%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1% %2% merel-stein.tex: The field generated by the points of small %3% prime order on an elliptic curve %4% %5% 15 April 2001 %6% %7% Authors: Loic Merel and William A. Stein %8% %9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%10\documentclass{article}11\hoffset=-0.075\textwidth12\textwidth=1.05\textwidth13\catcode`\@=1114\begin{document}1516\def\�#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi} % trema17\def\B#1{{\bf #1}} % bold18\def\lc{{\it loc.\thinspace{}cit.}} % loc. cit.19\def\mod#1{\ \hbox{{\rm mod}$#1$}} % modulo20\def\eps{\varepsilon}2122\font\titchap=cmr17 at 20pt % for the titles of chapters.23\font\pc=cmcsc10 % for the titles of sections, props, etc.2425\def\th#1{\noindent{\pc Theorem}\ #1. --- \ignorespaces} %Theorem 1.26\def\prop#1{\noindent{\pc Proposition}\ #1. --- \ignorespaces}%Proposition 1.27\def\Def#1{\noindent{\pc Definition}\ #1. --- \ignorespaces} %Definition 1.28\def\cor#1{\noindent{\pc Corollary}\ #1. --- \ignorespaces} %Corollary 1.29\def\conj#1{\noindent{\pc Conjecture}\ #1. --- \ignorespaces} %Conjecture 1.30\def\lem#1{\noindent{\it Lemma}\ #1. --- \ignorespaces} %Lemma 1.31\def\rem#1{\noindent{\it Remark}\ #1: \ignorespaces} %Remark 1.32\def\exe#1{\noindent{\it Example}\ #1: \ignorespaces} %Example 1.33\def\exr#1{\noindent{\it Exercise}\ #1: \ignorespaces} %Exercise 1.34\def\rems{\noindent{\it Remarks}: \ignorespaces} %Remarks 1.35\def\exes{\noindent{\it Examples}: \ignorespaces} %Examples 1.36\def\exrs{\noindent{\it Exercises}: \ignorespaces} %Exercises 1.37\def\thp{\noindent{\pc Theorem}. --- \ignorespaces} %Theorem 1.38\def\propp{\noindent{\pc Proposition}. --- \ignorespaces} %Proposition 1.39\def\Defp{\noindent{\pc Definition}. --- \ignorespaces} %Definition 1.40\def\corp{\noindent{\pc Corollary}. --- \ignorespaces} %Corollairy 1.41\def\conjp{\noindent{\pc Conjecture}. --- \ignorespaces} %Conjecture 1.42\def\lemp{\noindent{\it Lemma}. --- \ignorespaces} %Lemma 1.43\def\remp{\noindent{\it Remark}: \ignorespaces} %Remark 1.44\def\exep{\noindent{\it Example}: \ignorespaces} %Example 1.45\def\dm{\noindent{\it Proof}. --- \ignorespaces}46\def\raw{\longrightarrow}47\def\Hom{{\rm Hom}}48\def\Gal{{\rm Gal}}49\def\cP{{\cal P}}50\def\cO{{\cal O}}51\def\cI{{\cal I}}52\def\rp{{\rm Re}}53\def\ip{{\rm Im}}54\def\End{{\rm End}}5556\def\Agashe{{$[1]$}}57\def\Cremona{{$[2]$}}58\def\Merel{{$[3]$}}59\def\Mes{{$[4]$}}6061\def\change#1{[[{\bf Change:} #1]]}6263\centerline64{\titchap The field generated by the points of small}65\centerline66{\titchap prime order on an elliptic curve}67\medskip6869\bigskip\bigskip\bigskip70\centerline{\pc Lo\"\i c Merel {\rm and} William A.~Stein}71\bigskip\bigskip\bigskip727374\bigskip\bigskip\noindent75{\bf Introduction}76\bigskip7778Let $\bar\B Q$ be an algebraic closure of $\B Q$, and for any prime79number~$p$, denote by $\B Q(\mu_p)$ the cyclotomic subfield of $\bar\B80Q$ generated by the $p$th roots of unity.8182\bigskip83\th{}{\it84Let~$p$ be a prime. If there exists an elliptic curve~$E$ over $\B85Q(\mu_p)$ such that the points of order~$p$ of $E(\bar\B Q)$ are all86$\B Q(\mu_p)$-rational, then $p=2,3,5,13$ or $p>1000$.}87\bigskip8889The case $p=7$ was treated by Emmanuel Halberstadt. The90part of the theorem that concerns the case $p\equiv 3\!\!\pmod{4}$ is91given in~\Merel. In this paper, we give the details that permit our92treating the more difficult case in which $p\equiv 1 \!\!\pmod{4}$.93We treat this last case with the aid of Proposition~2 below, which is94not present in \lc.95The case $p=13$ is currently under investigation by Marusia Rebolledo,96as part of her Ph.D.{} thesis.9798\bigskip\noindent99{\bf 1. Counterexamples define points on $X_0(p)(\B Q(\sqrt{p}))$}100\bigskip101102First we recall some of the results and notation of \Merel.103Let $S_2(\Gamma_0(p))$ denote the space of cusp forms of weight~$2$ for104the congruence subgroup $\Gamma_0(p)$. Denote by $\B T$ the105subring of ${\rm End}\,S_2(\Gamma_0(p))$ generated by the106Hecke operators $T_n$ for all integers~$n$.107Let $f\in S_2(\Gamma_0(p))$ have $q$-expansion108$\sum_{n=1}^\infty a_nq^n$. When $\chi$ is a Dirichlet character,109denote by $L(f,\chi,s)$ the entire function which extends the110Dirichlet series $\sum_{n=1}^\infty a_n\chi(n)/n^s$.111112Let $S$ be the set of isomorphism classes of supersingular elliptic113curves in characteristic~$p$. Denote by $\Delta_S$ the group formed114by the divisors of degree~$0$ with support on~$S$. It is equipped with115a structure of $\B T$-module (induced, for example, from the action116of the Hecke correspondences on the fiber at~$p$ of the regular minimal117model of $X_0(p)$ over $\B Z$).118119Let $j\in\bar\B F_p-J_S$, where $J_S$ denotes the set of supersingular modular120invariants. We denote by $\iota_j$ the homomorphism of121groups $\Delta_S\raw \bar\B F_p$ that associates to $\sum_E n_E[E]$122the quantity123$\sum_E n_E/(j-j(E))$, where $j(E)$ denotes the modular invariant of~$E$.124125126One says that an element $j\in\B F_p$ is {\it anomalous}127if there exists an elliptic curve over $\B F_p$ with modular invariant~$j$128that possesses an $\B F_p$-rational point of order~$p$129(then necessarily $j\notin{}J_S$).130131Let~$p$ be a prime that is congruent to~$1$ modulo~$4$.132In the following proposition we prove, under a hypothesis on~$p$, that133if~$E$ is an elliptic curve over $\B Q(\mu_p)$ all of whose torsion is134$\B Q(\mu_p)$-rational, then for each subgroup $C\subset{}E(\bar\B Q)$135of order~$p$,136the point $(E,C)$ on $X_0(p)$ is defined over $\B Q(\sqrt{p})$. As we137will see in Proposition~2, this $\B Q(\sqrt{p})$-rationality138conclusion is contrary to fact, from which we conclude that such139elliptic curves~$E$ do not exist when the hypothesis on~$p$140is satisfied. In Section~3 we verify this hypothesis141for $p=11$ and $13 < p < 1000$.142143\bigskip144\prop{1}{\it Suppose that~$p$ is congruent to~$1$ modulo~$4$.145Suppose that for all anomalous146$j\in\B F_p$ and all147non-quadratic Dirichlet characters $\chi \colon (\B Z/p\B Z)^*\raw \B148C^*$, there exists $t_\chi\in \B T$ and149$\delta\in\Delta_S$ such that $L(f,\chi,1)\ne0$ for every newform150$f\in t_\chi S_2(\Gamma_0(p))$ and151$\iota_j(t_\chi\delta)\ne0$.152153Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the154points of order~$p$ of155$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.156Then for all subgroups~$C$ of order~$p$ of $E(\bar \B Q)$, there exists an157elliptic curve $E_C$ over $\B Q(\sqrt p)$ equipped with a158$\B Q(\sqrt p)$-rational subgroup $D_C$ of order~$p$, and159the pairs $(E,C)$ and $(E_C,D_C)$ are $\bar \B Q$-isomorphic.}160161\dm We prove the proposition using the results of~\Merel.162The hypothesis163$\iota_j(t_\chi\delta)\ne0$ forces $t_\chi\notin p\B T$164and, {\it a fortiori}, $t_\chi\ne0$; in addition,165the non-vanishing hypothesis on the $L$-series166forces the hypothesis $H_p(\chi)$ of \lc, introduction.167168By assumption, hypothesis $H_p(\chi)$ is satisfied for all169non-quadratic Dirichlet characters~$\chi$ of conductor~$p$.170Thus Corollary~3 of Proposition~6 of \lc{} implies that~$E$ has171potentially good reduction at the prime ideal172$\cP$ of $\B Z[\mu_p]$ that lies above~$p$.173174Denote by~$j$ the modular invariant of the fiber at~$\cP$ of the175N\'eron model of~$E$.176According to the corollary of Proposition~15 of \lc,177$j$ is anomalous.178179Let~$C$ be a subgroup of $E(\bar\B Q)$ of order~$p$.180By assumption~$E$ is an elliptic curve over~$\B Q(\mu_p)$ whose points181of order~$p$ are all $\B Q(\mu_p)$-rational, so182the pair $(E,C)$ defines a $\B Q(\mu_p)$-rational point~$P$183of the modular curve $X_0(p)$.184185Consider the morphism $\phi_{\chi}=\phi_{t_\chi}:X_0(p)\rightarrow J_0(p)$186obtained by composing the standard embedding of $X_0(p)$ into $J_0(p)$187with $t_{\chi}$. As in section 1.3 of \lc, $\phi_{\chi}$188extends to a map from the minimal regular model of $X_0(p)$ to the189N\'eron model of $J_0(p)$.190When $\iota_j(t_\chi\delta)\ne0$, this map is a formal191immersion at the point $P_{/\B F_p}$, according to \lc,192Proposition~4. The hypothesis that $L(f,\chi,1)\ne0$ for193every newform194$f\in t_\chi S_2(\Gamma_0(p))$, translates into $L(t_\chi J_0(p),195\chi,1)\ne0$, which in turn implies that the $\chi$-isotypical196component of197$t_\chi J_0(p)(\B Q(\mu_p))$ is finite (this is Kato's theorem, see the198discussion in section 1.5 of \lc).199We can then apply Corollary~1 of Proposition~6 of \lc. This proves200that~$P$ is201$\B Q(\sqrt p)$-rational, which translates into the conclusion of202Proposition~1.203204\bigskip205\rem{1}206Proposition~1 is true even under the weaker hypothesis207that $t_{\chi}$ lies in $\B T\otimes \B Z[\chi]$, which acts $\B208Z[\chi]$-linearly on modular forms.209210211\bigskip\bigskip\noindent212{\bf 2. Elliptic curves and quadratic fields}213214\bigskip215\prop{2}{\it Let~$p$ be a prime number $>5$ and congruent to~$1$216modulo~$4$. Let~$E$ be an elliptic curve over $\bar\B Q$.217There exists a subgroup~$C\subset{}E(\bar\B Q)$ of order~$p$218such that $(E,C)$ can not be defined over $\B Q(\sqrt{p})$.}219220\dm221We procede by contradiction, i.e., we assume that for222all cyclic subgroups~$C$ of order~$p$ of $E(\bar\B Q)$,223the pair $(E,C)$ can be defined over ${\B Q(\sqrt{p})}$.224We choose such a pair $(E_0,C_0)$ over ${\B Q(\sqrt{p})}$.225226Assume first that all twists of $E$ are quadratic, i.e.227that $j(E)$ is neither~$0$ nor $1728$.228We show that the group229$\Gal(\bar\B Q/{\B Q(\sqrt p)})$ acts by scalars230on the $\B F_p$-vector space $E_0(\bar\B Q)[p]$. For this it231suffices to show that all subgroups of order~$p$ of232$E_0(\bar\B Q)[p]$ are stable by $\Gal(\bar\B Q/{\B Q(\sqrt p)})$.233234Suppose $C_1$ is a cyclic subgroup of order~$p$ of $E_0(\bar\B Q)[p]$.235By assumption, there exists a quadratic twist $E_1$ of $E_0$ and236a cyclic subgroup $C_1'$ of $E_1(\bar\B Q)[p]$237that is defined over $\B Q(\sqrt{p})$, such that238the image of $C_1$ by the isomorphism $E_0\simeq E_1$ is $C'_1$.239Since $\Gal(\bar\B Q/{\B Q(\sqrt p)})$ leaves $C_1'$ stable and240the action of $\Gal(\bar\B Q/\B Q(\sqrt p))$241on $E_0(\bar\B Q)[p]$ is242a quadratic twist of the action on $E_1(\bar\B Q)[p]$,243we see that $\Gal(\bar\B Q/\B Q(\sqrt p))$ leaves $C_1$ stable.244Thus $\Gal(\bar\B Q/\B Q(\sqrt p))$ fixes all lines in245$E_0(\bar\B Q)[p]$, and hence246acts by scalars. Denote by~$\alpha$ the corresponding character247of $\Gal(\bar\B Q/\B Q(\sqrt p))$.248249Because of the Weil pairing, $\alpha^2$ coincides250with the cyclotomic character modulo~$p$, and it factors through251$\Gal(\B Q(\mu_p)/\B Q(\sqrt p))$. But, when252$p\equiv 1\!\!\pmod 4$, the group $\Gal(\B Q(\mu_p)/{\B Q(\sqrt p)})$ is of253even order, and the characters modulo~$p$ form a group generated by the254reduction modulo~$p$ of the cyclotomic character, which, therefore,255can not be a square.256257Next suppose that $j(E)=0$ or $j(E)=1728$. Indeed, in these258two cases~$E$ has259complex multiplication by an order of260$K=\B Q[\sqrt{-3}]$ or $\B Q[\sqrt{-1}]$. Let $d_K=3$ or $d_K=2$ in261these two cases respectively. Let $C$ be a subgroup of order $p$ of262$E(\bar\B Q)$. Consider263the map264$\rho_0 : \Gal(\bar \B Q/\B Q(\sqrt p))\longrightarrow{\rm265Aut}\,E_0(\bar\B Q)[p]$. Since $E$ has complex multiplication, the image266of $\rho_0$ has no element of order~$p$. Therefore, there are at least two267subgroups, including $C_0$, of order~$p$ of268$E(\bar\B Q)$ stable under the image of $\rho_0$. Call the other subgroup269$C_1$. Let $C_2$ be a subgroup of order $p$ of $E(\bar\B Q)$ which is270distinct from $C_0$ and $C_1$. The pair $(E,C_2)$ can be defined over271$\B Q(\sqrt p)$.272Therefore, there exists an extension field $K_2$ of $\B Q(\sqrt p)$,273whose degree $d_2$ divides $2d_K$, such that274the image of the restriction of $\rho_0$ to275$\Gal(\bar \B Q/K_2)$ leaves stable three distinct subgroups of276order~$p$ of $E_0(\bar\B Q)$, and therefore consists only of scalars.277If $d_2\le2$, one concludes as in the cases where278$j(E)\ne0$ and $j(E)\ne 1728$. We suppose now that $d_2>2$.279The projective image of $\rho_0$ has order $d_K$.280281Since~$E$ is an elliptic curve over $\bar \B Q$ with complex282multiplication by a field of class number one, there is a model283for~$E$ that is defined over $\B Q$. Consider the map284$\rho$ : $\Gal(\bar \B Q/\B Q)\longrightarrow{\rm Aut}\,E(\bar\B Q)[p]$.285By the theory of complex multiplication, the projective image of286$\rho$ has order $2(p+1)$ or $2(p-1)$. There exists a field extension~$L$287of degree dividing288$d_K$ of $\B Q(\sqrt p)$ such that the restrictions to289$\Gal(\bar \B Q/L)$ of the projective290images of $\rho$ and $\rho_0$ coincide.291Therefore one has $(p-1)|d_K^2$ or $(p+1)|d_K^2$.292This imposes $p=5$ and $d_K=2$.293294295296\bigskip\bigskip\noindent297{\bf 3. Verification of the hypothesis of Proposition~1}298\bigskip299Let~$p$ be a prime number. In this section we explain how we used300a computer to verify that the second hypothesis of Proposition~1 are301satisfied for $p=11$ and $13 < p < 1000$. (In the present paper,302this verification is only required for $p$ that are congruent to~$1$303modulo~$4$.)304305We first list the anomalous $j$-invariants $j\in\B F_p$. Since~$p$ is306fairly small in the range of our computations, we created this list by307simply enumerating all of the elliptic curves over $\B F_p$ and308counting the number of points on each curve. For example, when $p=31$309the anomalous $j$-invariants are $j=10,14$.310311Let~$\chi: \B Z/p\B Z\raw \B C$ be a non-quadratic312Dirichlet character, and denote by $\B Z[\chi]$ the313subring of $\B Q(\zeta_{p-1})$ generated by the image of~$\chi$.314Denote by $S_2(\Gamma_0(p);\B Z)$ the set of modular forms315$f\in S_2(\Gamma_0(p))$ whose Fourier expansion at the cusp~$\infty$316lies in $\B Z[[q]]$.317318We study the $\B T$-modules $\B T$, $\Delta_S$, and $S_2(\Gamma_0(p);\B Z)$.319After extension of scalars to~$\B Q$, these320are $\B T\otimes\B Q$-modules that are free of rank~$1$, of which the321irreducible sub-$\B T\otimes\B Q$ modules are the annihilators of the322minimal prime ideals of $\B T$. We compute a list of the minimal323prime ideals of $\B T$ by computing appropriate kernels and324characteristic polynomials of Hecke operators of small index on325$\Delta_S$, which we find using the graph method of Mestre and326Oesterl\'e \Mes{}.327328Having computed the minimal prime ideals of $\B T$, we verify that329some nontrivial ideal $\cI$ of $\B T$ (always a minimal prime330ideal in the range of our computations) simultaneously satisfies331the following three conditions:332\vskip 2ex3333341)335For each anomalous $j$-invariant, there exists $x\in\Delta_S$ such that336$\cI x=0$ and $\iota_j(x)\ne 0$.\vskip 1ex3373382) Each of the newforms~$f\in S_2(\Gamma_0(p))$ with339$\cI f=0$ satisfies $L(f,\chi,1)\ne 0$.340\vskip 1ex3413423) The image of~$\cI$ in the $\B T$-module $\B T/p\B T$343is a direct factor.\vskip 2ex344345Let $\cI$ be an ideal of $\B T$. Here is how we verify these conditions346for $\cI$.347348\bigskip349{\it \noindent Verification of condition 1.}350351We verified that $\cI$ satisfies the first condition by352finding a $\B T$-eigenvector~$v$ of $\Delta_S\otimes \bar\B Z$ that is353annihilated by $\cI$ and satisfies $\iota_j(v)\neq 0$ for all anomalous $j$-invariants. Because $\iota_j$354is a homomorphism, this implies the existence of~$x$ as in condition 1.355356\bigskip357{\it \noindent Verification of condition 2.}358359We verified the second condition using modular symbols.360Our method is purely algebraic, so we do not perform361any approximate computation of integrals.362Using the algorithm described in \Cremona, we compute the action of363the Hecke algebra $\B T$ on the space364$\Hom_{\B Q[\chi]}(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$. By intersecting365the kernels of appropriate elements of $\B T$, we find a basis366$\varphi_1,\ldots,\varphi_n$ for the subspace of367$\Hom_{\B Q[\chi]}(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$ that is368annihilated by~$\cI$. Let~$\Phi_{\cI}=\varphi_1\times \cdots \times369\varphi_n$ denote the linear map370$H_1(X_0(p);\B Q[\chi])\raw \B Q[\chi]^n$371defined by the $\varphi_i$.372373Let $\B T_{\B Q[\chi]} = \B T \otimes \B Q[\chi]$, where $\B Q[\chi]$374is the number field generated the image of~$\chi$.375The {\it $\chi$-twisted winding element} (denoted $\theta_\chi$ in376\Merel)377$$\B e_\chi = \sum_{a\in (\B Z/p\B Z)^*} \bar\chi(a)378\Big\{\infty,{a \over p}\Big\}$$379generates the {\it $\chi$-twisted winding submodule}380$\B T_{\B Q[\chi]}\cdot \B e_\chi$. To compute this submodule,381we use that $\B T$ is generated, even as a $\B Z$-module,382by $T_1,T_2,\ldots, T_b$, for any $b\geq (p+1)/6$383(see \Agashe).384385\bigskip386\lem 3387{\it Let $\cI$ be a minimal prime ideal of~$\B T$, and388let $\chi:(\B Z/N\B Z)^*\raw \B C^*$389be a nontrivial Dirichlet character.390Then the dimension of the $\B Q[\chi]$-vector space $\Phi_{\cI}(\B T_{\B391Q[\chi]}392\cdot393\B e_\chi)$ is equal to the cardinality of the set of newforms~$f$ such394that395$\cI f=0$ and $L(f,\chi,1) \neq 0$.396}397398\dm399We have400$$\dim_{\B Q[\chi]} \Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_\chi)401= \dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi).$$402This dimension is invariant upon changing the basis403$\varphi_1,\ldots, \varphi_n$ used to define $\Phi_{\cI}$.404In particular, over $\B C$ there is a basis405$\varphi_1',\ldots, \varphi_n'$ so that the resulting406map $\Phi_{\cI}'$ satisfies407$$\Phi_{\cI}'(x) =408\Bigl(\rp(\int_x f^{(1)}), \ip(\int_x f^{(1)}),409\ldots,410\rp(\int_x f^{(d)}),\ip(\int_x f^{(d)})\Bigr),$$411where $f^{(1)}, \ldots, f^{(d)}$ are the Galois conjugates412of a newform~$f^{(1)}=\sum a_n^{(1)} q^n$ such that $\cI f^{(1)}=0$.413Furthermore, $\Phi_{\cI}'$ is a $\B T_{\B C}$-module homomorphism414if we declare that $\B T_{\B C}$ acts on $\B R^{2d} = \B C^d$ via415$$T_n(x_1,y_1, \ldots, x_d, y_d) =416T_n(z_1,\ldots,z_d) = (a_n^{(1)} z_1,\ldots, a_n^{(d)}z_d),$$417where $z_j = x_j + i y_j$ and418the $a_n^{(j)}$ are Fourier coefficients of the $f^{(j)}$.419420As explained in Section 2.2 of~\Merel,421$\int_{\B e_\chi} f = *\cdot L(f,\chi,1)$, where~$*$422is some nonzero real or pure-imaginary complex number,423according to whether $\chi(-1)$ equals~$1$ or~$-1$,424respectively.425Combining this observation with the equality426$$\dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi)427= \dim_{\B C} (\B T_{\B C}\cdot \Phi_{\cI}(\B e_\chi)),$$428and that the image of $\B T_{\B C}$ in $\End(\B C^d)$ is429equal to the diagonal matrices, proves the asserted equality.430431\bigskip432\rem{2} The dimension of $\Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_{\chi})$433is unchanged if~$\chi$ is434replaced by a Galois-conjugate character.435436\bigskip437438In practice, computations over the cyclotomic field $\B Q[\chi]$ are439extremely expensive. Fortunately, for our application it suffices to440give a lower bound on the dimension appearing in the lemma. Such a441bound can be efficiently obtained by instead computing the reductions442of~$\Phi$,~$\chi$, and the $\chi$-twisted winding submodule modulo a443suitable maximal ideal of the ring of integers of $\B Q[\chi]$ that444splits completely; this amounts to performing the above linear algebra445over a relatively small finite field $\B F_\ell$ where~$\ell$ is446congruent to~$1$ modulo $p-1$.447448\bigskip449450\rem{3} For every newform~$f$ in $S_2(\Gamma_0(p))$, with $p\leq 1000$,451and every mod~$p$ Dirichlet character~$\chi$, we found that452$L(f,\chi,1)\neq 0$ if and only if453$L(f^{\sigma},\chi,1)\neq 0$ for all conjugates $f^{\sigma}$454of~$f$.455More generally, for any~$f$ and~$\chi$, this equivalence holds if456$\B Q[\chi]$ is linearly disjoint from the457field $K_f=(\B T/\cI)\otimes\B Q$.458The first few primes459for which there is a form~$f$ and a mod~$p$ character~$\chi$460such that the linear disjointness hypothesis fails are461$p=31, 113, 127$, and $191$.462The analogue of this nonvanishing observation is false if we instead consider463newforms on $\Gamma_1(p)$ and allow~$\chi$ to be arbitrary.464For example, let~$f$465be one of the two Galois-conjugate newforms in $S_2(\Gamma_1(13))$.466Then there is a character $\chi:(\B Z/7\B Z)^*\raw \B C^*$ of467order~$3$ such that $L(f,\chi,1) = 0$ and $L(f^{\sigma},\chi,1)\neq 0$.468469\bigskip470471{\it \noindent Verification of condition 3.}472473The third condition is satisfied for all $p<10000$, except possibly474$p = 389$,475because we have verified that the discriminant of $\B T$ is476prime to~$p$ for all such $p\neq 389$,477so the ring $\B T/p\B T$ is semisimple.478The discriminant computation was carried out by the second author479as follows.480Using the method of \Mes{}, we computed discrimininants of characteristic481polynomials mod~$p$ of the Hecke operators $T_2$, $T_3$, $T_5$, and $T_7$.482In the few cases when all four of these characteristic polynomials had483discriminant equal to~$0$ mod~$p$, we resorted to modular symbols to484compute several more characteristic polynomials until we found one485having nonzero discriminant modulo~$p$.486487We consider the remaining case $p=389$ in detail. There are exactly488five minimal prime ideals of $\B T$, which we denote $\cP_1$, $\cP_2$,489$\cP_3$, $\cP_6$, and $\cP_{20}$, where the quotient field of $\B490T/\cP_i$ has dimension~$i$. The discriminant of the characteristic491polynomial of $T_2$ is exactly divisible by $389$. Since the field of492fractions of $\B T/\cP_{20}$ has discriminant divisible by $389$, we493see that $389$ is not the residue characteristic of any congruence494prime. Let $\cO_i = \B T/\cP_{i}$. The natural map $\B T \rightarrow495\prod \cO_i$ has finite kernel and cokernel each of order coprime to496$389$, so $\B T / 389 \B T \cong \prod \cO_i/389 \cO_i$. The497nonquadratic characters $\chi:(\B Z/p\B Z)^*\rightarrow \B C^*$ have498orders $1, 4, 97, 193, 388$. We must verify that for each of these499degrees, one of the ideals $\cP_i$ satisfies conditions 1--3. We500check as above that conditions 1--3 for~$\chi$ of order~$4$ are501satisfied by $\cP_2$ and conditions 1--3 for~$\chi$ of order greater502than~$4$ are satisfied by $\cP_1$. When~$\chi$ is the trivial503character, conditions~1--3 are satisfied only by $\cP_{20}$.504505\bigskip506{\it \noindent Summary.}507508For each prime $p<1000$ different than $2,3,5,7, 13$, we509verified the existence of an ideal that satisfies the three conditions510given above, as follows. We consider each Galois conjugacy class of511non-quadratic characters~$\chi$. We find a single newform~$f$ such512that $L(f,\chi,1)\ne 0$ for all conjugates of~$f$ and of~$\chi$. Then513we let $\cI$ be the annihilator of~$f$, and try to verify condition~1514for {\it all} of the anamolous $j$-invariants in $\B F_p$.515When the three conditions are satisfied for an ideal~$\cI$ of~$\B T$,516there exists $t_\chi\in\B T$ that is annihilated by $\cI$ and is the517inverse image of a projector of $\B T/p\B T$ on the complement of518$\cI+p\B T$. Putting $\delta=x$, one has519$\iota_j(t_\chi \delta)=\iota_j(\delta)\ne0$520(because $\iota_j$ takes its values in521characteristic~$p$, it follows that $\delta$ is annihilated by~$\cI$ and522$t_\chi\in 1+p\B T+\cP$).523Every newform $f\in t_\chi S_2(\Gamma_0(p))$ satisfies524$\cI f=0$, and therefore, by our second condition, $L(f,\chi,1)\ne0$.525The pair $(t_\chi,\delta)$ then satisfies the conditions required by526Proposition~1.527528\bigskip529\noindent530{\it Acknowledgment}: We would like to thank Barry Mazur for531providing us several useful comments.532533\bigskip\bigskip534535\vskip 1in536537\centerline{\pc Bibliography}538\bigskip539540\begin{enumerate}541\item[]{\Agashe} {\pc A. Agashe},542{\it On invisible elements of the Tate-Shafarevich group},543C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), no. 5, 369--374.544\vskip 1ex545546\item[]{\Cremona} {\pc J. 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