chi J_0(p),\chi,1)\ne0$
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implies that the $\chi$-isotypical component of $t_\chi J_0(p)(\B
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Q(\mu_p))$ is finite (this is Kato's theorem, see the discussion in
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{\it loc. cit.} section 1.5). We can then apply Corollary~1 of
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Proposition~6 of {\it loc. cit}. This proves that $P$ is $\B Q(\sqrt
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p)$-rational; this translates into the conclusion of Proposition~1.
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\bigskip\bigskip\noindent
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{\bf 2. A lemma about elliptic curves}
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\bigskip
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\prop{2}{\it Let~$p$ be a prime number that is 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 cyclic subgroup~$C$ of order~$p$ of
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$E(\bar\B Q)[p]$, such that for all elliptic curves~$E'$ over
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${\B Q(\sqrt{p})}$
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equipped with a ${\B Q(\sqrt p)}$-rational subgroup~$C'$, the
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pairs $(E,C)$ and $(E', C')$ are not $\bar\B Q$-isomorphic.
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}
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\dm
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We procede by contradiction.
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Let $E_0$ be an elliptic curve over $\B Q(\sqrt p)$
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that is $\bar\B Q$ isomorphic to $E$ (it exists by hypothesis).
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We first show that the subgroup
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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 $E_0(\bar\B Q)[p]$.
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Denote by $X(p)$ the algebraic curve over $\B Q$ that parametrizes classes
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(fine because $p>2$) of generalized elliptic curves equipped with an
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embedding
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$\pi$ : $(\B Z/p\B Z)^2\raw E[p]$.
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Consider the morphism (of algebraic varieties
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over $\B Q$) $\phi$ :
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$X(p)\raw X_0(p)^{\B P^1(\B F_p)}$ that to $(E,\pi)$ associates
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$\prod_{t\in\B P^1(\B F_p)}(E,\pi(t))$.
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Denote by $X_\Delta(p)$ the image of~$\phi$.
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The covering (of algebraic curves over $\B Q$)
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$\phi'$ :
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$X(p)\raw X_\Delta(p)$ is Galois with Galois group isomorphic to
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$\B F_p^*$ (the action being deduced from the scalar action of
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$\B F_p^*$ on $E[p]$).
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Let $\pi_0$ be an embedding $(\B Z/p\B Z)^2\raw E_0[p]$.
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Denote by~$P$ the $\bar\B Q$-rational point of $X(p)$ deduced from
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$(E_0,\pi_0)$.
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Its image by~$\phi$ is $\B Q(\sqrt p)$-rational by hypothesis.
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We have then a character $\alpha$ : $\Gal(\bar\B Q/\B Q(\sqrt p))\raw\B
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F_p^*$ such that $\sigma(P)=\alpha(\sigma).P$ ($\sigma\in \Gal(\bar\B Q/\B
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Q(\sqrt p))$). In other words, $\Gal(\bar\B Q/\B Q(\sqrt p))$ acts
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by scalars on $E_0(\bar\B Q)[p]$ via the character~$\alpha$.
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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;
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it thus can not be a square.
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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.
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We now indicate how to computationally
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verify that the hypothesis of Proposition~1 are
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satisfied. We denote by $S_2(\Gamma_0(p))$ the group formed by the cuspidal
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modular forms of weight~$2$ for the congruence subgroup
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$\Gamma_0(p)$ which have integer Fourier coefficients.
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By a newform we mean a newform in the
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$\bar\B Q$-vector space generated by $S_2(\Gamma_0(p))$.
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We first list the modular $j$-invariants that present an anomaly;
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one is only interested in these.
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Let~$\chi$ be a Dirichlet character $(\B Z/p\B Z)^*\raw \B C^*$ and
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suppose $j\in\B P^1(\B F_p)$ presents an anomaly.
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We study the following three $\B T$-modules: $\B T$, $\Delta_S$, and
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$S_2(\Gamma_0(p))$. After extension of scalars to~$\B Q$,
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these are $\B T\otimes\B Q$-modules that are free of rank~$1$, of which
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the irreducible sub-$\B T$ modules are the annihilators of the
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minimal prime ideals of $\B T$.
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We establish a list of the minimal prime ideals of $\B T$.
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From an algorithmic point of view,
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these are generated by the
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minimal polynomials of small degree of Hecke operators of small index;
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we find these polynomials by utilizing the
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graph method of Mestre and Oesterl\'e \MO.
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For each ideal $\cP$ of $\B T$, we can determine if
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the following three conditions are satisfied:
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\noindent i)
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There exists $x\in\Delta_S$ such that
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$\cP x=0$ and $\iota_j(x)\ne 0$
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for all anomalous~$j$.
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\noindent
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ii) All newforms~$f$ such that $\cP f=0$ satisfy $L(f,\chi,1)\ne 0$.
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\noindent
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iii) The image of~$\cP$ in the $\B T$-module $\B T/p\B T$
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is a direct factor.
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We study the case when~$\cP$ is a minimal prime.
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The first condition is studied by the graph method of Mestre and Oesterl\'e.
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The second condition is verified using the theory of modular symbols
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(without recourse to the calculation of integrals).
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The third condition is verified for $p<1000$ and $p\ne 389$,
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because one of us has verified that the discriminant of
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$\B T$ is prime to~$p$ and so the ring $\B T/p\B T$ is semi-simple.
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In fact, we verified this for $p<10000$; this was accomplished by
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computing discrimininants of
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characteristic polynomials mod~$p$ of $T_p$, for $p\leq 7$,
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using the method of graphs. If all four characteristic polynomials
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had discriminant equal to~$0$ mod~$p$, we resorted to modular symbols
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to compute several more characteristic polynomials until one is
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found with nonzero discriminant mod~$p$.
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For all prime numbers~$p$ different than
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$2,3,5,7, 13$ and $389$, we thus
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verified the existence of a minimal prime ideal
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that satisfies the three conditions given above.
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In the case when $p=389$,
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we find $\cP$ in the following way.
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There exists two minimal prime ideals $\cP_1$ and $\cP_2$
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that satisfy the first two conditions.
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Because the discriminant of~$\B T$ has $p$-adic valuation~$1$,
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the image of at least one of the ideals
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$\cP_1$, $\cP_2$ and $\cP_1\cap\cP_2$ is a direct factor of
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$\B T/p\B T$. We choose~$\cP$ to be one of these that is
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appropriate.
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When our three conditions are satisfied for an ideal~$\cP$
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of~$\B T$, there exists
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$t_\chi\in\B T$ which is annihilated by $\cP$ and
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is the inverse image of a projector of
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$\B T/p\B T$ on the complement of $\cP+p\B T$.
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Putting $\delta=x$, one has $\iota_j(t_\chi \delta)=\iota_j(\delta)\ne0$
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(because $\iota_j$ takes its values in characteristic~$p$,
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$\delta$ is annihilated by~$\cP$ and $t_\chi\in 1+p\B T+\cP$).
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The assertion $L(t_\chi J_0(p),\chi,1)\ne0$ follows
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easily from the second condition,
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since $L(t_\chi J_0(p),\chi,s)$ is the product of
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$L(f,\chi,s)$, where~$f$ runs over the
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newforms that are not annihilated by~$t_\chi$.
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The pair $(t_\chi,\delta)$ thus satisfy the required conditions.
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\bigskip\bigskip
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\centerline{\pc Bibliography}
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\bigskip
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\item{\Mazur}{\pc B. Mazur},
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{\it Modular curves and the Eisenstein ideal},
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Pub. math. de l'IHES {\bf 47}, 33--186, {\oldstyle 1977}.
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\item{\Merel}{\pc L. Merel},
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{\it Sur la nature non cyclotomique des points d'ordre fini des courbes
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elliptiques},
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Pr\'epublication , {\oldstyle
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1999}.
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\item{\MO}{\pc J.-F. Mestre} and {\pc J. Oesterl\'e},
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{\it Courbes elliptiques de conducteur premier}, Manuscrit non publi\'e.
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\end