% modsymbols.tex \documentclass[11pt]{article} \usepackage{graphicx} \usepackage{psfrag} \hoffset=-0.04\textwidth \textwidth=1.08\textwidth \voffset=-0.04\textheight \textheight=1.08\textheight \bibliographystyle{amsplain} \include{macros} \usepackage{amsmath} \title{The Modular Curve $X_0(389)$:\\ {\large Discriminants, Ranks, Shafarevich-Tate Groups, and Weierstrass Points}} \author{William Stein} \date{April 2002} \renewcommand{\T}{\mathbf{T}} \begin{document} \maketitle \section{Introduction} Let $N$ be a positive integer, and let $X_0(N)$ be the compactified coarse moduli space that classifies pairs $(E,C)$ where $E$ is an elliptic curve and $C$ is a cyclic subgroup of order~$N$. The space $X_0(N)$ has a canonical structure of algebraic curve over~$\Q$, and its properties have been very well studied during the last forty years. For example, Breuil, Conrad, Diamond, Taylor, and Wiles proved that every elliptic curve over~$\Q$ is a quotient of some $X_0(N)$. The smallest~$N$ such that the Jacobian of $X_0(N)$ has positive Mordell-Weil rank is~$37$, and Zagier studied the genus-two curve $X_0(37)$ in depth in his paper~\cite{zagier:modular}. From this viewpoint, the next modular curve deserving intensive investigation is $X_0(389)$, which is the first modular curve whose Jacobian has Mordell-Weil rank larger than that predicted by the signs in the functional equations of the $L$-series attached to simple factors of its Jacobian; in fact, $389$ is the smallest conductor of an elliptic curve with Mordell-Weil rank~$2$. Note that $389$ is prime and $X_0(389)$ has genus $g=32$, which is much larger than the genus~$2$ of $X_0(37)$, which makes explicit investigation more challenging. Work of Kolyvagin \cite{kolyvagin:weil, kolyvagin:subclass} and Gross-Zagier \cite{gross-zagier} has completely resolved the rank assertion of the Birch and Swinnerton-Dyer conjecture (see, e.g., \cite{tate:bsd}) for elliptic curves~$E$ with $\ord_{s=1} L(E,s) \leq 1$. The lowest-conductor elliptic curve~$E$ that doesn't submit to the work of Kolyvagin and Gross-Zagier is the elliptic curve~$E$ of conductor~$389$ mentioned in the previous paragraph. At present we don't even have a conjectural natural construction of a finite-index subgroup of $E(\Q)$ analogous to that given by Gross and Zagier for rank~$1$ (but see Mazur's work on universal norms, which might be used to construct $E(\Q)\tensor\Z_p$ for some auxiliary prime~$p$). Inspired by the above observations, and with an eye towards providing helpful data for anyone trying to generalize the work of Gross, Zagier, and Kolyvagin, in this paper we compute everything we can about the modular curve $X_0(389)$. Some of the computations of this paper have already proved important in several other papers: the discriminant of the Hecke algebra attached to $X_0(389)$ plays a roll in \cite{ribet:torsion}, the verification of condition 3 in \cite{merel-stein}, and the remark after Theorem~1 of \cite{gordon-ono:vis}; also, the arithmetic of $J_0(389)$ provides a key example in \cite[\S4.2]{agashe-stein:visibility}. Finally, this paper serves as an entry in an ``encyclopaedia, atlas or hiker's guide to modular curves'', in the spirit of N.~Elkies (see \cite[pg.~22]{elkies:ffield}). We hilight several surprising ``firsts'' that occur at level~$389$. The discriminant of the Hecke algebra attached to $S_2(\Gamma_0(389))$ has the apparently unusual property that it is divisible by~$p=389$ (see Section~\ref{sec:disc_div}). Also $N=389$ is the smallest integer such that the order of vanishing of $L(J_0(N),s)$ at $s=1$ is larger than predicted by the functional equations of eigenforms (see Section~\ref{sec:mwranks}). The author conjectures that $N=389$ is the smallest level such that an optimal newform factor of $J_0(N)$ appears to have Shafarevich-Tate group with nontrivial odd part (see Section~\ref{sec:sha}). Atkin conjectures that $389$ is the largest prime such that the cusp of $X_0^+(389)$ fails to be a Weierstrass point (see Section~\ref{sec:atkin}). \noindent{\bf Acknowledgement.} Noam Elkies, Ken Ribet, Matt Baker. Hendrik Lenstra, for suggesting a method to compute discriminants of Hecke algebra efficiently (see Section~\ref{sec:hecke_algebra}). \section{Factors of $J_0(389)$} To each newform $f\in S_2(\Gamma_0(389))$, Shimura \cite{shimura:factors} associated a quotient $A_f$ of $J_0(389)$, and $J_0(389)$ is isogeneous to the product $\prod A_f$, where the product runs over the $\Gal(\Qbar/\Q)$-conjugacy classes of newforms. Moreover, because $389$ is prime each factor $A_f$ cannot be decomposed further up to isogeny, even over $\Qbar$ (see \cite{ribet:endo}). \subsection{Newforms of Level $389$} There are five $\Gal(\Qbar/\Q)$-conjugacy classes of newforms in $S_2(\Gamma_0(389))$. The first class corresponds to the unique elliptic curve of conductor $389$, and its $q$-exansion begins $$ f_1 = q - {2}q^{2} - {2}q^{3} + {2}q^{4} - {3}q^{5} + {4}q^{6} - {5}q^{7} + q^{9} + {6}q^{10} + \cdots. $$ The second has coefficients in the quadratic field $\Q(\sqrt{2})$, and has $q$-expansion $$ f_2 = q + {\sqrt{2}}q^{2}+({\sqrt{2} - {2}})q^{3} - q^{5}+({{-2}\sqrt{2} + {2}})q^{6}+ \cdots. $$ The third has coefficients in the cubic field generated by a root $\alpha$ of $x^3-4x-2$: $$ f_3 = q + {\alpha}q^{2} - {\alpha}q^{3}+({\alpha^{2} - {2}})q^{4}+({-\alpha^{2} + 1})q^{5} - {\alpha^{2}}q^{6} +\cdots. $$ The fourth has coefficients that generate the degree-six field defined by a root $\beta$ of $x^6+3x^5-2x^4-8x^3+2x^2+4x-1$ and $q$-expansion $$ f_4 = q + {\beta}q^{2}+({\beta^{5} + {3}\beta^{4} - {2}\beta^{3} - {8}\beta^{2} + \beta + {2}})q^{3}+\cdots. $$ The fifth and final newform (up to conjugacy) has coefficients that generate the degree $20$ field defined by a root of \begin{eqnarray*} f_5 &=& x^{20} - 3x^{19} - 29x^{18} + 91x^{17} + 338x^{16} - 1130x^{15} - 2023x^{14} + 7432x^{13} \\ && + 6558x^{12} - 28021x^{11} - 10909x^{10} + 61267x^{9} + 6954x^8 - 74752x^7 \\ && + 1407x^6+ 46330x^5 - 1087x^4 - 12558x^3 - 942x^2 + 960x + 148. \end{eqnarray*} \subsubsection{Congruences} The vertices in Figure~\ref{fig:cong} correspond to the newforms $f_i$; there is an edge between $f_i$ and $f_j$ labeled~$p$ if there is a maximal ideal $\wp\mid p$ of the field generated by the Fourier coefficients of $f_i$ and $f_j$ such that $f_i \con f_j \pmod{\wp}$. \begin{figure} \begin{center} \psfrag{f1}{$f_1$} \psfrag{f2}{$f_2$} \psfrag{f3}{$f_3$} \psfrag{f4}{$f_4$} \psfrag{f5}{$f_5$} \psfrag{2 }{$2$} \psfrag{2 }{$2$} \psfrag{2,5}{$2,5$} \psfrag{3 }{$3$} \psfrag{31}{$31$} \psfrag{2}{$2$} \psfrag{2 }{$2$} \psfrag{2 }{$2$} \psfrag{2 }{$2$} \includegraphics{cong.epsi} \caption{Congruences Between Newforms\label{fig:cong}} \end{center} \end{figure} \comment{ > D := SortDecomposition(NewformDecomposition(CuspidalSubspace(ModularSymbols(389,2)))); S := CuspidalSubspace(ModularForms(389,2)); f := [* *]; for i in [1..5] do Append(~f,Newform(S,i)); end for; for i in [1..5] do for j in [i+1..5] do F := Factorization(#CongruenceGroup(Parent(f[i]),Parent(f[j]),200)); if #F gt 0 then printf "%o--%o [label=\"", i,j; for i in [1..#F] do printf "%o",F[i][1]; if i lt #F then printf ","; end if; end for; printf "\"];\n"; end if; end for; end for; shell-> ps2epsi cong.ps cong.epsi } \subsection{Isogeny Structure} We deduce from the above determination of the newforms in $S_2(\Gamma_0(389))$ that $J_0(389)$ is $\Q$-isogenous to a product of $\Qbar$-simple abelian varieties $$ J \sim A_1\cross A_2 \cross A_3 \cross A_4 \cross A_{5}. $$ View the duals $A_i^{\vee}$ of the $A_i$ as abelian subvarieties of $J_0(389)$. Using modular symbols as in \cite[\S3.4]{agashe-stein:bsd} we find that, for $i\neq j$, a prime~$p$ divides $\#(A_i^{\vee}\intersect A_j^{\vee})$ if and only if $f_i\con f_j\pmod{\wp}$ for some prime $\wp\mid p$ (recall that the congruence primes are given in Figure~\ref{fig:cong} above). \subsection{Mordell-Weil Ranks}\label{sec:mwranks} Suppose $f\in S_2(\Gamma_0(N))$ is a newform of some level~$N$. The functional equation for $L(f,s)$ implies that $\ord_{s=1}L(f,s)$ is odd if and only if the sign of the eigenvalue of the Atkin-Lehner involution $W_N$ on~$f$ is $+1$. \begin{proposition}\label{prop:minanrank} If $f\in S_2(\Gamma_0(N))$ is a newform of level $N<389$, then $\ord_{s=1}L(f,s)$ is either$0$ or~$1$. \end{proposition} \begin{proof} The proof amounts to a large computation, which divides into two parts: \begin{enumerate} \item Verify, for each newform~$f$ of level $N<389$ such that $W_N(f) = -f$, that $L(f,1) = *\int_{0}^{i\infty} f(z) dz$ (for some nonzero $*$) is nonzero. This is a purely algebraic computation involving modular symbols. \item Verify, for each newform~$f$ of level $N<389$ such that $W_N(f) = f$, that $L'(f,1)\neq 0$ (see \cite[\S4.1]{empirical}, which points to \cite[\S2.11,\S2.13]{cremona:algs}). We do this by approximating an infinite series that converges to $L'(f,1)$ and noting that the value we get is far from~$0$. \end{enumerate} \end{proof} Thus $N=389$ is the smallest level such that the $L$-series of some factor $A_f$ of $J_0(N)$ has order of vanishing higher than that which is forced by the sign in the functional equation. %The elliptic curve~$A_1$ of rank~$2$ is the lowest-conductor elliptic %curve having rank~$>1$, because every elliptic curve over~$\Q$ is modular %\cite{breuil-conrad-diamond-taylor}, and \cite{cremona:algs} doesn't contain %any elliptic curve of conductor $<389$ having rank~$>1$. %The $L$-function corresponding to~$A_{5}$ does not vanish at $s=1$, %so $A_{5}$ has analytic rank~$0$, and hence algebraic rank~$0$, by %the theorem of Kolyvagin and Logachev. %The genus of $X^+=X/w_{389}$ is $g^{+} =11$, and $J_0(389)^{-}$ %is isogeneous to $A_{1} \cross A_{5}.$ \begin{proposition}\label{prop:anrank} The following table summarizes the dimensions and Mordell-Weil ranks (over the image of the Hecke ring) of the newform factors of $J_0(N)$: \begin{center} \begin{tabular}{|l|c|c|c|c|c|}\hline \text{\rm }& $A_1$ & $A_2$ & $A_3$ & $A_4$ & $A_{5}$\\\hline \text{\rm Dimension}& $1$ & $2$ & $3$ & $6$ & $20$\\\hline \text{\rm Rank} & $2$ & $1$ & $1$ & $1$ & $0$\\\hline \end{tabular} \end{center} \end{proposition} \begin{proof} The elliptic curve $A_1$ is $389A$ in Cremona's tables, which is the elliptic curve of smallest conductor having rank $2$. For $A_{5}$ we directly compute whether or not the $L$-function vanishes using modular symbols, by taking an inner product with the winding element $e_w=-\{0,\infty\}$. We find that the $L$-function does not vanish. By Kolyvagin-Logachev, it follows that $A_{5}$ has Mordell-Weil rank $0$. For each of the other three factors, the sign of the functional equation is odd, so the analytic ranks are odd. As in the proof of Proposition~\ref{prop:minanrank}, we verify that the analytic rank is~$1$ in each case. By work of Gross, Zagier, and Kolyvagin it follows that the ranks are~$1$. \end{proof} %The elliptic curve $A_1$ has minimal Weierstrass model % $$y^2+y=x^3+x^2-2x.$$ %It satisfies $A_1(\Q)=\Z\oplus \Z$. The modular %degree is $40$. \comment{\subsection{Characteristic polynomials} The characteristic polynomials of the first two Hecke operators are as follows: \begin{eqnarray*} T_2&=&(x + 2)({x^{2} - {2}})({x^{3} - {4}x - {2}})(x^{6} + {3}x^{5} - {2}x^{4} - {8}x^{3} + {2}x^{2} + {4}x - 1)\\ &&(x^{20} - {3}x^{19} - {29}x^{18} + {91}x^{17} + {338}x^{16} - {1130}x^{15} - {2023}x^{14}+ {7432}x^{13}+ {6558}x^{12}\\ &&- {28021}x^{11} - {10909}x^{10} + {61267}x^{9} + {6954}x^{8}- {74752}x^{7} + {1407}x^{6} + {46330}x^{5}\\ && - {1087}x^{4} - {12558}x^{3} - {942}x^{2} + {960}x + {148})\\ T_3&=&(x + 2)(x^2 + 4x + 2)(x^3 -4x + 2)(x^6 + 5x^5 + 4x^4 -13x^3 -21x^2 -6x + 1)\\ &&(x^{20} -11x^{19} + 19x^{18} + 204x^{17} -845x^{16} -781x^{15} + 8883x^{14} -6177x^{13} -40916x^{12}\\ &&+ 63058x^{11} + 85034x^{10} -215618x^9 -46920x^8 + 342529x^7 -84612x^6 -241030x^5 \\ &&+ 112365x^4 + 51018x^3 -28526x^2 + 3560x -100) \end{eqnarray*} } \section{The Hecke algebra}\label{sec:hecke_algebra} \subsection{The Discriminant is Divisible By $p$} \label{sec:disc_div} Let $N$ be a positive integer. The Hecke algebra $\T\subset\End(S_2(\Gamma_0(N)))$ is the subring generated by all Hecke operators $T_n$ for $n=1,2,3,\ldots$. We are concerned with the {\em discriminant} of the trace pairing $(t,s)\mapsto \Tr(ts)$. When~$N$ is prime, $\T_\Q=\T\tensor_\Z\Q$ is a product $K_1\cross\cdots\cross K_n$ of totally real number fields. Let $\tilde{\T}$ denote the integral closure of~$\T$ in $\T_\Q$; note that $\tilde{\T}=\prod \O_i$ where~$\O_i$ is the ring of integers of $K_i$. Then $\disc(\T)=[\tilde{\T}:\T]\cdot \prod_{i=1}^n\disc(K_i)$. %One can view the primes dividing $\prod_{i=1}^n\disc(K_i)$ as being %associated to singularities of the irreducible components of $\Spec(\T)$, %and the primes dividing $[\tilde{\T}:\T]$ are associated to intersections %of irreducible components, or, equivalently, congruences between eigenforms. \begin{proposition} The discriminant of the Hecke algebra associated to $S_2(\Gamma_0(389))$ is $$ 2^{53}\cdot{}3^4\cdot{}5^6\cdot{}31^2\cdot{}37\cdot{}389 \cdot{}3881\cdot{}215517113148241\cdot{}477439237737571441. $$ \end{proposition} \begin{proof} By \cite{agashe-stein:schoof-appendix}, the Hecke algebra~$\T$ is generated as a $\Z$-module by $T_1,T_2,\ldots T_{65}$. %Then $\disc(\T)$ can be quickly computed once one finds %a $\Z$-basis for~$\T$, which can be accomplished as follows. To compute $\disc(\T)$, we proceed as follows. First, compute the space $\sS_2(\Gamma_0(389))$ of cuspidal modular symbols, which is a faithful $\T$-module. Choose a random element $x\in \sS_2(\Gamma_0(389))_+$ of the $+1$-quotient of the cuspidal modular symbols, then compute the images $v_1=T_1(x), v_2=T_2(x), \ldots, v_{65}=T_{65}(x)$. If these don't span a space of dimension $32=\rank_\Z \T$ choose a new random element~$x$ and repeat. Using the Hermite Normal Form, find a $\Z$-basis $b_1,\ldots, b_{32}$ for the $\Z$-span of $v_1,\ldots, v_{65}$. The trace pairing on $\T$ induces a trace pairing on the $v_i$, and hence on the $b_i$. Then $\disc(\T)$ is the discriminant of this pairing on the $b_i$. The reason we embed $\T$ in $\sS_2(\Gamma_0(389))_+$ as $\T{}x$ is because directly finding a $\Z$-basis for $\T$ would involve computing the Hermite Norm Form of a list of $65$ vectors in a $1024$-dimensional space, which is unnecessarily difficult (though possible). \end{proof} We compute this discriminant by applying the definition of discriminant to a matrix representation of the first~$65$ Hecke operators $T_1,\ldots, T_{65}$. Matrices representing these Hecke operators were computed using the modular symbols algorithms described in \cite{cremona:algs}. [Sturm, {\em On the congruence of modular forms}]. %\end{proof} %It might be possible to choose a %bound smaller than~$65$ by directly considering a basis of %modular forms. %\comment{ A \pari{} program which can usually compute the discriminant %of a commutative $\Z$-module represented by matrices is included as an %appendix. } \comment{ We should also note that by computing successive greatest common divisors of discriminants of characteristic polynomials of Hecke operators $T_p$, one can get a fairly good multiplicative upper bound on the discriminant. If $D$ is the bound computed by computing successive greatest common divisors of discriminants of characteristic polynomials of Hecke operators $T_p$, until the gcd stabilized $15$ times in a row, then $D$ is $2^{10}$ times the correct discriminant (in the case $N=389$, of course).} In the case of $X_0(389)$, $\T\tensor \Q = K_1\cross K_2\cross K_3\cross K_6\cross K_{20},$ where~$K_d$ has degree~$d$ over~$\Q$. We have \begin{eqnarray*} K_1&=&\Q, \\ K_2&=&\Q(\sqrt{2})\\ K_3&=&\Q(\beta) , \quad \beta^3-4\beta-2=0,\\ K_6&=&\Q(\gamma), \quad \gamma^6+3\gamma^5-2\gamma^4-8\gamma^3+2\gamma^2+4\gamma-1=0,\\ K_{20}&=&\Q(\delta), \quad \delta^{20}-3\delta^{19}-29\delta^{18}+91\delta^{17}+338\delta^{16}-1130\delta^{15}-2023\delta^{14}+7432\delta^{13}\\ &&\qquad +6558\delta^{12}-28021\delta^{11}-10909\delta^{10}+61267\delta^9 +6954\delta^8-74752\delta^7\\ &&\qquad +1407\delta^6+46330\delta^5 -1087\delta^4-12558\delta^3-942\delta^2+960\delta+148=0. \end{eqnarray*} \comment{Note that the given generators of these fields are the eigenvalues $a_2$ of the corresponding eigenforms.} The discriminants of the $K_i$ are \begin{center} \begin{tabular}{|c|c|c|c|}\hline $K_1$ & $K_2$ & $K_3$ & $K_6$ \\\hline 1 & $2^3$ & $\quad 2^2\cdot 37$ & $\quad 5^3\cdot 3881$ \\\hline \end{tabular} \end{center} and $$ \disc(K_{20}) = 2^{14}\cdot 5\cdot 389 \cdot 215517113148241\cdot 477439237737571441. $$ Observe that the discriminant of $K_{20}$ is divisible by~$389$. The product of the discriminants is $$2^{19}\cdot 5^4\cdot 37\cdot 389\cdot 3881\cdot 215517113148241\cdot 477439237737571441.$$ This differs from the exact discriminant by a factor of $2^{34}\cdot 3^4\cdot 5^2\cdot 31^2$, so the index of $\T$ in its normalization is $$[\tilde{\T}:\T]=2^{17}\cdot 3^2\cdot 5\cdot 31.$$ Notice that $389$ does not divide this index, and that $389$ is not a ``congruence prime'', so~$389$ does not divide any modular degrees. \begin{question} Is there a newform optimal quotient $A_f$ of $J_0(p)$ such that~$p$ divides the modular degree of $A_f$? (No, if $p<14000$.) \end{question} \comment{% this is wrong since the Hecke algebra doesn't split as a product! Away from the primes $211$ and $65011$, $a_2$ actually generates the ring $\Z(f_{20})=\Z[a_1,a_2,\ldots]$ generated by the Fourier coefficients of one of the degree $20$ eigenforms. The discriminant of the order generated by $a_2$ divided by the discriminant of the maximal order in $K_{20}$ is $2^{44}\cdot 5^2\cdot 211^2\cdot 65011^2$. Thus the discriminant of $\T$ must be divisible by $$2^{63}\cdot 5^6\cdot 37\cdot 389\cdot 3881\cdot 215517113148241\cdot 477439237737571441.$$ } \subsection{Congruences Primes in $S_{p+1}(\Gamma_0(1))$} K.~Ono asked the following question, in connection with Theorem~1 of~\cite{gordon-ono:vis}. \begin{question} Let~$p$ be a prime. Is~$p$ ever a congruence prime on $S_{p+1}(\Gamma_0(1))$? More precisely, if~$K$ is the number field generated by all the eigenforms of weight $p+1$ on $\Gamma_0(1)$, can there be a prime ideal~$\wp\mid p$ for which $ f \con g \pmod{\wp} $ for distinct eigenforms $f, g \in S_{p+1}(\Gamma_0(1))$? \end{question} The answer is ``yes''. There is a standard relationship between $S_{p+1}(\Gamma_0(1))$ and $S_2(\Gamma_0(p))$. As noted in Section~\ref{sec:disc_div}, $p=389$ is a congruence prime for $S_2(\Gamma_0(389))$, so we investigate $S_{389+1}(\Gamma_0(1))$. \begin{proposition} There exist distinct newforms $f, g\in S_{389+1}(\Gamma_0(1))$ and a prime $\wp$ of residue characteristic $389$ such that $f\con g \pmod{\wp}$. \end{proposition} \begin{proof} We compute the characteristic polynomial~$f$ of the Hecke operator $T_2$ on $S_{389+1}(\Gamma_0(1))$ using nothing more than \cite[Ch.~VII]{serre:arithmetic}. We find that~$f$ factors modulo $389$ as follows: \begin{eqnarray*} \fbar&=&(x + 2)(x + 56)(x + 135)(x + 158)(x + 175)^2(x + 315)(x + 342)(x^2 + 387)\\ &&(x^2 + 97x + 164)(x^2 + 231x + 64)(x^2 + 286x + 63)\\ &&(x^5 + 88x^4 + 196x^3 + 113x^2 + 168x + 349)\\ &&(x^{11} + 276x^{10} + 182x^9 + 13x^8 + 298x^7 + 316x^6 + 213x^5 \\ &&\qquad+ 248x^4 + 108x^3 + 283x^2 + x + 101) \end{eqnarray*} Moreover,~$f$ is irreducible and $389\mid\mid \disc(f)$, so the square factor $(x+175)^2$ implies that~$389$ is ramified in the degree-$32$ field $L$ generated by a single root of $f$. Thus there are exactly $31$ distinct homomorphisms from the ring of integers of $L$ to $\Fbar_{389}$. That is, there are exactly $31$ ways to reduce the $q$-expansion of a newform in $S_{390}(\Gamma_0(1))$ to obtain a $q$-expansion in $\Fbar_{389}[[q]]$. Let~$K$ be the field generated by all eigenvalues of the $32$ newforms $g_1, \ldots g_{32} \in S_{390}(\Gamma_0(1))$, and let $\wp$ be a prime of $\O_K$ lying over $389$. Then the subset $\{g_1 \pmod{\wp}, g_2 \pmod{\wp}, \ldots, g_{32}\pmod{\wp}\}$ of $\Fbar_{389}[[q]]$ has cardinality at most $31$, so there exists $i \neq j$ such that $g_i \con g_j \pmod{\wp}$. \end{proof} \section{Supersingular Points in Characteristic $389$} \subsection{The Supersingular $j$-invariants In Characteristic $389$} Let $\alpha$ be a root of $\alpha^2+95\alpha+20$. Then the $33=g(X_0(389))+1$ supersingular $j$-invariants in $\F_{389^2}$ are $$ \begin{array}{l} 0, 7, 16, 17, 36, 121, 154, 220, 318, 327, 358, 60\alpha + 22, 68\alpha + 166, 80\alpha + 91, 86\alpha + 273, \\ 93\alpha + 333, 123\alpha + 350, 123\alpha + 375, 129\alpha + 247, 131\alpha + 151, 160\alpha + 321, 176\alpha + 188,\\ 213\alpha + 195, 229\alpha + 292, 258\alpha + 154, 260\alpha + 51, 266\alpha + 335, 266\alpha + 360, 296\alpha + 56, \\ 303\alpha + 272, 309\alpha + 271, 321\alpha + 319, 329\alpha + 157. \end{array} $$ Figure~\ref{fig:graph_t2} contains the graph of the Hecke operator $T_2$ computed using method of Mestre and Oesterl\'{e} \cite{mestre:graphs}. Here, if $T_2(j) = \sum_{i} j_i$, then there is an edge from the node~$j$ to each of the nodes~$j_i$. Thus, for example, $$ T_2([318]) = [0] + [93\alpha+333] + [296\alpha+56]. $$ \begin{center} \begin{figure} \psfrag{1}{\small\!\!\!\!\!\!\!\!\!\!\!$0$} \psfrag{2}{\small\!\!\!\!\!\!\!\!\!\!\!$7$} \psfrag{3}{\small\!\!\!\!\!\!\!\!\!\!\!$16$} \psfrag{4}{\small\!\!\!\!\!\!\!\!\!\!\!$17$} \psfrag{5}{\small\!\!\!\!\!\!\!\!\!\!\!$36$} \psfrag{6}{\small\!\!\!\!\!\!\!\!\!\!\!$121$} \psfrag{7}{\small\!\!\!\!\!\!\!\!\!\!\!$154$} \psfrag{8}{\small\!\!\!\!\!\!\!\!\!\!\!$220$} \psfrag{9}{\small\!\!\!\!\!\!\!\!\!\!\!$318$} \psfrag{10}{\small\!\!\!\!\!\!\!\!\!\!\!$327$} \psfrag{11}{\small\!\!\!\!\!\!\!\!\!\!\!$358$} \psfrag{12}{\small\!\!\!\!\!\!\!\!\!\!\!$60\alpha + 22$} \psfrag{13}{\small\!\!\!\!\!\!\!\!\!\!\!$68\alpha + 166$} \psfrag{14}{\small\!\!\!\!\!\!\!\!\!\!\!$80\alpha + 91$} \psfrag{15}{\small\!\!\!\!\!\!\!\!\!\!\!$86\alpha + 273$} \psfrag{16}{\small\!\!\!\!\!\!\!\!\!\!\!$93\alpha + 333$} \psfrag{17}{\small\!\!\!\!\!\!\!\!\!\!\!$123\alpha + 350$} \psfrag{18}{\small\!\!\!\!\!\!\!\!\!\!\!$123\alpha + 375$} \psfrag{19}{\small\!\!\!\!\!\!\!\!\!\!\!$129\alpha + 247$} \psfrag{20}{\small\!\!\!\!\!\!\!\!\!\!\!$131\alpha + 151$} \psfrag{21}{\small\!\!\!\!\!\!\!\!\!\!\!$160\alpha + 321$} \psfrag{22}{\small\!\!\!\!\!\!\!\!\!\!\!$176\alpha + 188$} \psfrag{23}{\small\!\!\!\!\!\!\!\!\!\!\!$213\alpha + 195$} \psfrag{24}{\small\!\!\!\!\!\!\!\!\!\!\!$229\alpha + 292$} \psfrag{25}{\small\!\!\!\!\!\!\!\!\!\!\!$258\alpha + 154$} \psfrag{26}{\small\!\!\!\!\!\!\!\!\!\!\!$260\alpha + 51$} \psfrag{27}{\small\!\!\!\!\!\!\!\!\!\!\!$266\alpha + 335$} \psfrag{28}{\small\!\!\!\!\!\!\!\!\!\!\!$266\alpha + 360$} \psfrag{29}{\small\!\!\!\!\!\!\!\!\!\!\!$296\alpha + 56$} \psfrag{30}{\small\!\!\!\!\!\!\!\!\!\!\!$303\alpha + 272$} \psfrag{31}{\small\!\!\!\!\!\!\!\!\!\!\!$309\alpha + 271$} \psfrag{32}{\small\!\!\!\!\!\!\!\!\!\!\!$321\alpha + 319$} \psfrag{33}{\small\!\!\!\!\!\!\!\!\!\!\!$329\alpha + 157$} \includegraphics[height=0.98\textheight,width=1.1\textwidth]{graph_t2.epsi} %\caption{Figure~\ref{fig:graph_t2}: Graph of Hecke Operator $T_2$} \caption{Graph of the Hecke Operator $T_2$\label{fig:graph_t2}} \end{figure} \end{center} \comment{ > S := SupersingularInvariants(389); > k := GF(389^2); > P := Sort([Reverse(Eltseq(a)) : a in S]); > B := [GF(389^2)!Reverse(a) : a in P]; > X := SupersingularModule(389); > X`ss_points := [ : x in B]; > T2 := HeckeOperator(X,2); > Attach("graph.m"); > Digraph(T2); ... shell-> dot -Tps t2.dot -o graph_t2.ps; ps2epsi graph_t2.ps graph_t2.epsi } \subsection{Gross Triple-Product Business} \section{Miscellaneous} \subsection{The Shafarevich-Tate Group}\label{sec:sha} Using visibility theory \cite[\S4.2]{agashe-stein:visibility}, one sees that $\#\Sha(A_5)$ is divisible by an odd prime, because $$ (\Z/5\Z)^2 \ncisom A_1(\Q)/5 A_1(\Q) \subset \Sha(A_{5}). $$ Additional computations suggest the following conjecture. \begin{conjecture} $N=389$ is the smallest level such that there is an optimal newform quotient $A_f$ of $J_0(N)$ with $\#\Sha(A_f)$ divisible by an odd prime. \end{conjecture} \subsection{Weierstrass Points on $X_0^+(p)$}\label{sec:atkin} Oliver Atkin has conjectured that $389$ is the largest prime such that the cusp on $X_0^+(389)$ fails to be a Weierstrass point. He verified that the cusp of $X_0^+(389)$ is not a Weierstrass point but that the cusp of $X_0^+(p)$ is a Weierstrass point for all primes~$p$ such that $389 = tr(Ti*Tj). \\ The word ``usually'' is used in the description because the program \\ can fail if the matrices aren't ``generic enough.'' See \\ the ``coords'' function. {nrows(M)=matsize(M)[1];} {ncols(M)=matsize(M)[2];} \\ coords: make a matrix whose columns are the col's columns of all of the Ti {coords(col, i,j)=matrix(nrows(T[1]),ncols(T),i,j, T[j][i,col]);} \\ Find an integral basis for the module. \\ The columns of the matrix returned give the \\ linear combination a1*T1+a2*T2+...+an*Tn, ai in Z, \\ which give a system of generators for the module. {intbasis( i,j,n,C,h,U,M)= n=nrows(T[1]); i=1; C=coords(i); while(matrank(C)ncols(T), print("Error, not enough generators."); return(0);); C=coords(i);); print("The ", i,"th columns coordinatize the module."); print("Computing mathnf."); U=mathnf(C,2)[2]; print("Extracting integral basis."); \\ take exactly those columns of U, so that when multiplied \\ by C, the result is =/= 0. M=matrix(nrows(U),n,i,j,0); j=1; for(i=1,ncols(U), if(C*U[,i]!=0, if(j>n,print("ERROR: PARI's mathnf is incorrect.");return(0);); M[,j]=U[,i]; j++) ); if(j=j,trace(T[i]*T[j]),0)); for(i=1,nrows(B),for(j=i+1,ncols(B),B[i,j]=B[j,i])); B;} \\ A = computed using intbasis; an integral basis of the Hecke algebra, \\ expressed on the T[i]. \\ B = computed using Bmat; matrix of inner products (tr(T[i]*T[j])). {DiscMat(A,B)=mattranspose(A)*B*A;} \end{verbatim}