% merel-stein_v6.tex1% 12 February 200123\magnification=12004\pretolerance=2005\tolerance=4006\brokenpenalty=20078\catcode`\@=1191011\def\�#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi} % trema12\def\B#1{{\bf #1}} % bold13\def\lc{{\it loc.\thinspace{}cit.}} % loc. cit.14\def\mod#1{\ \hbox{{\rm mod}$#1$}} % modulo15\def\eps{\varepsilon}1617\font\titchap=cmr17 at 20pt % for the titles of chapters.18\font\pc=cmcsc10 % for the titles of sections, props, etc.1920\def\th#1{\noindent{\pc Theorem}\ #1. --- \ignorespaces} %Theorem 1.21\def\prop#1{\noindent{\pc Proposition}\ #1. --- \ignorespaces}%Proposition 1.22\def\Def#1{\noindent{\pc Definition}\ #1. --- \ignorespaces} %Definition 1.23\def\cor#1{\noindent{\pc Corollary}\ #1. --- \ignorespaces} %Corollary 1.24\def\conj#1{\noindent{\pc Conjecture}\ #1. --- \ignorespaces} %Conjecture 1.25\def\lem#1{\noindent{\it Lemma}\ #1. --- \ignorespaces} %Lemma 1.26\def\rem#1{\noindent{\it Remark}\ #1: \ignorespaces} %Remark 1.27\def\exe#1{\noindent{\it Example}\ #1: \ignorespaces} %Example 1.28\def\exr#1{\noindent{\it Exercise}\ #1: \ignorespaces} %Exercise 1.29\def\rems{\noindent{\it Remarks}: \ignorespaces} %Remarks 1.30\def\exes{\noindent{\it Examples}: \ignorespaces} %Examples 1.31\def\exrs{\noindent{\it Exercises}: \ignorespaces} %Exercises 1.32\def\thp{\noindent{\pc Theorem}. --- \ignorespaces} %Theorem 1.33\def\propp{\noindent{\pc Proposition}. --- \ignorespaces} %Proposition 1.34\def\Defp{\noindent{\pc Definition}. --- \ignorespaces} %Definition 1.35\def\corp{\noindent{\pc Corollary}. --- \ignorespaces} %Corollairy 1.36\def\conjp{\noindent{\pc Conjecture}. --- \ignorespaces} %Conjecture 1.37\def\lemp{\noindent{\it Lemma}. --- \ignorespaces} %Lemma 1.38\def\remp{\noindent{\it Remark}: \ignorespaces} %Remark 1.39\def\exep{\noindent{\it Example}: \ignorespaces} %Example 1.40\def\dm{\noindent{\it Proof}. --- \ignorespaces}41\def\raw{\longrightarrow}42\def\Hom{{\rm Hom}}43\def\Gal{{\rm Gal}}44\def\cP{{\cal P}}45\def\cO{{\cal O}}46\def\cI{{\cal I}}47\def\rp{{\rm Re}}48\def\ip{{\rm Im}}49\def\End{{\rm End}}5051\def\Agashe{{$[1]$}}52\def\Cremona{{$[2]$}}53\def\Merel{{$[3]$}}54\def\Mes{{$[4]$}}5556\def\change#1{[[{\bf Change:} #1]]}5758\centerline59{\titchap The field generated by the points of small}60\centerline61{\titchap prime order on an elliptic curve}62\medskip6364\bigskip\bigskip\bigskip65\centerline{\pc Lo\"\i c Merel {\rm and} William A.~Stein}66\bigskip\bigskip\bigskip676869\bigskip\bigskip\noindent70{\bf Introduction}71\bigskip7273Let~$p$ be a prime number.74Let $\bar\B Q$ be an algebraic closure of $\B Q$, and75denote by $\B Q(\mu_p)$ the cyclotomic subfield of $\bar\B Q$76generated by the $p$th roots of unity.77Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the78points of order~$p$ of79$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.80\bigskip81\th{}{\it One has $p=2,3,5,13$ or $p>1000$.}82\bigskip8384The case $p=7$ was treated by Emmanuel Halberstadt. The85part of the theorem that concerns the case $p\equiv 3\!\!\pmod{4}$ is86given in~\Merel. In this paper, we give the details that permit our87treating the more difficult case in which $p\equiv 1 \!\!\pmod{4}$.88We treat this last case with the aid of Proposition~2 below, which is89not present in \lc.90The case $p=13$ is currently under investigation by Marusia Rebolledo,91as part of her Ph.D.{} thesis.9293\bigskip\noindent94{\bf 1. We recall the results of \Merel}95\bigskip96\change{Change the title to something like97``Counterexamples define points on $X_0(p)(\B Q(\sqrt{p}))$''.}9899Let $S_2(\Gamma_0(p))$ denote the space of cusp forms of weight~$2$ for100the congruence subgroup $\Gamma_0(p)$. Denote by $\B T$ the101subring of ${\rm End}\,S_2(\Gamma_0(p))$ generated by the102Hecke operators $T_n$ for all integers~$n$.103Let $f\in S_2(\Gamma_0(p))$ have $q$-expansion104$\sum_{n=1}^\infty a_nq^n$. When $\chi$ is a Dirichlet character,105denote by $L(f,\chi,s)$ the entire function which extends the106Dirichlet series $\sum_{n=1}^\infty a_n\chi(n)/n^s$.107108Let $S$ be the set of isomorphism classes of supersingular elliptic109curves in characteristic~$p$. Denote by $\Delta_S$ the group formed110by the divisors of degree~$0$ with support on~$S$. It is equipped with111a structure of $\B T$-module (induced, for example, from the action112of the Hecke correspondences on the fiber at~$p$ of the regular minimal113model of $X_0(p)$ over $\B Z$).114115Let $j\in\bar\B F_p-J_S$, where $J_S$ denotes the set of supersingular modular116invariants. We denote by $\iota_j$ the homomorphism of117groups $\Delta_S\raw \bar\B F_p$ that associates to $\sum_E n_E[E]$118the quantity119$\sum_E n_E/(j-j(E))$, where $j(E)$ denotes the modular invariant of~$E$.120121122One says that an element $j\in\B F_p$ is {\it anomalous}123if there exists an elliptic curve over $\B F_p$ with modular invariant~$j$124that possesses an $\B F_p$-rational point of order~$p$125(then necessarily $j\notin{}J_S$).126127\change{128Let~$p$ be a prime that is congruent to~$1$ modulo~$4$.129In the following proposition we prove, under a hypothesis on~$p$, that130if~$E$ is an elliptic curve over $\B Q(\mu_p)$ all of whose torsion is131$\B Q(\mu_p)$-rational, then for each subgroup $C\subset{}E(\B Q)$132of order~$p$,133the point $(E,C)$ on $X_0(p)$ is defined over $\B Q(\sqrt{p})$. As we134will see in Proposition~2, this $\B Q(\sqrt{p})$-rationality135conclusion is contrary to fact, from which we conclude that such136elliptic curves~$E$ do not exist when the hypothesis on~$p$137is satisfied. In Section~3 we verify this hypothesis138for $p=11$ and $13 < p < 1000$.139}140141\bigskip142\prop{1}{\it Suppose that~$p$ is congruent to~$1$ modulo~$4$.143Suppose that for all anomalous144$j\in\B F_p$ and all145non-quadratic Dirichlet characters $\chi \colon \B Z/p\B Z\raw \B C$,146there exists $t_\chi\in \B T$ and147$\delta\in\Delta_S$ such that $L(f,\chi,1)\ne0$ for every newform148$f\in t_\chi S_2(\Gamma_0(p))$ and149$\iota_j(t_\chi\delta)\ne0$.150151Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the152points of order~$p$ of153$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.154Then for all subgroups~$C$ of order~$p$ of $E(\bar \B Q)$, there exists an155elliptic curve $E_C$ over $\B Q(\sqrt p)$ equipped with a156$\B Q(\sqrt p)$-rational subgroup $D_C$ of order~$p$, and157the pairs $(E,C)$ and $(E_C,D_C)$ are $\bar \B Q$-isomorphic.}158159\dm We prove the proposition using the results of~\Merel.160The hypothesis161$\iota_j(t_\chi\delta)\ne0$ forces $t_\chi\notin p\B T$162and, {\it a fortiori}, $t_\chi\ne0$; in addition,163the non-vanishing hypothesis on the $L$-series164forces the hypothesis $H_p(\chi)$ of \lc, introduction.165166\change{167{\bf (the following paragraph)}168By 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~$p$.}173174According to Corollary~3 of Proposition~6 of \lc,~$E$ has175potentially good reduction at the prime ideal176$\cP$ of $\B Z[\mu_p]$ that lies above~$p$177once we know that178hypothesis $H_p(\chi)$ is satisfied for all179non-quadratic Dirichlet characters~$\chi$ of conductor~$p$180(this is the case by hypothesis).181182Denote by~$j$ the modular invariant of the fiber at~$\cP$ of the183N\'eron model of~$E$.184According to the corollary of Proposition~15 of \lc,185$j$ is anomalous.186187Let~$C$ be a subgroup of $E(\bar\B Q)$ of order~$p$.188By assumption~$E$ is an elliptic curve over~$\B Q(\mu_p)$ whose points189of order~$p$ are all $\B Q(\mu_p)$-rational, so190the pair $(E,C)$ defines a $\B Q(\mu_p)$-rational point~$P$191of the modular curve $X_0(p)$.192193Consider the morphism $\phi_{\chi}=\phi_{t_\chi}:X_0(p)\rightarrow J_0(p)$194obtained by composing the standard embedding of $X_0(p)$ into $J_0(p)$195with $t_{\chi}$. As in section 1.3 of \lc, $\phi_{\chi}$196extends to a map from the minimal regular model of $X_0(p)$ to the197N\'eron model of $J_0(p)$.198When $\iota_j(t_\chi\delta)\ne0$, this map is a formal199immersion at the point $P_{/\B F_p}$, according to \lc,200Proposition~4. The hypothesis that $L(f,\chi,1)\ne0$ for201every newform202$f\in t_\chi S_2(\Gamma_0(p))$, translates into $L(t_\chi J_0(p),203\chi,1)\ne0$, which in turn implies that the $\chi$-isotypical204component of205$t_\chi J_0(p)(\B Q(\mu_p))$ is finite (this is Kato's theorem, see the206discussion in section 1.5 of \lc).207\change{Mazur asked us to verify that section 1.5 of Merel's other208paper properly refers to Scholl's paper.}209We can then apply Corollary~1 of Proposition~6 of \lc. This proves210that~$P$ is211$\B Q(\sqrt p)$-rational, which translates into the conclusion of212Proposition~1.213214\bigskip215216\change{Delete this remark.}217\rem{1} In this proposition we content ourself with a stronger hypothesis218than the one generally used in \lc: the Hecke operator219$t_\chi$ is required to belong to $\B T$ and not to $\B T\otimes\B220Z[\chi]$.221222\bigskip\bigskip\noindent223{\bf 2. Elliptic curves and quadratic fields}224225\bigskip226\prop{2}{\it Let~$p$ be a prime number that is congruent to~$1$227modulo~$4$. Let~$E$ be an elliptic curve over $\bar\B Q$.228\change{There exists a subgroup~$C\subset{}E(\B Q)$ of order~$p$229such that $(E,C)$ can not be defined over $\B Q(\sqrt{p})$.}230There exists a cyclic subgroup~$C$ of order~$p$ of231$E(\bar\B Q)[p]$, such that for all elliptic curves~$E'$ over232${\B Q(\sqrt{p})}$233equipped with a ${\B Q(\sqrt p)}$-rational subgroup~$C'$, the234pairs $(E,C)$ and $(E', C')$ are not $\bar\B Q$-isomorphic.235}236237\dm238We procede by contradiction, i.e., we assume that for239all cyclic subgroups~$C$ of order~$p$ of $E(\bar\B Q)$,240the pair $(E,C)$ can be defined over ${\B Q(\sqrt{p})}$.241We choose such a pair $(E_0,C_0)$ over ${\B Q(\sqrt{p})}$.242243Assume first that all twists of $E$ are quadratic, i.e.,244that $j(E)$ is neither~$0$ nor $1728$.245We show that the group246$\Gal(\bar\B Q/{\B Q(\sqrt p)})$ acts by scalars247on the $\B F_p$-vector space $E_0(\bar\B Q)[p]$. For this it248suffices to show that all subgroups of order~$p$ of249$E_0(\bar\B Q)[p]$ are stable by $\Gal(\bar\B Q/{\B Q(\sqrt p)})$.250251\change{I reworded the following paragraph.}252Suppose $C_1$ is a cyclic subgroup of order~$p$ of $E_0(\bar\B Q)[p]$.253By assumption, there exists a quadratic twist $E_1$ of $E_0$ and254a cyclic subgroup $C_1'$ of $E_1(\bar\B Q)[p]$255that is defined over $\B Q(\sqrt{p})$, such that256the image of $C_1$ by the isomorphism $E_0\simeq E_1$ is $C'_1$.257Since $\Gal(\bar\B Q/{\B Q(\sqrt p)})$ leaves $C_1'$ stable and258the action of $\Gal(\bar\B Q/\B Q(\sqrt p))$259on $E_0(\bar\B Q)[p]$ is260a quadratic twist of the action on $E_1(\bar\B Q)[p]$,261we see that $\Gal(\bar\B Q/\B Q(\sqrt p))$ leaves $C_1$ stable.262Thus $\Gal(\bar\B Q/\B Q(\sqrt p))$ fixes all lines in263$E_0(\bar\B Q)[p]$, and hence264acts by scalars. Denote by~$\alpha$ the corresponding character265of $\Gal(\bar\B Q/\B Q(\sqrt p))$.266267Because of the Weil pairing, $\alpha^2$ coincides268with the cyclotomic character modulo~$p$, and it factors through269$\Gal(\B Q(\mu_p)/\B Q(\sqrt p))$. But, when270$p\equiv 1\!\!\pmod 4$, the group $\Gal(\B Q(\mu_p)/{\B Q(\sqrt p)})$ is of271even order, and the characters modulo~$p$ form a group generated by the272reduction modulo~$p$ of the cyclotomic character, which, therefore,273can not be a square.274275\change{I don't understand this argument.}276Next suppose that $j(E)=0$ or $j(E)=1728$. Indeed, in these277two cases~$E$ has278complex multiplication by an order $R_K$ of $K=\B Q[\sqrt{-1}]$ or279$\B Q[\sqrt{-3}]$.280Consider the map281$\rho : \Gal(\bar \B Q/\B Q(\sqrt p))\longrightarrow{\rm Aut}\,E_0(\bar\B Q)[p]$.282%Suppose $\rho(\Gal(\bar \B Q/K(\sqrt p)))$ contains an element of order~$p$.283Let $L_p$ be the ray class field of conductor~$p$ of~$K$.284It contains $\B Q(\sqrt p)$ since $p\equiv 1\!\!\!\pmod 4$.285By the theory286of complex multiplication,287$\rho(\Gal(\bar \B Q/L_p))$ is trivial. By class field theory, $\Gal(L_p/K)$288has no element of order~$p$, since~$K$ has class number~$1$. Therefore289$\rho(\Gal(\bar \B Q/\B Q(\sqrt p)))$ has no elements of order~$p$.290Since it is contained in the Borel subgroup of ${\rm Aut}\,E_0(\bar\B Q)[p]$291which stabilizes $C_0$, it is an abelian group. By the theory of complex292multiplication, it is the semi-direct product of $\Gal(L_p/K(\sqrt p))$ and293$\Gal(K(\sqrt p)/\B Q(\sqrt p))\simeq\Gal(\B C/\B R)$. Such a group294is not abelian since295$\Gal(L_p/K(\sqrt p))$ is not a $2$-group, hence the contradiction.296297298\bigskip\bigskip\noindent299{\bf 3. Verification of the hypothesis of Proposition~1}300\bigskip301Let $p$ be a prime number. In this section we explain how we used302a computer to verify that the hypothesis of Proposition~1 are satisfied303for $p=11$ and $13 < p < 1000$.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 $S_2(\Gamma_0(p);\B Z)$ be 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$\Hom(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$. By intersecting the kernels365of appropriate elements of $\B T$, we find a basis366$\varphi_1,\ldots,\varphi_n$ for the subspace of367$\Hom(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$ that is annihilated by~$\cI$.368Let~$\Phi_{\cI}=\varphi_1\times \cdots \times \varphi_n$ denote the linear map369$H_1(X_0(p);\B Q[\chi])\raw \B Q[\chi]^n$370defined by the $\varphi_i$.371372Let $\B T_{\B Q[\chi]} = \B T \otimes \B Q[\chi]$, where $\B Q[\chi]$373is the number field generated the image of~$\chi$.374The {\it $\chi$-twisted winding element} (denoted $\theta_\chi$ in375\Merel)376$$\B e_\chi = \sum_{a\in (\B Z/p\B Z)^*} \bar\chi(a)377\Big\{\infty,{a \over p}\Big\}$$378generates the {\it $\chi$-twisted winding submodule}379$\B T_{\B Q[\chi]}\cdot \B e_\chi$. To compute this submodule,380we use that $\B T$ is generated, even as a $\B Z$-module,381by $T_1,T_2,\ldots, T_b$, for any $b\geq (p+1)/6$382(see \Agashe).383384\bigskip385\lem 3386{\it Let $\cI$ be a minimal prime ideal of~$\B T$, and387let $\chi:(\B Z/N\B Z)^*\raw \B C^*$388be a nontrivial Dirichlet character.389Then the dimension of $\Phi_{\cI}(\B T_{\B Q[\chi]} \cdot \B e_\chi)$ is390equal to the cardinality of the set of newforms~$f$ such that391$\cI f=0$ and $L(f,\chi,1) \neq 0$.392}393394\dm395We have396$$\dim_{\B Q[\chi]} \Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_\chi)397= \dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi).$$398This dimension is invariant upon changing the basis399$\varphi_1,\ldots, \varphi_n$ used to define $\Phi_{\cI}$.400In particular, over $\B C$ there is a basis401$\varphi_1',\ldots, \varphi_n'$ so that the resulting402map $\Phi_{\cI}'$ satisfies403$$\Phi_{\cI}'(x) =404\Bigl(\rp(\int_x f^{(1)}), \ip(\int_x f^{(1)}),405\ldots,406\rp(\int_x f^{(d)}),\ip(\int_x f^{(d)})\Bigr),$$407where $f^{(1)}, \ldots, f^{(d)}$ are the Galois conjugates408of a newform~$f^{(1)}=\sum a_n^{(1)} q^n$ such that $\cI f^{(1)}=0$.409Furthermore, $\Phi_{\cI}'$ is a $\B T_{\B C}$-module homomorphism410if we declare that $\B T_{\B C}$ as acts on $\B R^{2d} = \B C^d$ via411$$T_n(x_1,y_1, \ldots, x_d, y_d) =412T_n(z_1,\ldots,z_d) = (a_n^{(1)} z_1,\ldots, a_n^{(d)}z_d),$$413where $z_j = x_j + i y_j$ and414the $a_n^{(j)}$ are Fourier coefficients of the $f^{(j)}$.415416As explained in Section 2.2 of~\Merel,417$\int_{\B e_\chi} f = *\cdot L(f,\chi,1)$, where~$*$418is some nonzero real or pure-imaginary complex number,419according to whether $\chi(-1)$ equals~$1$ or~$-1$,420respectively.421Combining this observation with the equality422$$\dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi)423= \dim_{\B C} (\B T_{\B C}\cdot \Phi_{\cI}(\B e_\chi)),$$424and that the image of $\B T_{\B C}$ in $\End(\B C^d)$ is425equal to the diagonal matrices, proves the asserted equality.426427\bigskip428\rem{2} The dimension of $\Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_{\chi})$429is unchanged if~$\chi$ is430replaced by a Galois-conjugate character.431432\bigskip433434In practice, computations over the cyclotomic field $\B Q[\chi]$ are435extremely expensive. Fortunately, for our application it suffices to436give a lower bound on the dimension appearing in the lemma. Such a437bound can be efficiently obtained by instead computing the reductions438of~$\Phi$,~$\chi$, and the $\chi$-twisted winding submodule modulo a439suitable maximal ideal of the ring of integers of $\B Q[\chi]$ that440splits completely; this amounts to performing the above linear algebra441over a relatively small prime finite field $\B F_\ell$ such442that~$\ell$ is congruent to~$1$ modulo $p-1$.443444\bigskip445446\rem{3} For every newform~$f$ in $S_2(\Gamma_0(p))$, with $p\leq 1000$,447and every mod~$p$ Dirichlet character~$\chi$, we found that448$L(f,\chi,1)\neq 0$ if and only if449$L(f^{\sigma},\chi,1)\neq 0$ for all conjugates $f^{\sigma}$450of~$f$.451More generally, for any~$f$ and~$\chi$, this equivalence holds if452$\B Q[\chi]$ is linearly disjoint from the453field $K_f=(\B T/\cI)\otimes\B Q$.454The first few primes455for which there is a form~$f$ and a mod~$p$ character~$\chi$456such that the linear disjointness hypothesis fails are457$p=31, 113, 127$, and $191$.458The analogue of this nonvanishing observation is false if we instead consider459newforms on $\Gamma_1(p)$ and allow~$\chi$ to be arbitrary.460For example, let~$f$461be one of the two Galois-conjugate newforms in $S_2(\Gamma_1(13))$.462Then there is a character $\chi:(\B Z/7\B Z)^*\raw \B C^*$ of463order~$3$ such that $L(f,\chi,1) = 0$ and $L(f^{\sigma},\chi,1)\neq 0$.464465\bigskip466467{\it \noindent Verification of condition 3.}468469The third condition is satisfied for all $p<10000$, except possibly470$p = 389$,471because we have verified that the discriminant of $\B T$ is472prime to~$p$ for all such $p\neq 389$,473so the ring $\B T/p\B T$ is semisimple.474The discriminant computation was carried out by the second author475as follows.476Using the method of \Mes{}, we computed discrimininants of characteristic477polynomials mod~$p$ of the Hecke operators $T_2$, $T_3$, $T_5$, and $T_7$.478In the few cases when all four of these characteristic polynomials had479discriminant equal to~$0$ mod~$p$, we resorted to modular symbols to480compute several more characteristic polynomials until we found one481having nonzero discriminant modulo~$p$.482483We consider the remaining case $p=389$ in detail. There are exactly484five minimal prime ideals of $\B T$, which we denote $\cP_1$, $\cP_2$,485$\cP_3$, $\cP_6$, and $\cP_{20}$, where the quotient field of $\B486T/\cP_i$ has dimension~$i$. The discriminant of the characteristic487polynomial of $T_2$ is exactly divisible by $389$. Since the field of488fractions of $\B T/\cP_{20}$ has discriminant divisible by $389$, we489see that $389$ is not the residue characteristic of any congruence490prime. Let $\cO_i = \B T/\cP_{i}$. The natural map $\B T \rightarrow491\prod \cO_i$ has finite kernel and cokernel each of order coprime to492$389$, so $\B T / 389 \B T \cong \prod \cO_i/389 \cO_i$. The493nonquadratic characters $\chi:(\B Z/p\B Z)^*\rightarrow \B C^*$ have494orders $1, 4, 97, 193, 388$. We must verify that for each of these495degrees, one of the ideals $\cP_i$ satisfies conditions 1--3. We496check as above that conditions 1--3 for~$\chi$ of order~$4$ are497satisfied by $\cP_2$ and conditions 1--3 for~$\chi$ of order greater498than~$4$ are satisfied by $\cP_1$. When~$\chi$ is the trivial499character, conditions~1--3 are satisfied only by $\cP_{20}$.500501\bigskip502{\it \noindent Summary.}503504For each prime $p<1000$ different than $2,3,5,7, 13$, we505verified the existence of an ideal that satisfies the three conditions506given above, as follows. For each~$p$, we consider each Galois conjugacy class of507non-quadratic characters~$\chi$. We find a single newform~$f$ such508that $L(f,\chi,1)\ne 0$ for all conjugates of~$f$ and of~$\chi$. Then509we let $\cI$ be the annihilator of~$f$, and try to verify condition~1510for {\it all} of the anamolous $j$-invariants in $\B F_p$.511When the three conditions are satisfied for an ideal~$\cI$ of~$\B T$,512there exists $t_\chi\in\B T$ that is annihilated by $\cI$ and is the513inverse image of a projector of $\B T/p\B T$ on the complement of514$\cI+p\B T$. Putting $\delta=x$, one has515$\iota_j(t_\chi \delta)=\iota_j(\delta)\ne0$516(because $\iota_j$ takes its values in517characteristic~$p$, it follows that $\delta$ is annihilated by~$\cI$ and518$t_\chi\in 1+p\B T+\cP$).519Every newform $f\in t_\chi S_2(\Gamma_0(p))$ satisfies520$\cI f=0$, and therefore, by our second condition, $L(f,\chi,1)\ne0$.521The pair $(t_\chi,\delta)$ then satisfies the conditions required by522Proposition~1.523524525526\bigskip\bigskip527528\vskip .5in529530\centerline{\pc Bibliography}531\bigskip532533\item{\Agashe}{\pc A. Agashe},534{\it On invisible elements of the Tate-Shafarevich group},535C. R. 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