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\def\sh{\mathop{\rm sh}\nolimits} %sinus 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\Argth{\mathop{\rm Arg\,th}\nolimits} %Arg tangente hyperbolique
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\def\RE{\mathop{\rm Re}\nolimits} %Partie r=C8elle
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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=3Dcmr17 at 20pt % pour les titres de chapitres
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,etc.
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\def\moinsgras{\mathrel{\hbox{\bf -\kern-1pt -}}} %moins gras
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\def\th#1{\noindent{\pc Theorem}\ #1. --- \ignorespaces} %Th=C8or=CBme 1=
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=2E-
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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=C8finition 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 Remarks}: \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=C8or=CBme .=
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-
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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=C8finition .-
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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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-
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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\raw{\longrightarrow}
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\def\Hom{{\rm Hom}}
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\def\Sha{{\cyr Sh}}
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\def\Gal{{\rm Gal}}
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\def\cP{{\cal P}}
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\def\cP{{\cal P}}
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\def\End{{\rm End}}
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%\def\Mazur{{$[1]$}}
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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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\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~$p$ be a prime number.
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Let $\bar\B Q$ be an algebraic closure of $\B Q$, and
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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 $p=3D2,3,5, 13$ or $p>1000$.}
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\bigskip
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We note that the case $p=3D7$ 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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We did not thoroughly study the case $p=3D13$.
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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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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 Hecke operators.
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Let $f\in S_2(\Gamma_0(p))$ have $q$-expansion
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$\sum_{n=3D1}^\infty a_nq^n$. 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 $\sum_{n=3D1}^\infty a_n\chi(n)/n^s$.
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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 modul=
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ar
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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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\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 $\chi$ : $(\B Z/p\B Z)^*\raw \B C$,
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there exists $t_\chi\in \B T$ and
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$\delta\in\Delta_S$ such that $L(f,\chi,1)\ne0$ for every newform
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$f\in t_\chi S_2(\Gamma_0(p))$ and
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=20 $\iota_j(t_\chi\delta)\ne0$.
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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 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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$j$ 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 $\phi_{\chi}=3D\phi_{t_\chi}$ (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 $L(f,\chi,1)\ne0$ 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 {\it loc. cit.} section 1.5). We can then apply
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Corollary~1 of Proposition~6 of {\it loc. cit}. This proves 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{} In this proposition we content ourself with a stronger hypothesis
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than the one generally used in {\it loc. cit}: the Hecke operator
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$t_\chi$ is required to belong to $\B T$ and not to $\B T\otimes\B
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Z[\chi]$.
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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 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, {\it i.e.} we assume that for
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all cyclic subgroup $C$ 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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We show first that such a situation prevents that $j(E)=3D0$ or
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$j(E)=3D1728$. Indeed in those cases
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$E$ has complex multiplication by an order $R_K$ of $K=3D\B Q[\sqrt{-1}]$ or
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$\B Q[\sqrt{-3}]$.
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Consider the map $\rho$ :
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$\Gal(\bar \B Q/\B Q(\sqrt p))\longrightarrow{\rm Aut}\,E_0(\bar\B Q)[p]$.
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If its image had an element of order $p$, so would have
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$\rho(\Gal(\bar \B Q/K(\sqrt p)))$. Let $L_p$ be the class field of ray $p$
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of $K$. It contains $\B Q(\sqrt p)$ since $p\equiv 1\pmod 4$.
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By the theory
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of complex multiplication,
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$\rho(\Gal(\bar \B Q/L_p))$ is trivial. By class field theory, $\Gal(L_p/K)$
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has no element of order $p$, since $K$ has class number $1$. Therefore
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$\rho(\Gal(\bar \B Q/\B Q(\sqrt p)))$ has no element of order $p$.
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Since it is contained in the Borel subgroup of ${\rm Aut}\,E_0(\bar\B Q)[p]$
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which stabilizes $C_0$, it is an abelian group. By the theory of complex
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multiplication, it is the semi-direct product of $\Gal(L_p/K(\sqrt p))$ and
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$\Gal(K(\sqrt p)/\B Q(\sqrt p))\simeq\Gal(\B C/\B R)$. Such a group
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is not abelian since
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$\Gal(L_p/K(\sqrt p))$ is not a $2$-group, hence the contradiction.
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We can now assume that all twists of $E$ are quadratic.
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We now 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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Let $C_1$ be 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 $E_1(\bar\B Q)[p]$ of order $p$, 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 the Galois action on $E_1(\bar\B Q)[p]$ is the one obtained
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by twisting by a character the action on $E_0(\bar\B Q)[p]$.
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The group $\Gal(\bar\B Q/{\B Q(\sqrt p)})$ leaves $C_1$ stable.
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That implies that $\Gal(\bar\B Q/\B Q(\sqrt p))$ acts
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by scalars on $E_0(\bar\B Q)[p]$. Denote by ~$\alpha$ the corresponding
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character 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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\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 to use a
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computer to verify that the hypothesis of Proposition~1 are satisfied.
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We have carried out this verification for $p=3D11$ and $13 < p < 1000$.
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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 $p=3D31$
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the anomalous $j$-invariants are $j=3D10,14$.
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Let~$\chi: (\B Z/p\B Z)^*\raw \B C^*$ denote a non-quadratic
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Dirichlet character and
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suppose $j\in\B P^1(\B F_p)$ is anomalous.
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We study the following three $\B T$-modules: $\B T$, $\Delta_S$, and
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$S_2(\Gamma_0(p);\B Z)$. 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 $\B T$ (usually a minimal prime
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ideal) simultaneously satisfies the following three conditions:
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\vskip 2ex
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1)
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There exists $x\in\Delta_S$ such that
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=20 $\cI x=3D0$ 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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$\cI f=3D0$ 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 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 $\iota_j(v)\neq 0$. Because $\iota_j$
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is a homomorphism, this implies the existence of an~$x$ as in condition 1.
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\bigskip
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{\it \noindent Condition 2.}
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We verified the second condition using modular symbols.
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Our method is purely algebraic, avoiding the need to 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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$\Hom(H_1(X_0(p);\B Q(\chi)),\B Q(\chi))$. By intersecting the kernels
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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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$\Hom(H_1(X_0(p);\B Q(\chi)),\B Q(\chi))$ that is annihilated by~$\cI$.
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Let~$\Phi_{\cI}=3D(\varphi_1,\ldots,\varphi_n)$ denote the linear map
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=20 $H_1(X_0(p);\B Q(\chi))\raw \B Q(\chi)^n$
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defined by the $\varphi_i$.
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Let $\B T_{\B Q(\chi)} =3D \B T \otimes \B Q(\chi)$, 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 by $\theta_\chi$ in
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\Merel)
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=20 $$\B e_\chi =3D \sum_{a\in (\B Z/p\B Z)^*} \bar\chi(a)
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=20 \Big\{\infty,{a \over p}\Big\},$$
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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
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\lem 3
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{\it Let $\cI$ be a minimal prime ideal of~$\B T$, and
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let $\chi:(\B Z/p\B Z)^*\raw \B C^*$
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be a nontrivial Dirichlet character.
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Then the dimension of $\Phi_{\cI}(\B T_{\B Q(\chi)} \cdot \B e_\chi)$ is
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equal to the cardinality of the set of newforms~$f$ such that
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$\cI f=3D0$ and $L(f,\chi,1) \neq 0$.
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}
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\dm
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Since $\B C$ is faithfully flat over $\B Q(\chi)$,
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we have
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$$\dim_{\B Q(\chi)} \Phi_{\cI}(\B T_{\B Q(\chi)}\cdot \B e_\chi)
453
=20 =3D \dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi).$$
454
This dimension is invariant upon changing the basis
455
$\varphi_1,\ldots, \varphi_n$ used to define $\Phi_{\cI}$.
456
In particular, over $\B C$ there is a basis
457
$\varphi_1',\ldots, \varphi_n'$ so that the resulting
458
map $\Phi_{\cI}'$ satisfies
459
$$\Phi_{\cI}'(x) =3D
460
\Bigl(\rp(\int_x f^{(1)}), \ip(\int_x f^{(1)}),
461
\ldots,
462
=20 \rp(\int_x f^{(d)}),\ip(\int_x f^{(d)})\Bigr),$$
463
where $f^{(1)}, \ldots, f^{(d)}$ are the Galois conjugates
464
of a newform~$f^{(1)}=3D\sum a_n^{(1)} q^n$ such that $\cI f^{(1)}=3D0$.
465
=46urthermore, $\Phi_{\cI}'$ is a $\B T_{\B C}$-module homomorphism
466
if we declare that $\B T_{\B C}$ as acts on $\B R^{2d} =3D \B C^d$ via
467
$$T_n(x_1,y_1, \ldots, x_d, y_d) =3D
468
=20 T_n(z_1,\ldots,z_d) =3D (a_n^{(1)} z_1,\ldots, a_n^{(d)}z_d),$$
469
where $z_j =3D x_j + i y_j$ and
470
the $a_n^{(j)}$ are Fourier coefficients of the $f^{(j)}$.
471
472
As explained in Section 2.2 of~\Merel,
473
$\int_{\B e_\chi} f =3D *\cdot L(f,\chi,1)$, where~$*$
474
is some nonzero real or pure-imaginary complex number,
475
according to whether $\chi(-1)$ equals~$1$ or~$-1$,
476
respectively.
477
Combining this observation with the equality
478
=20 $$\dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi)
479
=20 =3D \dim_{\B C} (\B T_{\B C}\cdot \Phi_{\cI}(\B e_\chi)),$$
480
and that the image of $\B T_{\B C}$ in $\End(\B C^d)$ is
481
equal to the diagonal matrices, gives the equality asserted
482
in the lemma.
483
484
\bigskip
485
\rem {} The dimension of $\Phi_{\cI}(\B T_{\B Q(\chi)}\cdot \B e_{\chi})$
486
is unchanged if~$\chi$ is
487
replaced by a Galois-conjugate character. Also, the
488
conductor of~$\chi$ can be arbitrary.
489
490
\bigskip
491
492
In practice, computations over the cyclotomic field $\B Q(\chi)$ are
493
extremely expensive. Fortunately, for our application it suffices to
494
give a lower bound on the dimension appearing in the lemma. Such a
495
bound can be efficiently obtained by instead computing the reductions
496
of~$\Phi$,~$\chi$, and the $\chi$-twisted winding submodule modulo a
497
suitable maximal ideal of the ring of integers of $\B Q(\chi)$ that
498
splits completely; this amounts to performing the above linear algebra
499
over a relatively small prime finite field $\B F_\ell$ such
500
that~$\ell$ is congruent to~$1$ modulo $p-1$.
501
502
\bigskip
503
504
\rem {} For every newform~$f$ in $S_2(\Gamma_0(p))$, with $p\leq 1000$,
505
and every mod~$p$ Dirichlet character~$\chi$, we found that
506
$L(f,\chi,1)\neq 0$ if and only if
507
$L(f^{\sigma},\chi,1)\neq 0$ for all conjugates $f^{\sigma}$
508
of~$f$.
509
More generally, for any~$f$ and~$\chi$, this equivalence holds if
510
$\B Q(\chi)$ is linearly disjoint from the
511
field $K_f=3D(\B T/\cI)\otimes\B Q$.
512
The first few primes
513
for which there is a form~$f$ and a mod-$p$ character~$\chi$
514
such that the linear disjointness hypothesis fails are
515
$p=3D31, 113, 127$, and $191$.
516
517
In general, the equivalence need not hold. For example,
518
let~$f$ be one of the two Galois-conjugate weight-$2$ newforms on
519
$X_1(13)$ and $\chi:(\B Z/7\B Z)^*\raw \B C^*$ be one of the
520
two conjugate characters of order~$3$. Then for one choice
521
of~$\chi$, we have $L(f,\chi,1) =3D 0$ but
522
$L(f^{\sigma},\chi,1)\neq 0$.
523
524
\bigskip
525
526
{\it \noindent Condition 3.}
527
528
The third condition is satisfied for all $p<10000$, except $p =3D 389$,
529
because one of us has verified that the discriminant of $\B T$ is
530
prime to~$p$, so the ring $\B T/p\B T$ is semi-simple. This was
531
accomplished by computing discrimininants of characteristic
532
polynomials mod~$p$ of the Hecke operators $T_2$, $T_3$, $T_5$, $T_7$.
533
In the few cases when all four of these characteristic
534
polynomials had discriminant equal to~$0$ mod~$p$, we resorted to
535
modular symbols to compute several more characteristic polynomials
536
until we found one with nonzero discriminant mod~$p$.
537
538
In the case when $p=3D389$, we find $\cI$ in the following way. There
539
exists at least two minimal prime ideals $\cP_1$ and $\cP_2$ that satisfy
540
the conditions 1 and 2. Therefore the ideal $\cP_1\cap \cP_2$ satisfy
541
also the conditions 1 and 2. Because the discriminant of~$\B T$ has
542
$p$-adic valuation~$1$, the image of at least one of the ideals $\cP_1$,
543
$\cP_2$ and $\cP_1\cap\cP_2$ is a direct factor of $\B T/p\B T$. We
544
choose~$\cI$ to be one of these that is appropriate.
545
546
547
\bigskip
548
{\it \noindent Conclusion.}
549
550
=46or all prime numbers~$p<1000$ different than $2,3,5,7, 13$, we
551
verified the existence of an ideal that satisfies the three conditions
552
given above. For each~$p$, we consider each Galois conjugacy class of
553
non-quadratic characters~$\chi$. We find a single newform~$f$ such
554
that $L(f,\chi,1)\ne 0$ for all conjugates of~$f$ and of $\chi$. Then
555
we let $\cI$ be the annihilator of~$f$, and try to verify condition~1
556
for {\it all} of the anamolous $j$-invariants in $\B F_p$.
557
558
When the three conditions are satisfied for an ideal~$\cI$ of~$\B T$,
559
there exists $t_\chi\in\B T$ which is annihilated by $\cI$ and is the
560
inverse image of a projector of $\B T/p\B T$ on the supplement of
561
$\cI+p\B T$. Putting $\delta=3Dx$, one has
562
$\iota_j(t_\chi \delta)=3D\iota_j(\delta)\ne0$
563
(because $\iota_j$ takes its values in
564
characteristic~$p$, $\delta$ is annihilated by~$\cI$ and
565
$t_\chi\in 1+p\B T+\cP$).
566
Every newform $f\in t_\chi S_2(\Gamma_0(p))$ satisfies
567
$\cI f=3D0$, and therefore, by our second condition, $L(f,\chi,1)\ne0$.
568
The pair $(t_\chi,\delta)$ thus satisfies the conditions required by
569
Proposition~1.
570
571
572
573
\bigskip\bigskip
574
575
576
577
\centerline{\pc Bibliography}
578
\bigskip
579
580
\item{\Agashe}{\pc A. Agashe},
581
{\it On invisible elements of the Tate-Shafarevich group},
582
C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), no. 5, 369--374.
583
584
\item{\Cremona}{\pc J. Cremona},
585
{\it Algorithms for modular elliptic curves},
586
second ed., Cambridge University Press, Cambridge,
587
{\oldstyle 1997}.
588
\vskip 1ex
589
590
\item{\Merel}{\pc L. Merel},
591
{\it Sur la nature non cyclotomique des points d'ordre fini des courbes
592
elliptiques},
593
To appear in Duke Math. Journal.
594
\vskip 1ex
595
596
\item{\Mes}{\pc J.-F. Mestre}, {\it La m\'ethode des graphes.
597
Exemples et applications}, Proceedings of the international
598
conference on class numbers and fundamental units of algebraic number
599
fields (Katata), 217--242, {\oldstyle 1986}.
600
\end
601
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