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Author: William A. Stein
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\title{The Modular Curve $X_0(389)$:\\
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{\large Discriminants, Ranks, Shafarevich-Tate Groups, and
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Weierstrass Points}}
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\author{William Stein}
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\date{April 2002}
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\renewcommand{\T}{\mathbf{T}}
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\begin{document}
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\maketitle
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\section{Introduction}
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Let $N$ be a positive integer, and let $X_0(N)$ be the compactified
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coarse moduli space that classifies pairs $(E,C)$ where $E$ is an
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elliptic curve and $C$ is a cyclic subgroup of order~$N$. The space
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$X_0(N)$ has a canonical structure of algebraic curve over~$\Q$, and
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its properties have been very well studied during the last forty years.
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For example, Breuil, Conrad, Diamond, Taylor, and Wiles proved that
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every elliptic curve over~$\Q$ is a quotient of some $X_0(N)$.
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The smallest~$N$ such that the Jacobian of $X_0(N)$ has positive
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Mordell-Weil rank is~$37$, and Zagier studied the genus-two curve
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$X_0(37)$ in depth in his paper~\cite{zagier:modular}. From this
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viewpoint, the next modular curve deserving intensive investigation is
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$X_0(389)$, which is the first modular curve whose Jacobian has
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Mordell-Weil rank larger than that predicted by the signs in the
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functional equations of the $L$-series attached to simple factors of
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its Jacobian; in fact, $389$ is the smallest conductor of an elliptic
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curve with Mordell-Weil rank~$2$. Note that $389$ is prime and
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$X_0(389)$ has genus $g=32$, which is much larger than the genus~$2$
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of $X_0(37)$, which makes explicit investigation more challenging.
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Work of Kolyvagin \cite{kolyvagin:weil, kolyvagin:subclass} and
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Gross-Zagier \cite{gross-zagier} has completely resolved the rank
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assertion of the Birch and Swinnerton-Dyer conjecture (see, e.g.,
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\cite{tate:bsd}) for elliptic curves~$E$ with $\ord_{s=1} L(E,s) \leq
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1$. The lowest-conductor elliptic curve~$E$ that doesn't submit to
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the work of Kolyvagin and Gross-Zagier is the elliptic curve~$E$ of
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conductor~$389$ mentioned in the previous paragraph. At present we
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don't even have a conjectural natural construction of a finite-index
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subgroup of $E(\Q)$ analogous to that given by Gross and Zagier for
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rank~$1$ (but see Mazur's work on universal norms, which might be used
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to construct $E(\Q)\tensor\Z_p$ for some auxiliary prime~$p$).
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Inspired by the above observations, and with an eye towards providing
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helpful data for anyone trying to generalize the work of Gross,
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Zagier, and Kolyvagin, in this paper we compute everything we can
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about the modular curve $X_0(389)$. Some of the computations of this
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paper have already proved important in several other papers: the
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discriminant of the Hecke algebra attached to $X_0(389)$ plays a roll
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in \cite{ribet:torsion}, the verification of condition 3 in
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\cite{merel-stein}, and the remark after Theorem~1 of
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\cite{gordon-ono:vis}; also, the arithmetic of $J_0(389)$ provides a
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key example in \cite[\S4.2]{agashe-stein:visibility}. Finally, this
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paper serves as an entry in an ``encyclopaedia, atlas or hiker's
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guide to modular curves'', in the spirit of N.~Elkies
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(see \cite[pg.~22]{elkies:ffield}).
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We hilight several surprising ``firsts'' that occur at level~$389$.
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The discriminant of the Hecke algebra attached to $S_2(\Gamma_0(389))$
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has the apparently unusual property that it is divisible by~$p=389$
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(see Section~\ref{sec:disc_div}). Also $N=389$ is the smallest integer such
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that the order of vanishing of $L(J_0(N),s)$ at $s=1$ is larger than
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predicted by the functional equations of eigenforms (see Section~\ref{sec:mwranks}).
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The author conjectures that $N=389$ is the smallest
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level such that an optimal newform factor of $J_0(N)$
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appears to have Shafarevich-Tate group with nontrivial odd
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part (see Section~\ref{sec:sha}).
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Atkin conjectures that $389$ is the largest prime such that
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the cusp of $X_0^+(389)$ fails to be a
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Weierstrass point (see Section~\ref{sec:atkin}).
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\noindent{\bf Acknowledgement.}
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Noam Elkies, Ken Ribet, Matt Baker.
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Hendrik Lenstra, for suggesting a method to compute discriminants
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of Hecke algebra efficiently (see Section~\ref{sec:hecke_algebra}).
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\section{Factors of $J_0(389)$}
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To each newform $f\in S_2(\Gamma_0(389))$, Shimura \cite{shimura:factors}
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associated a quotient $A_f$ of $J_0(389)$, and
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$J_0(389)$ is isogeneous to the product $\prod A_f$,
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where the product runs over the $\Gal(\Qbar/\Q)$-conjugacy classes of newforms.
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Moreover, because $389$ is prime each factor $A_f$ cannot be decomposed further
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up to isogeny, even over $\Qbar$ (see \cite{ribet:endo}).
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\subsection{Newforms of Level $389$}
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There are five $\Gal(\Qbar/\Q)$-conjugacy classes of newforms in $S_2(\Gamma_0(389))$.
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The first class corresponds to the unique elliptic curve of conductor $389$, and its
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$q$-exansion begins
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$$
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f_1 = q - {2}q^{2} - {2}q^{3} + {2}q^{4} - {3}q^{5} + {4}q^{6} - {5}q^{7} + q^{9} + {6}q^{10} + \cdots.
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$$
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The second has coefficients in the quadratic field $\Q(\sqrt{2})$, and has $q$-expansion
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$$
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f_2 = q + {\sqrt{2}}q^{2}+({\sqrt{2} - {2}})q^{3} - q^{5}+({{-2}\sqrt{2} + {2}})q^{6}+ \cdots.
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$$
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The third has coefficients in the cubic field generated by a root $\alpha$ of
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$x^3-4x-2$:
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$$
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f_3 = q + {\alpha}q^{2} - {\alpha}q^{3}+({\alpha^{2} - {2}})q^{4}+({-\alpha^{2} + 1})q^{5} - {\alpha^{2}}q^{6} +\cdots.
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$$
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The fourth has coefficients that generate the degree-six field defined by
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a root $\beta$ of $x^6+3x^5-2x^4-8x^3+2x^2+4x-1$ and $q$-expansion
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$$
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f_4 = q + {\beta}q^{2}+({\beta^{5} + {3}\beta^{4} - {2}\beta^{3} - {8}\beta^{2} + \beta + {2}})q^{3}+\cdots.
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$$
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The fifth and final newform (up to conjugacy) has coefficients that generate
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the degree $20$ field defined by a root of
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\begin{eqnarray*}
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f_5 &=& x^{20} - 3x^{19} - 29x^{18} + 91x^{17} + 338x^{16} - 1130x^{15} - 2023x^{14} + 7432x^{13} \\
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&& + 6558x^{12} - 28021x^{11} - 10909x^{10} + 61267x^{9} + 6954x^8 - 74752x^7 \\
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&& + 1407x^6+ 46330x^5 - 1087x^4 - 12558x^3 - 942x^2 + 960x + 148.
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\end{eqnarray*}
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\subsubsection{Congruences}
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The vertices in Figure~\ref{fig:cong} correspond to the newforms $f_i$; there is an edge between
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$f_i$ and $f_j$ labeled~$p$ if there is a maximal ideal $\wp\mid p$ of the field generated
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by the Fourier coefficients of $f_i$ and $f_j$ such that $f_i \con f_j \pmod{\wp}$.
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\begin{figure}
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\begin{center}
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\psfrag{f1}{$f_1$}
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\psfrag{f2}{$f_2$}
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\psfrag{f3}{$f_3$}
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\psfrag{f4}{$f_4$}
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\psfrag{f5}{$f_5$}
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\psfrag{2 }{$2$}
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\psfrag{2 }{$2$}
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\psfrag{2,5}{$2,5$}
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\psfrag{3 }{$3$}
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\psfrag{31}{$31$}
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\psfrag{2}{$2$}
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\psfrag{2 }{$2$}
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\psfrag{2 }{$2$}
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\psfrag{2 }{$2$}
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\includegraphics{cong.epsi}
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\caption{Congruences Between Newforms\label{fig:cong}}
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\end{center}
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\end{figure}
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\comment{
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> D := SortDecomposition(NewformDecomposition(CuspidalSubspace(ModularSymbols(389,2))));
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S := CuspidalSubspace(ModularForms(389,2));
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f := [* *]; for i in [1..5] do Append(~f,Newform(S,i)); end for;
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for i in [1..5] do
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for j in [i+1..5] do
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F := Factorization(#CongruenceGroup(Parent(f[i]),Parent(f[j]),200));
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if #F gt 0 then
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printf "%o--%o [label=\"", i,j;
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for i in [1..#F] do
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printf "%o",F[i][1];
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if i lt #F then
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printf ",";
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end if;
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end for;
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printf "\"];\n";
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end if;
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end for;
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end for;
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shell-> ps2epsi cong.ps cong.epsi
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}
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\subsection{Isogeny Structure}
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We deduce from the above determination of the newforms in $S_2(\Gamma_0(389))$
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that $J_0(389)$ is $\Q$-isogenous to a product of $\Qbar$-simple abelian varieties
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$$
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J \sim A_1\cross A_2 \cross A_3 \cross A_4 \cross A_{5}.
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$$
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View the duals $A_i^{\vee}$ of the $A_i$ as abelian subvarieties of $J_0(389)$.
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Using modular symbols as in \cite[\S3.4]{agashe-stein:bsd}
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we find that, for $i\neq j$, a prime~$p$
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divides $\#(A_i^{\vee}\intersect A_j^{\vee})$ if and only if $f_i\con f_j\pmod{\wp}$
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for some prime $\wp\mid p$ (recall that the congruence primes
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are given in Figure~\ref{fig:cong} above).
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\subsection{Mordell-Weil Ranks}\label{sec:mwranks}
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Suppose $f\in S_2(\Gamma_0(N))$ is a newform of some level~$N$.
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The functional equation for $L(f,s)$ implies that $\ord_{s=1}L(f,s)$
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is odd if and only if the sign of the eigenvalue of the Atkin-Lehner
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involution $W_N$ on~$f$ is $+1$.
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\begin{proposition}\label{prop:minanrank}
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If $f\in S_2(\Gamma_0(N))$ is a newform of level $N<389$, then
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$\ord_{s=1}L(f,s)$ is either$0$ or~$1$.
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\end{proposition}
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\begin{proof}
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The proof amounts to a large computation, which divides into two parts:
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\begin{enumerate}
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\item Verify, for each newform~$f$ of level $N<389$ such that
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$W_N(f) = -f$, that $L(f,1) = *\int_{0}^{i\infty} f(z) dz$ (for some nonzero $*$)
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is nonzero. This is a purely algebraic computation involving modular symbols.
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\item Verify, for each newform~$f$ of level $N<389$ such that
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$W_N(f) = f$, that $L'(f,1)\neq 0$ (see \cite[\S4.1]{empirical},
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which points to \cite[\S2.11,\S2.13]{cremona:algs}).
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We do this by approximating an infinite
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series that converges to $L'(f,1)$ and noting that the value we
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get is far from~$0$.
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\end{enumerate}
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\end{proof}
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Thus $N=389$ is the smallest level such that the $L$-series of
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some factor $A_f$ of $J_0(N)$ has order of vanishing higher than
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that which is forced by the sign in the functional equation.
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%The elliptic curve~$A_1$ of rank~$2$ is the lowest-conductor elliptic
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%curve having rank~$>1$, because every elliptic curve over~$\Q$ is modular
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%\cite{breuil-conrad-diamond-taylor}, and \cite{cremona:algs} doesn't contain
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%any elliptic curve of conductor $<389$ having rank~$>1$.
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%The $L$-function corresponding to~$A_{5}$ does not vanish at $s=1$,
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%so $A_{5}$ has analytic rank~$0$, and hence algebraic rank~$0$, by
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%the theorem of Kolyvagin and Logachev.
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%The genus of $X^+=X/w_{389}$ is $g^{+} =11$, and $J_0(389)^{-}$
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%is isogeneous to $A_{1} \cross A_{5}.$
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\begin{proposition}\label{prop:anrank}
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The following table summarizes the dimensions and Mordell-Weil ranks
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(over the image of the Hecke ring) of the newform factors of $J_0(N)$:
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\begin{center}
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\begin{tabular}{|l|c|c|c|c|c|}\hline
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\text{\rm }& $A_1$ & $A_2$ & $A_3$ & $A_4$ & $A_{5}$\\\hline
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\text{\rm Dimension}& $1$ & $2$ & $3$ & $6$ & $20$\\\hline
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\text{\rm Rank} & $2$ & $1$ & $1$ & $1$ & $0$\\\hline
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\end{tabular}
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\end{center}
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\end{proposition}
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\begin{proof}
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The elliptic curve $A_1$ is $389A$ in Cremona's tables, which is
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the elliptic curve of smallest conductor having rank $2$.
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For $A_{5}$ we directly compute whether or not the $L$-function
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vanishes using modular symbols, by taking an inner product with
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the winding element $e_w=-\{0,\infty\}$. We find that the $L$-function
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does not vanish. By Kolyvagin-Logachev, it follows that $A_{5}$ has
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Mordell-Weil rank $0$.
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For each of the other three factors, the sign of the functional
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equation is odd, so the analytic ranks are odd.
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As in the proof of Proposition~\ref{prop:minanrank},
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we verify that the analytic rank is~$1$ in each case. By work of
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Gross, Zagier, and Kolyvagin it follows that the ranks are~$1$.
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\end{proof}
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%The elliptic curve $A_1$ has minimal Weierstrass model
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% $$y^2+y=x^3+x^2-2x.$$
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%It satisfies $A_1(\Q)=\Z\oplus \Z$. The modular
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%degree is $40$.
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\comment{\subsection{Characteristic polynomials}
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The characteristic polynomials of the first two Hecke operators are
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as follows:
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\begin{eqnarray*}
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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)\\
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&&(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}\\
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&&- {28021}x^{11} - {10909}x^{10} + {61267}x^{9} + {6954}x^{8}- {74752}x^{7} + {1407}x^{6} + {46330}x^{5}\\
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&& - {1087}x^{4} - {12558}x^{3} - {942}x^{2} + {960}x + {148})\\
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T_3&=&(x + 2)(x^2 + 4x + 2)(x^3 -4x + 2)(x^6 + 5x^5 + 4x^4 -13x^3 -21x^2 -6x + 1)\\
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&&(x^{20} -11x^{19} + 19x^{18} + 204x^{17} -845x^{16} -781x^{15} + 8883x^{14} -6177x^{13} -40916x^{12}\\
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&&+ 63058x^{11} + 85034x^{10} -215618x^9 -46920x^8 + 342529x^7 -84612x^6 -241030x^5 \\
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&&+ 112365x^4 + 51018x^3 -28526x^2 + 3560x -100)
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\end{eqnarray*}
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}
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\section{The Hecke algebra}\label{sec:hecke_algebra}
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\subsection{The Discriminant is Divisible By $p$}
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\label{sec:disc_div}
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Let $N$ be a positive integer.
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The Hecke algebra $\T\subset\End(S_2(\Gamma_0(N)))$ is
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the subring generated by all Hecke operators $T_n$ for $n=1,2,3,\ldots$.
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We are concerned with the {\em discriminant} of the
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trace pairing $(t,s)\mapsto \Tr(ts)$.
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When~$N$ is prime, $\T_\Q=\T\tensor_\Z\Q$ is a product
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$K_1\cross\cdots\cross K_n$ of totally real number fields.
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Let $\tilde{\T}$ denote the integral closure of~$\T$ in $\T_\Q$;
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note that $\tilde{\T}=\prod \O_i$ where~$\O_i$ is the ring
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of integers of $K_i$.
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Then $\disc(\T)=[\tilde{\T}:\T]\cdot \prod_{i=1}^n\disc(K_i)$.
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%One can view the primes dividing $\prod_{i=1}^n\disc(K_i)$ as being
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%associated to singularities of the irreducible components of $\Spec(\T)$,
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%and the primes dividing $[\tilde{\T}:\T]$ are associated to intersections
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%of irreducible components, or, equivalently, congruences between eigenforms.
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\begin{proposition}
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The discriminant of the Hecke algebra associated to
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$S_2(\Gamma_0(389))$ is
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$$
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2^{53}\cdot{}3^4\cdot{}5^6\cdot{}31^2\cdot{}37\cdot{}389
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\cdot{}3881\cdot{}215517113148241\cdot{}477439237737571441.
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$$
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\end{proposition}
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\begin{proof}
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By \cite{agashe-stein:schoof-appendix}, the Hecke algebra~$\T$
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is generated as a $\Z$-module by $T_1,T_2,\ldots T_{65}$.
309
%Then $\disc(\T)$ can be quickly computed once one finds
310
%a $\Z$-basis for~$\T$, which can be accomplished as follows.
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To compute $\disc(\T)$, we proceed as follows.
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First, compute the space $\sS_2(\Gamma_0(389))$ of
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cuspidal modular symbols, which is a faithful $\T$-module.
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Choose a random element $x\in \sS_2(\Gamma_0(389))_+$ of the $+1$-quotient
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of the cuspidal modular symbols, then compute the
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images $v_1=T_1(x), v_2=T_2(x), \ldots, v_{65}=T_{65}(x)$. If these don't span
317
a space of dimension $32=\rank_\Z \T$ choose a new random element~$x$
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and repeat. Using the Hermite Normal Form, find a $\Z$-basis $b_1,\ldots, b_{32}$ for
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the $\Z$-span of $v_1,\ldots, v_{65}$. The trace pairing on $\T$ induces a trace
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pairing on the $v_i$, and hence on the $b_i$. Then $\disc(\T)$ is the discriminant
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of this pairing on the $b_i$. The reason we embed $\T$ in $\sS_2(\Gamma_0(389))_+$
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as $\T{}x$ is because directly finding a $\Z$-basis for $\T$ would involve computing
323
the Hermite Norm Form of a list of $65$ vectors in a $1024$-dimensional space, which
324
is unnecessarily difficult (though possible).
325
\end{proof}
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328
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We compute this discriminant by applying the definition of
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discriminant to a matrix representation of the first~$65$ Hecke
331
operators $T_1,\ldots, T_{65}$. Matrices representing these
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Hecke operators were computed using the modular symbols algorithms
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described in \cite{cremona:algs}.
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[Sturm, {\em On the congruence of modular forms}].
335
%\end{proof}
336
337
%It might be possible to choose a
338
%bound smaller than~$65$ by directly considering a basis of
339
%modular forms.
340
%\comment{ A \pari{} program which can usually compute the discriminant
341
%of a commutative $\Z$-module represented by matrices is included as an
342
%appendix. }
343
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\comment{
345
We should also note that by computing successive greatest common divisors of
346
discriminants of characteristic polynomials of Hecke operators $T_p$, one
347
can get a fairly good multiplicative upper bound on the discriminant.
348
If $D$ is the bound computed by computing successive greatest common divisors of
349
discriminants of characteristic polynomials of Hecke operators $T_p$, until the
350
gcd stabilized $15$ times in a row, then $D$ is $2^{10}$ times the correct
351
discriminant (in the case $N=389$, of course).}
352
353
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In the case of $X_0(389)$,
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$\T\tensor \Q = K_1\cross K_2\cross K_3\cross K_6\cross K_{20},$
356
where~$K_d$ has degree~$d$ over~$\Q$.
357
We have
358
\begin{eqnarray*}
359
K_1&=&\Q, \\
360
K_2&=&\Q(\sqrt{2})\\
361
K_3&=&\Q(\beta) , \quad \beta^3-4\beta-2=0,\\
362
K_6&=&\Q(\gamma),
363
\quad \gamma^6+3\gamma^5-2\gamma^4-8\gamma^3+2\gamma^2+4\gamma-1=0,\\
364
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}\\
365
&&\qquad +6558\delta^{12}-28021\delta^{11}-10909\delta^{10}+61267\delta^9 +6954\delta^8-74752\delta^7\\
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&&\qquad +1407\delta^6+46330\delta^5
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-1087\delta^4-12558\delta^3-942\delta^2+960\delta+148=0.
368
\end{eqnarray*}
369
370
\comment{Note that the given generators of these fields are the eigenvalues
371
$a_2$ of the corresponding eigenforms.}
372
373
The discriminants of the $K_i$ are
374
\begin{center}
375
\begin{tabular}{|c|c|c|c|}\hline
376
$K_1$ & $K_2$ & $K_3$ & $K_6$ \\\hline
377
1 & $2^3$ & $\quad 2^2\cdot 37$ & $\quad 5^3\cdot 3881$ \\\hline
378
\end{tabular}
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\end{center}
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and
381
$$
382
\disc(K_{20}) = 2^{14}\cdot 5\cdot 389 \cdot 215517113148241\cdot 477439237737571441.
383
$$
384
Observe that the discriminant of $K_{20}$ is divisible by~$389$.
385
The product of the discriminants is
386
$$2^{19}\cdot 5^4\cdot 37\cdot 389\cdot 3881\cdot 215517113148241\cdot
387
477439237737571441.$$
388
This differs from the exact discriminant by a factor
389
of $2^{34}\cdot 3^4\cdot 5^2\cdot 31^2$, so
390
the index of $\T$ in its normalization is
391
$$[\tilde{\T}:\T]=2^{17}\cdot 3^2\cdot 5\cdot 31.$$
392
Notice that $389$ does not divide this index, and that $389$ is not
393
a ``congruence prime'', so~$389$ does not divide any
394
modular degrees.
395
396
\begin{question}
397
Is there a newform optimal quotient $A_f$ of $J_0(p)$
398
such that~$p$ divides the modular degree of $A_f$?
399
(No, if $p<14000$.)
400
\end{question}
401
402
\comment{% this is wrong since the Hecke algebra doesn't split as a product!
403
Away from the primes $211$ and $65011$, $a_2$ actually generates the
404
ring $\Z(f_{20})=\Z[a_1,a_2,\ldots]$ generated by the Fourier
405
coefficients of one of the degree $20$ eigenforms. The discriminant
406
of the order generated by $a_2$ divided by the discriminant of the
407
maximal order in $K_{20}$ is $2^{44}\cdot 5^2\cdot 211^2\cdot 65011^2$.
408
Thus the discriminant of $\T$ must be divisible by
409
$$2^{63}\cdot 5^6\cdot 37\cdot 389\cdot 3881\cdot 215517113148241\cdot
410
477439237737571441.$$
411
}
412
413
\subsection{Congruences Primes in $S_{p+1}(\Gamma_0(1))$}
414
K.~Ono asked the following question, in connection with
415
Theorem~1 of~\cite{gordon-ono:vis}.
416
\begin{question}
417
Let~$p$ be a prime. Is~$p$ ever a congruence prime on $S_{p+1}(\Gamma_0(1))$?
418
More precisely, if~$K$ is the number field generated by all the
419
eigenforms of weight $p+1$ on $\Gamma_0(1)$, can there be a prime
420
ideal~$\wp\mid p$ for which
421
$
422
f \con g \pmod{\wp}
423
$
424
for distinct eigenforms $f, g \in S_{p+1}(\Gamma_0(1))$?
425
\end{question}
426
427
The answer is ``yes''.
428
There is a standard relationship between
429
$S_{p+1}(\Gamma_0(1))$ and $S_2(\Gamma_0(p))$. As noted
430
in Section~\ref{sec:disc_div}, $p=389$ is a
431
congruence prime for $S_2(\Gamma_0(389))$, so we investigate
432
$S_{389+1}(\Gamma_0(1))$.
433
434
\begin{proposition}
435
There exist distinct newforms $f, g\in S_{389+1}(\Gamma_0(1))$
436
and a prime $\wp$ of residue characteristic $389$ such that
437
$f\con g \pmod{\wp}$.
438
\end{proposition}
439
\begin{proof}
440
We compute the characteristic polynomial~$f$ of the Hecke operator $T_2$
441
on $S_{389+1}(\Gamma_0(1))$ using nothing
442
more than \cite[Ch.~VII]{serre:arithmetic}.
443
We find that~$f$ factors modulo $389$ as follows:
444
\begin{eqnarray*}
445
\fbar&=&(x + 2)(x + 56)(x + 135)(x + 158)(x + 175)^2(x + 315)(x + 342)(x^2 + 387)\\
446
&&(x^2 + 97x + 164)(x^2 + 231x + 64)(x^2 + 286x + 63)\\
447
&&(x^5 + 88x^4 + 196x^3 + 113x^2 + 168x + 349)\\
448
&&(x^{11} + 276x^{10} + 182x^9 + 13x^8 + 298x^7 + 316x^6 + 213x^5 \\
449
&&\qquad+ 248x^4 + 108x^3 + 283x^2 + x + 101)
450
\end{eqnarray*}
451
Moreover,~$f$ is irreducible and $389\mid\mid \disc(f)$, so
452
the square factor $(x+175)^2$ implies that~$389$
453
is ramified in the degree-$32$ field $L$ generated by a single root of $f$.
454
Thus there are exactly $31$ distinct homomorphisms from the ring of integers of $L$ to
455
$\Fbar_{389}$. That is, there are exactly $31$ ways to reduce the $q$-expansion of a
456
newform in $S_{390}(\Gamma_0(1))$ to obtain a
457
$q$-expansion in $\Fbar_{389}[[q]]$.
458
Let~$K$ be the field generated by all eigenvalues of the $32$
459
newforms $g_1, \ldots g_{32} \in S_{390}(\Gamma_0(1))$, and let $\wp$
460
be a prime of $\O_K$ lying over $389$.
461
Then the subset
462
$\{g_1 \pmod{\wp}, g_2 \pmod{\wp}, \ldots, g_{32}\pmod{\wp}\}$
463
of $\Fbar_{389}[[q]]$ has cardinality at most $31$, so
464
there exists $i \neq j$ such that
465
$g_i \con g_j \pmod{\wp}$.
466
\end{proof}
467
468
469
\section{Supersingular Points in Characteristic $389$}
470
\subsection{The Supersingular $j$-invariants In Characteristic $389$}
471
Let $\alpha$ be a root of $\alpha^2+95\alpha+20$. Then the
472
$33=g(X_0(389))+1$ supersingular $j$-invariants in $\F_{389^2}$ are
473
$$
474
\begin{array}{l}
475
0, 7, 16, 17, 36, 121, 154, 220, 318, 327, 358, 60\alpha + 22, 68\alpha + 166, 80\alpha + 91, 86\alpha + 273, \\
476
93\alpha + 333, 123\alpha + 350, 123\alpha + 375, 129\alpha + 247, 131\alpha + 151, 160\alpha + 321, 176\alpha + 188,\\
477
213\alpha + 195, 229\alpha + 292, 258\alpha + 154, 260\alpha + 51, 266\alpha + 335, 266\alpha + 360, 296\alpha + 56, \\
478
303\alpha + 272, 309\alpha + 271, 321\alpha + 319, 329\alpha + 157.
479
\end{array}
480
$$
481
Figure~\ref{fig:graph_t2} contains the graph of the Hecke operator $T_2$ computed
482
using method of Mestre and Oesterl\'{e} \cite{mestre:graphs}.
483
Here, if $T_2(j) = \sum_{i} j_i$, then there is an edge from
484
the node~$j$ to each of the nodes~$j_i$.
485
Thus, for example,
486
$$
487
T_2([318]) = [0] + [93\alpha+333] + [296\alpha+56].
488
$$
489
\begin{center}
490
\begin{figure}
491
\psfrag{1}{\small\!\!\!\!\!\!\!\!\!\!\!$0$}
492
\psfrag{2}{\small\!\!\!\!\!\!\!\!\!\!\!$7$}
493
\psfrag{3}{\small\!\!\!\!\!\!\!\!\!\!\!$16$}
494
\psfrag{4}{\small\!\!\!\!\!\!\!\!\!\!\!$17$}
495
\psfrag{5}{\small\!\!\!\!\!\!\!\!\!\!\!$36$}
496
\psfrag{6}{\small\!\!\!\!\!\!\!\!\!\!\!$121$}
497
\psfrag{7}{\small\!\!\!\!\!\!\!\!\!\!\!$154$}
498
\psfrag{8}{\small\!\!\!\!\!\!\!\!\!\!\!$220$}
499
\psfrag{9}{\small\!\!\!\!\!\!\!\!\!\!\!$318$}
500
\psfrag{10}{\small\!\!\!\!\!\!\!\!\!\!\!$327$}
501
\psfrag{11}{\small\!\!\!\!\!\!\!\!\!\!\!$358$}
502
\psfrag{12}{\small\!\!\!\!\!\!\!\!\!\!\!$60\alpha + 22$}
503
\psfrag{13}{\small\!\!\!\!\!\!\!\!\!\!\!$68\alpha + 166$}
504
\psfrag{14}{\small\!\!\!\!\!\!\!\!\!\!\!$80\alpha + 91$}
505
\psfrag{15}{\small\!\!\!\!\!\!\!\!\!\!\!$86\alpha + 273$}
506
\psfrag{16}{\small\!\!\!\!\!\!\!\!\!\!\!$93\alpha + 333$}
507
\psfrag{17}{\small\!\!\!\!\!\!\!\!\!\!\!$123\alpha + 350$}
508
\psfrag{18}{\small\!\!\!\!\!\!\!\!\!\!\!$123\alpha + 375$}
509
\psfrag{19}{\small\!\!\!\!\!\!\!\!\!\!\!$129\alpha + 247$}
510
\psfrag{20}{\small\!\!\!\!\!\!\!\!\!\!\!$131\alpha + 151$}
511
\psfrag{21}{\small\!\!\!\!\!\!\!\!\!\!\!$160\alpha + 321$}
512
\psfrag{22}{\small\!\!\!\!\!\!\!\!\!\!\!$176\alpha + 188$}
513
\psfrag{23}{\small\!\!\!\!\!\!\!\!\!\!\!$213\alpha + 195$}
514
\psfrag{24}{\small\!\!\!\!\!\!\!\!\!\!\!$229\alpha + 292$}
515
\psfrag{25}{\small\!\!\!\!\!\!\!\!\!\!\!$258\alpha + 154$}
516
\psfrag{26}{\small\!\!\!\!\!\!\!\!\!\!\!$260\alpha + 51$}
517
\psfrag{27}{\small\!\!\!\!\!\!\!\!\!\!\!$266\alpha + 335$}
518
\psfrag{28}{\small\!\!\!\!\!\!\!\!\!\!\!$266\alpha + 360$}
519
\psfrag{29}{\small\!\!\!\!\!\!\!\!\!\!\!$296\alpha + 56$}
520
\psfrag{30}{\small\!\!\!\!\!\!\!\!\!\!\!$303\alpha + 272$}
521
\psfrag{31}{\small\!\!\!\!\!\!\!\!\!\!\!$309\alpha + 271$}
522
\psfrag{32}{\small\!\!\!\!\!\!\!\!\!\!\!$321\alpha + 319$}
523
\psfrag{33}{\small\!\!\!\!\!\!\!\!\!\!\!$329\alpha + 157$}
524
\includegraphics[height=0.98\textheight,width=1.1\textwidth]{graph_t2.epsi}
525
%\caption{Figure~\ref{fig:graph_t2}: Graph of Hecke Operator $T_2$}
526
\caption{Graph of the Hecke Operator $T_2$\label{fig:graph_t2}}
527
\end{figure}
528
\end{center}
529
\comment{
530
> S := SupersingularInvariants(389);
531
> k<a> := GF(389^2);
532
> P := Sort([Reverse(Eltseq(a)) : a in S]);
533
> B := [GF(389^2)!Reverse(a) : a in P];
534
> X := SupersingularModule(389);
535
> X`ss_points := [<x,x> : x in B];
536
> T2 := HeckeOperator(X,2);
537
> Attach("graph.m");
538
> Digraph(T2);
539
...
540
541
shell-> dot -Tps t2.dot -o graph_t2.ps; ps2epsi graph_t2.ps graph_t2.epsi
542
}
543
544
\subsection{Gross Triple-Product Business}
545
546
\section{Miscellaneous}
547
548
\subsection{The Shafarevich-Tate Group}\label{sec:sha}
549
Using visibility theory \cite[\S4.2]{agashe-stein:visibility},
550
one sees that $\#\Sha(A_5)$ is divisible by an odd prime, because
551
$$
552
(\Z/5\Z)^2 \ncisom A_1(\Q)/5 A_1(\Q) \subset \Sha(A_{5}).
553
$$
554
Additional computations suggest the following conjecture.
555
\begin{conjecture}
556
$N=389$ is the smallest level such that there is an optimal newform
557
quotient $A_f$ of $J_0(N)$ with $\#\Sha(A_f)$
558
divisible by an odd prime.
559
\end{conjecture}
560
561
562
\subsection{Weierstrass Points on $X_0^+(p)$}\label{sec:atkin}
563
Oliver Atkin has conjectured that $389$ is the largest prime such that
564
the cusp on $X_0^+(389)$ fails to be a Weierstrass point. He verified
565
that the cusp of $X_0^+(389)$ is not a Weierstrass point but
566
that the cusp of $X_0^+(p)$ is a Weierstrass point for all primes~$p$
567
such that $389<p\leq 883$ (see, e.g., \cite[pg.39]{elkies:ffield}).
568
In addition, the author has extended the verification of Atkin's conjecture
569
for all primes $<3000$. Explicitly, this involves computing a reduced-echelon
570
basis for the subspace of $S_2(\Gamma_0(p))$ where the Atkin-Lehner involution
571
$W_p$ acts as $+1$, and comparing the largest valuation of an element of this
572
basis with the dimension of the subspace. These numbers differ exactly when
573
the cusp is a Weierstrass point.
574
575
576
\comment{
577
function IsWP(p)
578
M := CuspidalSubspace(ModularSymbols(p,2,+1));
579
W := AtkinLehnerSubspace(M,p,+1);
580
d := Dimension(W);
581
Q := qExpansionBasis(W,d+5);
582
return Coefficient(Q[#Q],d) eq 0;
583
end function;
584
}
585
586
\subsection{A Property of the Plus Part of the Integral Homology}
587
For any positive integer~$N$, let $H^+(N) = H_1(X_0(N),\Z)^+$ be the
588
$+1$ eigen-submodule for the action of complex conjugation on the
589
integral homology of $X_0(N)$. Then $H^+(N)$ is a module over the
590
Hecke algebra~$\T$. Let
591
$$
592
F^+(N) = \coker\left(H^+(N) \cross \Hom(H^+(N),\Z) \ra \Hom(\T,\Z)\right)
593
$$
594
where the map sends $(x,\vphi)$ to the homomorphism
595
$t\mapsto \vphi(tx)$. Then $\#F^+(p)\in \{1,2,4\}$ for
596
all primes $p<389$, but $\#F^+(389) = 8$.
597
598
\subsection{The Field Generated By Points of
599
Small Prime Order On an Elliptic Curve}
600
The prime $389$ arises in a key way in the verification of condition~3
601
in \cite{merel-stein}.
602
603
604
\bibliography{biblio}
605
606
\end{document}
607
608
609
\section{PARI code}
610
The following is a \pari{} program that might be helpful
611
to the reader interested in computing discriminants of $\Z$-modules.
612
In retrospect it looks ``unfinished''.
613
\begin{verbatim}
614
\\ disc.gp -- Usually compute the discriminant of a finite commutative
615
\\ Z-module represented by matrices T1,...,Tn, such that
616
\\ the inner product is <Ti,Tj> = tr(Ti*Tj).
617
\\ The word ``usually'' is used in the description because the program
618
\\ can fail if the matrices aren't ``generic enough.'' See
619
\\ the ``coords'' function.
620
{nrows(M)=matsize(M)[1];}
621
{ncols(M)=matsize(M)[2];}
622
\\ coords: make a matrix whose columns are the col's columns of all of the Ti
623
{coords(col, i,j)=matrix(nrows(T[1]),ncols(T),i,j, T[j][i,col]);}
624
\\ Find an integral basis for the module.
625
\\ The columns of the matrix returned give the
626
\\ linear combination a1*T1+a2*T2+...+an*Tn, ai in Z,
627
\\ which give a system of generators for the module.
628
{intbasis(
629
i,j,n,C,h,U,M)=
630
n=nrows(T[1]); i=1; C=coords(i);
631
while(matrank(C)<n,
632
i++; if(i>ncols(T), print("Error, not enough generators."); return(0););
633
C=coords(i););
634
print("The ", i,"th columns coordinatize the module.");
635
print("Computing mathnf.");
636
U=mathnf(C,2)[2];
637
print("Extracting integral basis.");
638
\\ take exactly those columns of U, so that when multiplied
639
\\ by C, the result is =/= 0.
640
M=matrix(nrows(U),n,i,j,0);
641
j=1;
642
for(i=1,ncols(U),
643
if(C*U[,i]!=0,
644
if(j>n,print("ERROR: PARI's mathnf is incorrect.");return(0););
645
M[,j]=U[,i]; j++)
646
);
647
if(j<n,print("ERROR: PARI's mathnf is producing invalid results."));
648
M;
649
}
650
{Bmat( i=0,j=0)=
651
B=matrix(ncols(T),ncols(T),i,j,if(i>=j,trace(T[i]*T[j]),0));
652
for(i=1,nrows(B),for(j=i+1,ncols(B),B[i,j]=B[j,i]));
653
B;}
654
\\ A = computed using intbasis; an integral basis of the Hecke algebra,
655
\\ expressed on the T[i].
656
\\ B = computed using Bmat; matrix of inner products (tr(T[i]*T[j])).
657
{DiscMat(A,B)=mattranspose(A)*B*A;}
658
\end{verbatim}
659
660