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Author: William A. Stein
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%From: "Edward F. Schaefer" <[email protected]>
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%Hi all, below is what I sent to Mathematics of Computation,
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%\section{Introduction}
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%\label{intro}
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% \normalsize
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%Cheers, Ed
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\documentclass[12pt]{amsart}
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\usepackage{amscd}
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\newfont{\cyr}{wncyr10 scaled \magstep1}
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\newcommand{\Sh}{\hbox{\cyr Sh}}
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\newcommand{\C}{{\mathbf C}}
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\newcommand{\Q}{{\mathbf Q}}
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\newcommand{\Qbar}{\overline{\Q}}
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%\newcommand{\GalQ}{{\Gal}(\Qbar/\Q)}
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\newcommand{\JJ}{{\mathcal J}}
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\newcommand{\DD}{{\mathcal D}}
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\newcommand{\tors}{_{\text{tors}}}
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\newtheorem{prop}[theorem]{Proposition}
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\theoremstyle{definition}
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\newtheorem{conj}{Conjecture}
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\begin{document}
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\title[Modular Jacobians]{Empirical evidence for the Birch and
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Swinnerton-Dyer conjectures for
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modular Jacobians of genus~2 curves}
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\author{E.\ Victor Flynn}
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\address{Department of Mathematical Sciences, University of
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Liverpool, P.O.Box 147,
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Liverpool L69 3BX, England}
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\email{evflynn@liverpool.ac.uk}
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\author{Franck Lepr\'{e}vost}
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\address{CNRS Equipe d'arithm\'etique, Institut de Math\'ematiques de Paris,
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Universit\'e Paris 6,
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Tour 46-56, 5\`eme \'etage, Case 247,
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2-4 place Jussieu, F-75252 Paris cedex 05, France}
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\email{leprevot@math.jussieu.fr}
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\author{Edward F.\ Schaefer}
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\address{Department of Mathematics and Computer Science \\
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Santa Clara University \\ Santa Clara, CA 95053, USA}
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\email{eschaefe@math.scu.edu}
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\author{William A.\ Stein}
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\address{Department of Mathematics \\ University of California
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at Berkeley \\ Berkeley, CA 94720, USA}
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\email{was@math.berkeley.edu}
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\author{Michael Stoll}
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\address{Mathematisches Institut, Universit\"{a}tsstr.\ 1, D-40225
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D\"{u}sseldorf, Germany}
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\email{stoll@math.uni-duesseldorf.de}
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\author{Joseph L.\ Wetherell}
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\address{Department of Mathematics, University of Southern California,
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1042 W.\ 36th Place, Los Angeles, CA 90089-1113, USA}
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\email{jlwether@alum.mit.edu}
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\subjclass{Primary 11G40; Secondary 11G10, 11G30, 14H25, 14H40,14H45}
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\keywords{Birch and Swinnerton-Dyer conjecture, genus~2, Jacobian, modular
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abelian variety}
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\thanks{The first author thanks the Nuffield Foundation
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(Grant SCI/180/96/71/G) for financial support.
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The second author did some of the research at
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the Max-Planck Institut f\"ur Mathematik and
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the Technische Universit\"at Berlin.
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The third author thanks the National Security Agency (Grant
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MDA904-99-1-0013).
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The fourth author was supported by a Sarah M. Hallam fellowship.
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The fifth author did some of the research at
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the Max-Planck-Institut f\"ur Mathematik.
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The sixth author thanks the National Science Foundation
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(Grant DMS-9705959).
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The authors had useful conversations with John Cremona, Qing Liu,
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Karl Rubin and
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Peter Swinnerton-Dyer and are grateful to Xiangdong Wang and Michael
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M\"{u}ller for making data available to them.}
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\date{August 11, 1999}
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\begin{abstract}
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This paper provides empirical evidence for the Birch and
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Swinnerton-Dyer conjectures for modular Jacobians of genus~2 curves.
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The second of these conjectures relates six quantities associated to
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a Jacobian over the rational numbers. One of these
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six quantities is
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the size of the Shafarevich-Tate group.
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Unable to compute that, we
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computed the five other quantities and solved for the last one. In
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all 32~cases, the result is very close to an integer that is a power
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of~2. In addition, this power of~2 agrees with the size of the
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2-torsion of the Shafarevich-Tate group, which we could compute.
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\end{abstract}
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\maketitle
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\markboth{FLYNN, LEPR\'{E}VOST, SCHAEFER, STEIN, STOLL, AND WETHERELL}%
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{GENUS~2 BIRCH AND SWINNERTON-DYER CONJECTURE}
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% \pagebreak
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\section{Introduction}
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\label{intro}
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% \normalsize
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% \baselineskip=18pt
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The conjectures of Birch and Swinnerton-Dyer, originally stated
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for elliptic curves over~$\Q$, have been a constant source of
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motivation for the study of elliptic curves, with the ultimate
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goal being to find a proof.
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This has resulted not only in a better
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theoretical understanding, but also in the development of better
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algorithms for computing the analytic and arithmetic
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invariants that are so intriguingly related by them. We now know
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that the first and, up to a non-zero rational factor, the
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second conjecture hold for modular elliptic curves over~$\Q$
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\footnote{It has recently been announced by
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Brueil, Conrad, Diamond and Taylor that they have extended Wiles'
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results and shown
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that all elliptic curves over~$\Q$ are modular.}
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in the
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analytic rank~0 and~1 cases (see \cite{GZ,Ko,Wal1,Wal2}).
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Furthermore,
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a number of people have provided numerical evidence for the
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conjectures for a large number of elliptic curves; see
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for example~\cite{BSD,Ca,Cr}.
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By now, our theoretical and algorithmic knowledge of curves of
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genus~2 and their Jacobians has reached a state that makes it
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possible to conduct similar investigations. The Birch and
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Swinnerton-Dyer conjectures have been generalized to arbitrary
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abelian varieties over number fields by Tate~\cite{Ta}. If
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$J$ is the Jacobian of a genus~2 curve over $\Q$,
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then the first conjecture
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states that the order of vanishing of the $L$-series of the Jacobian at
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$s=1$ (the {\em analytic rank}) is equal to the Mordell-Weil rank of the
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Jacobian. The second conjecture is that
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\begin{equation} \label{eqn1}
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\lim\limits_{s \to 1} (s-1)^{-r} L(J,s) =
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\Omega \cdot {\rm Reg} \cdot \prod\limits_{p} c_{p}
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\cdot \#\Sh(J,\Q ) \cdot (\#J(\Q)\tors)^{-2} \,.
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\end{equation}
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In this equation, $L(J,s)$ is the $L$-series of the Jacobian
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$J$, and $r$ is its analytic rank. We use $\Omega$ to denote the
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integral over $J(\R)$ of a particular differential 2-form; the
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precise choice of this differential is described in
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Section~\ref{Omega}. ${\rm Reg}$ is the regulator of $J(\Q)$. For
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primes $p$, we use $c_{p}$ to denote the size of $J(\Q_p)/J^0(\Q_p)$,
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where $J^0(\Q_p)$ is defined in Section~\ref{Tamagawa}. We let
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$\Sh(J,\Q)$ be the Shafarevich-Tate group of $J$ over $\Q$, and we let
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$J(\Q)\tors$ denote the torsion subgroup of $J(\Q)$.
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As in the case of elliptic curves, the first conjecture assumes
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that the $L$-series can be analytically continued to $s = 1$,
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and the second conjecture additionally assumes that the
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Shafarevich-Tate group is finite. Neither of these assumptions is
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known to hold for arbitrary genus~2 curves. The analytic
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continuation of the $L$-series, however, is known to exist for
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modular abelian varieties over~$\Q$, where an abelian
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variety is called {\em modular} if it is a quotient of the Jacobian~$J_0(N)$
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of the modular curve~$X_0(N)$ for some level~$N$. For simplicity,
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we will also call a genus~2 curve {\em modular} when its Jacobian is
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modular in this sense. So it is certainly a good idea to look
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at modular genus~2 curves over~$\Q$, since we then at least know that the
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statement of the first conjecture makes sense. Moreover, for many modular
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abelian varieties it is also known that the Shafarevich-Tate group
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is finite, therefore the statement of the second conjecture also
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makes sense. As it turns out, all of our examples belong to this
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class. An additional benefit of choosing modular genus~2 curves is
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that one can find lists of such curves in the literature.
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In this article, we provide empirical evidence for the Birch and
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Swinnerton-Dyer conjectures for such modular genus~2 curves. Since there
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is no known effective way of computing the size of the Shafarevich-Tate
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group, we computed the other five terms in equation~\eqref{eqn1}
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(in two different ways, if possible). This required several different
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algorithms, some of which were developed or improved while we were
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working on this paper. If one of these algorithms
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is already well described in the literature, then we simply cite it.
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Otherwise, we describe it here in some detail (in particular,
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algorithms for computing $\Omega$ and
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$c_p$).
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For modular abelian varieties associated to newforms whose
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$L$-series have analytic rank~0 or~1, the first Birch and Swinnerton-Dyer
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conjecture has been proven. In such cases, the
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Shafarevich-Tate group is also known to be finite and the second conjecture
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has been proven, up to a non-zero rational factor. This all
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follows {}from results in
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\cite{GZ,KL,Wal1,Wal2}.
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In our examples, all of the analytic
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ranks are either~0 or~1. Thus we already know that the first
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conjecture holds. Since the Jacobians we consider are associated to a
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quadratic conjugate pair of newforms, the analytic rank of the
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Jacobian is twice the analytic rank of either newform (see \cite{GZ}).
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The second Birch and Swinnerton-Dyer conjecture has not been proven
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for the cases we consider. In order to verify equation~\eqref{eqn1},
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we computed the five terms other than $\#\Sh(J,\Q)$ and solved for
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$\#\Sh(J,\Q)$. In each case, the value is an integer to within the
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accuracy of our calculations. This number is a power of~2, which
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coincides with the independently computed size of the 2-torsion
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subgroup of~$\Sh(J,\Q)$. Hence, we have verified the second
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Birch and Swinnerton-Dyer conjecture for our curves at least
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numerically, if we assume that the Shafarevich-Tate group consists
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of 2-torsion only. (This is an ad hoc assumption based only
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on the fact that we do not know better.) See Section~\ref{Shah} for
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circumstances under which the verification is exact.
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The curves are listed in Table~\ref{table1},
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and the numerical results can be found in Table~\ref{table2}.
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\section{The Curves}
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\label{curves}
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Each of the genus~2 curves we consider is related to the Jacobian
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$J_0(N)$ of the modular curve $X_0(N)$ for some level $N$. When only
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one of these genus~2 curves arises {}from a given level $N$, then we
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denote this curve by $C_N$; when there are two curves coming {}from level
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$N$ we use the notation $C_{N,A}$, $C_{N,B}$. The relationship
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of, say, $C_N$ to $J_0(N)$ depends on the source. Briefly, {}from
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Hasegawa \cite{Hs} we obtain quotients of $X_0(N)$ and {}from Wang
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\cite{Wan} we obtain curves whose Jacobians are quotients of $J_0(N)$.
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In both cases the Jacobian $J_N$ of $C_N$ is isogenous to a
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2-dimensional factor of $J_0(N)$. (When not referring to a specific
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curve, we will typically drop the subscript $N$ {}from $J$.)
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In this way we can also associate
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$C_N$ with a 2-dimensional subspace of $S_2(N)$, the space of cusp
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forms of weight~2 for $\Gamma_0(N)$.
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We now discuss the precise source of the genus~2 curves we will
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consider. Hasegawa \cite{Hs} has provided exact equations for all
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genus~2 curves which are quotients of $X_0(N)$ by a subgroup of the
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Atkin-Lehner involutions. There are 142 such curves. We are
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particularly interested in those where the Jacobian corresponds to a
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subspace of $S_2(N)$ spanned by a quadratic conjugate pair of
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newforms. There are 21 of these with level $N \leq 200$. For these
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curves we will provide evidence for the second conjecture. There are
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seven more such curves with $N > 200$. We can classify the other
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2-dimensional subspaces into four types. There are
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2-dimensional subspaces of oldforms that are irreducible under the
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action of the Hecke algebra. There are also 2-dimensional subspaces
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that are reducible under the action of the Hecke algebra and are
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spanned by two oldforms, two newforms or one of each. The Jacobians
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corresponding to the latter three kinds are always isogenous, over
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$\Q$, to the product of two elliptic curves. Given the small levels,
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these are elliptic curves for which Cremona \cite{Cr} has already
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provided evidence for the Birch and Swinnerton-Dyer conjectures. In
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Table~\ref{Hasegawa}, we describe the kind of cusp forms spanning the
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2-dimensional subspace and the signs of their functional equations
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{}from the level at which they are newforms. The analytic and
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Mordell-Weil ranks were always the smallest possible given those signs.
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The second set of curves was created by Wang \cite{Wan} and is further
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discussed in \cite{FM}. This set consists of 28 curves that were
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constructed by considering the spaces $S_2(N)$ with $N \leq 200$.
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Whenever a subspace spanned by a pair of quadratic conjugate newforms
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was found, these newforms were integrated to produce a quotient
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abelian variety~$A$ of $J_0(N)$. These quotients are {\em optimal} in the
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sense of \cite{Ma}, in that the kernel of the quotient map is
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connected.
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The period matrix for~$A$ was created using certain intersection
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numbers. When all of the intersection numbers have the same value,
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then the polarization on~$A$ induced {}from the canonical polarization
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of~$J_0(N)$ is equivalent to a principal polarization. (Two
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polarizations are {\em equivalent} if they differ by an integer multiple.)
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Conversely, every 2-dimensional optimal quotient of $J_0(N)$ in which
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the induced polarization is equivalent to a principal polarization is
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found in this way.
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Using theta functions, numerical approximations were found for the
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Igusa invariants of the abelian surfaces. These numbers coincide with
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rational numbers of fairly small height within the limits of the
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precision used for the computations. Wang then constructed curves
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defined over~$\Q$ whose Igusa invariants are the rational numbers
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found. (There is one abelian surface at level $N = 177$ for which Wang
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was not able to find a curve.) If we assume that these rational
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numbers are the true Igusa invariants of the abelian surfaces, then it
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follows that Wang's curves have Jacobians isomorphic, over~$\Qbar$, to
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the principally polarized abelian surfaces in his list. Since the
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classification given by these invariants is only up to isomorphism
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over~$\Qbar$, the Jacobians of Wang's curves are not necessarily
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isomorphic to, but can be twists of, the optimal quotients
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of~$J_0(N)$ over~$\Q$ (see below).
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There are four curves in Hasegawa's list which do not show up in
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Wang's list (they are listed in Table~\ref{table1} with an $H$ in the
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last column). Their Jacobians are quotients of~$J_0(N)$, but are not
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optimal quotients. It is likely that there are modular genus~2 curves
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which neither are Atkin-Lehner quotients of~$X_0(N)$ (in Hasegawa's
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sense) nor have Jacobians that are optimal quotients. These curves
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could be found by looking at the optimal quotient abelian surfaces and
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checking whether they are isogenous to a principally polarized abelian
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surface over $\Q$.
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For 17 of the curves in Wang's list, the 2-dimensional subspace
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spanned by the newforms is the same as that giving one of Hasegawa's
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curves. In all of those cases, the curve given by Wang's equation is
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isomorphic, over $\Q$, to that given by Hasegawa. This verifies Wang's
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equations for these 17 curves. They are listed in Table~\ref{table1}
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with $HW$ in the last column.
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The remaining eleven curves (listed in Table~\ref{table1} with a
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$W$ in the last column) derive from the other eleven optimal
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quotients in Wang's list. These are described in more detail in
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Section~\ref{bad11} below.
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With the exception of curves $C_{63}$, $C_{117,A}$ and $C_{189}$, the
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Jacobians of all of our curves are absolutely simple, and the
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canonically polarized Jacobians have automorphism groups of size two.
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We showed that these Jacobians are absolutely simple using an argument
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like those in \cite{Le,Sto1}. The automorphism group of the
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canonically polarized Jacobian of a hyperelliptic curve is isomorphic
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to the automorphism group of the curve (see \cite[Thm.\
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12.1]{Mi2}). Each automorphism of a hyperelliptic curve is induced by
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a linear fractional transformation on $x$-coordinates (see \cite[p.\
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1]{CF}). Each automorphism also permutes the six Weierstrass
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points. Once we believed we had found all of the automorphisms, we
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were able to show that there are no more by considering all linear
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fractional transformations sending three fixed Weierstrass points to
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any three Weierstrass points. In each case, we worked with sufficient
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accuracy to show that other linear fractional transformations did not
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permute the Weierstrass points.
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Let $\zeta_{3}$ denote a primitive third root of unity. The
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Jacobians of curves $C_{63}$, $C_{117,A}$ and $C_{189}$ are each
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isogenous to the product of two elliptic curves over $\Q(\zeta_3)$,
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though not over $\Q$, where they are simple. These genus~2 curves
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have automorphism groups of size 12. In the following table we list
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the curve at the left. On the right we give one of the elliptic
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curves which is a factor of its Jacobian. The second factor is the
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conjugate.
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\[
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\begin{array}{ll}
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C_{63}: & y^2 = x(x^2 + (9 - 12\zeta_{3})x - 48\zeta_{3}) \\
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C_{117,A}: & y^2 = x(x^2 - (12 + 27\zeta_{3})x - (48 + 48\zeta_{3})) \\
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C_{189}: & y^2 = x^3 + (66 - 3\zeta_{3})x^2 + (342 + 81\zeta_{3})x
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+ 105 + 21\zeta_{3}
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\end{array}
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\]
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Note that these three Jacobians are examples of abelian varieties
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`with extra twist' as discussed in~\cite{Cr2}, where they can be
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found in the tables on page~409.
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\subsection{Models for the Wang-only curves}
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\label{bad11}
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As we have already noted, a modular genus~2 curve may be found by
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either, both, or neither of Wang's and \linebreak
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Hasegawa's techniques.
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Hasegawa's method allows for the exact determination, over $\Q$, of
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the equation of any modular genus~2 curve it has found. On the other
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hand, if Wang's technique detects a modular genus~2 curve $C_N$, his
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method produces real approximations to a curve $C'_N$ which is defined
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over $\Q$ and is isomorphic to $C_N$ over $\Qbar$. We will call
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$C'_N$ a {\em twisted modular genus~2 curve}.
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In this section we attempt to determine equations for the eleven
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modular genus~2 curves detected by Wang but not by Hasegawa. If we
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assume that Wang's equations for the twisted modular genus~2 curves
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are correct, we find that we are able to determine the twists. In
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turn, this gives us strong evidence that Wang's equations for the
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twisted curves were correct. Undoing the twist, we determine probable
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equations for the modular genus~2 curves. We end by providing further
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evidence for the correctness of these equations.
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In what follows, we will use the notation of~\cite{Cr} and recommend
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it as a reference on the general results that we assume here and in
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Section~\ref{modular} and the appendix.
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Fix a level~$N$ and let
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$f(z) \in S_2(N)$. Then $f$ has a Fourier expansion
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\[ f(z) = \sum\limits_{n=1}^{\infty} a_{n} e^{2 \pi i n z}\,. \]
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For a newform~$f$, we have $a_1 \neq 0$; so we can normalize it by
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setting $a_1 = 1$. In our cases, the $a_n$'s are integers in a real
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quadratic field. For each prime~$p$ not dividing~$N$, the
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corresponding Euler factor of the $L$-series $L(f,s)$ is
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$1 - a_p p^{-s} + p^{1-2s}$. Let $N(a_p)$ and $Tr(a_p)$ denote the
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norm and trace of~$a_p$. The product of this Euler factor and its
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conjugate is
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$1 - Tr(a_p)\,p^{-s} + (N(a_p) + 2p)\,p^{-2s}
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- p\,Tr(a_p)\,p^{-3s} + p^2\,p^{-4s}$.
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Therefore, the characteristic
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polynomial of the $p$-Frobenius on the corresponding abelian variety
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over $\F_{p}$ is
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$x^4 - Tr(a_p)\,x^3 + (N(a_p) + 2p)\,x^2 - p\,Tr(a_p)\,x + p^2$.
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Let $C$ be a curve, over $\Q$, whose Jacobian, over $\Q$, comes {}from
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the space spanned by $f$ and its conjugate. Then we know that
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$p+1 - \#C(\F_{p}) = Tr(a_p)$ and
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$\frac{1}{2}(\#C(\F_{p})^{2} + \#C(\F_{p^2})) - (p+1)\# C(\F_{p}) - p =
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N(a_p)$ (see \cite[Lemma 3]{MS}).
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For the odd primes less than 200, not dividing $N$, we computed
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$\# C(\F_{p})$ and $\# C(\F_{p^2})$ for each curve given by one of
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Wang's equations. {}From these we could compute the characteristic
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polynomials of Frobenius and see if they agreed with those predicted
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by the $a_p$'s of the newforms.
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Of the eleven curves, the characteristic polynomials agreed for only
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four. In each of the remaining seven cases we found a twist of Wang's
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curve whose characteristic polynomials agreed with those predicted by
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the newform for all odd primes less than 200 not dividing $N$. Four
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of these twists were quadratic and three were of higher degree. It
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is these twists that appear in Table~\ref{table1}.
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We can provide further evidence that these equations are correct.
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For each curve given in Table~\ref{table1}, it is easy to determine
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the primes of singular reduction. In Section~\ref{Tamagawa} we will
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provide techniques for determining which of those primes divides the
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conductor of its Jacobian. In each case, the primes dividing the
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conductor of the Jacobian of the curve are exactly the primes
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dividing the level $N$; this is necessary. With the exception of
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curve $C_{188}$, all the curves come {}from odd levels. We used Liu's
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{\tt genus2reduction} program
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({\tt ftp://megrez.math.u-bordeaux.fr/pub/liu}) to compute the
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conductor of the curve. In each case (other than curve $C_{188}$),
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the conductor is the square of the level; this is also necessary. For
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curve $C_{188}$, the odd part of the conductor of the curve is the
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square of the odd part of the level.
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In addition, since the Jacobians of the Wang curves are optimal
478
quotients, we can compute~$k\cdot\Omega$ (where $k$ is the Manin constant,
479
conjectured to be 1)
480
using the newforms.
481
In each case, these agree (to within the accuracy of our computations)
482
with the $\Omega$'s computed using the equations for the curves.
483
We can also compute the value of~$c_p$ for optimal quotients from
484
the newforms, when $p$ exactly divides~$N$ and the eigenvalue of the
485
$p$th Atkin-Lehner involution is $-1$. When $p$ exactly divides~$N$
486
and the eigenvalue of the $p$th Atkin-Lehner involution is~$+1$, the
487
component group is either $0$, $\Z/2\Z$, or~$(\Z/2\Z)^2$. These results
488
are always in agreement with the values computed using the equations
489
for the curves. The algorithms based on the newforms are
490
described in Section~\ref{modular}, those based on the
491
equations of the curves are described in Section~\ref{algms}.
492
493
Lastly, we were able to compute the Mordell-Weil ranks of the Jacobians
494
of the curves given by ten of these eleven equations. In
495
each case it agrees with the analytic rank of the Jacobian,
496
as deduced {}from the newforms.
497
498
It should be noted that curve~$C_{125,B}$ is the $\sqrt{5}$-twist of
499
curve~$C_{125,A}$; the corresponding statement holds for the associated
500
2-dimensional subspaces of~$S_2(125)$. Since curve~$C_{125,A}$ is
501
a Hasegawa curve, this proves that the equation given in Table~\ref{table1}
502
for curve~$C_{125,B}$ is correct.
503
504
The $a_p$'s and other information concerning Wang's curves are
505
currently kept in a database at the Institut f\"{u}r experimentelle
506
Mathematik in Essen, Germany. Most recently, this database was under
507
the care of Michael M\"{u}ller. William Stein also keeps a database
508
of~$a_p$'s for newforms.
509
510
\begin{rem}
511
For the remainder of this paper we will assume that the equations for
512
the curves given in Table~\ref{table1} are correct; that is, that
513
they are equations for the curves whose Jacobians are isogenous
514
to a factor of~$J_0(N)$ in the way described above.
515
Some of the quantities can be computed either {}from the newform
516
or {}from the equation for the curve. We performed both computations
517
whenever possible, and view this duplicate effort as an attempt to
518
verify our implementation of the algorithms rather than an attempt
519
to verify the equations in Table~\ref{table1}. For most quantities,
520
one method or the other is not guaranteed to produce a value; in this
521
case, we simply quote the value {}from whichever method did succeed.
522
The reader who is disturbed by this philosophy should
523
ignore the Wang-only curves, since the equations for the Hasegawa
524
curves can be proven to be correct.
525
\end{rem}
526
527
528
\section{Algorithms for genus~2 curves}
529
\label{algms}
530
531
In this section, we describe the algorithms that are based on the
532
given models for the curves. We give algorithms that compute all
533
terms on the right hand side of equation~\eqref{eqn1}, with the
534
exception of the size of the Shafarevich-Tate group. We describe,
535
however, how to find the size of its 2-torsion subgroup.
536
537
\subsection{Torsion Subgroup}
538
\label{torsion}
539
540
The computation of the torsion subgroup of~$J(\Q)$ is straightforward.
541
We used the technique described in~\cite[pp.~78--82]{CF}.
542
This technique is not always effective, however. For an algorithm working
543
in all cases see~\cite{Sto3}.
544
545
\subsection{Mordell-Weil rank and $\Sh(J,\Q)[2]$}
546
\label{MW}
547
548
The group $J(\Q)$ is a finitely generated abelian group and so is
549
isomorphic to $\Z^{r} \oplus J(\Q)\tors$ for some $r$ called the
550
Mordell-Weil rank.
551
As noted above (see Section~\ref{intro}), we justifiably use
552
$r$ to denote both the analytic and Mordell-Weil ranks since they
553
agree for all curves in Table~\ref{table1}.
554
555
We used the algorithm described in \cite{FPS} to compute ${\rm
556
Sel}^{2}_{\rm fake}(J,\Q)$ (notation {}from \cite{PSc}), which is a
557
quotient of the 2-Selmer group ${\rm Sel}^{2}(J,\Q)$. More details
558
on this algorithm can be found in \cite{Sto2}. Theorem 13.2 of
559
\cite{PSc} explains how to get ${\rm Sel}^{2}(J,\Q)$ {}from ${\rm
560
Sel}^{2}_{\rm fake}(J,\Q)$. Let $M[2]$ denote the 2-torsion of an
561
abelian group $M$ and let dim$V$ denote the dimension of an $\F_{2}$
562
vector space $V$. We have
563
$\dim {\rm Sel}^{2}(J,\Q) = r + \dim J(\Q)[2] + \dim \Sh(J,\Q)[2]$.
564
In other words,
565
\[ \dim\, \Sh (J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2]. \]
566
567
It is interesting to note that in all 30 cases where
568
$\dim \Sh(J,\Q)[2] \le 1$, we were able to compute the Mordell-Weil rank
569
independently from the analytic rank.
570
The
571
cases where $\dim \Sh(J,\Q)[2] = 1$ are discussed in more
572
detail in Section~\ref{Shah}.
573
For both of the remaining cases we have $\dim \Sh(J,\Q)[2]=2$.
574
One of these cases is
575
$C_{125,B}$. For this curve we computed
576
${\rm Sel}^{\sqrt{5}}(J_{125,B},\Q)$
577
using the technique described in
578
\cite{Sc}. {}From this, we were able to determine that the Mordell-Weil
579
rank is 0 independently from the analytic rank.
580
For the other case,
581
$C_{133,A}$,
582
we could show that $r$ had to be either~0
583
or~2 {}from the equation, but we needed the analytic computation to
584
show that $r=0$.
585
586
\subsection{Regulator}
587
\label{reg}
588
589
When the Mordell-Weil rank is~0, then the regulator is~1. When the
590
Mordell-Weil rank is positive, then to compute the regulator, we
591
first need to find generators for $J(\Q)/J(\Q)\tors$. The regulator
592
is the determinant of the canonical height pairing matrix on this set
593
of generators. An algorithm for computing the generators and
594
canonical heights is given in~\cite{FS}; it was used to find
595
generators for $J(\Q)/J(\Q)\tors$ and to compute the regulators. In
596
that article, the algorithm for computing height constants at the
597
infinite prime is not clearly explained and there are some errors in
598
the examples. A clear algorithm for computing infinite height
599
constants is given in~\cite{Sto3}. In~\cite{Sto4}, some improvements of
600
the results and algorithms in~\cite{FS} and~\cite{Sto3} are discussed.
601
The regulators in Table~\ref{table2} have been double-checked using
602
these improved algorithms.
603
604
\subsection{Tamagawa Numbers}
605
\label{Tamagawa}
606
607
Let $\OO$ be the integer ring in~$K$ which will be $\Q_{p}$ or
608
$\Q_{p}\unr$ (the maximal unramified extension of $\Q_{p})$.
609
Let $\JJ$ be the N\'{e}ron model of~$J$ over~$\OO$.
610
Define $\JJ^{0}$ to be the open subgroup scheme of~$\JJ$ whose
611
generic fiber is isomorphic to~$J$ over~$K$ and whose special fiber
612
is the identity component of the closed fiber of~$\JJ$.
613
The group $\JJ^{0}(\OO)$ is isomorphic to a subgroup of~$J(K)$ which
614
we denote $J^{0}(K)$. The group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is
615
the component group of~$\JJ$ over~$\OO_{\Q_{p}\unr}$. We are
616
interested in computing $c_p = \#J(\Q_{p})/J^{0}(\Q_{p})$, which is
617
sometimes called the Tamagawa number.
618
Since N\'{e}ron models are stable under unramified base extension,
619
the $\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of
620
$J^{0}(\Q_{p}\unr)$ is~$J^{0}(\Q_{p})$.
621
Since $H^1(\Gal(\Q_{p}\unr/\Q_{p}), J^{0}(\Q_{p}\unr))$
622
is trivial (see~\cite[p.\ 58]{Mi1}) we see that the
623
$\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of
624
$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is $ J(\Q_{p})/J^{0}(\Q_{p})$.
625
626
There exist several discussions in the literature on constructing the
627
group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ starting with an integral
628
model of the underlying curve. For our purposes, we especially
629
recommend Silverman's book~\cite{Si}, Chapter~IV, Sections 4 and~7.
630
For a more detailed treatment, see~\cite[chap.\ 9]{BLR}. In these
631
books, one can find justifications for what we will do. While
632
constructing such groups, we ran into a number of difficulties that
633
we did not find described anywhere. For that reason, we will present
634
examples of such difficulties that arose, as well as our methods of
635
resolution. We do not claim that we will describe all situations
636
that could arise.
637
638
When computing $c_p$ we need a proper, regular model~$\CC$ for~$C$
639
over~$\Z_p$. Let $\Z_p\unr$ denote the ring of integers of~$\Q_p\unr$
640
and note that $\Z_p\unr$ is a pro-\'etale Galois extension
641
of~$\Z_p$ with Galois group
642
$\Gal(\Z_p\unr/\Z_p) = \Gal(\Q_p\unr/\Q_p)$.
643
It follows that giving a model for~$C$ over~$\Z_p$ is equivalent to
644
giving a model for~$C$ over~$\Z_p\unr$ that
645
is equipped with a Galois action. We have found it convenient to
646
always work with the latter description. Thus for us, giving a model
647
over~$\Z_p$ will always mean giving a model over~$\Z_p\unr$ together
648
with a Galois action.
649
650
In order to find a proper, regular model for~$C$ over~$\Z_p$,
651
we start with the models in Table~\ref{table1}. Technically, we
652
consider the curves to be the two affine pieces $y^2+g(x)y=f(x)$ and
653
$v^2 + u^3 g(1/u)v = u^6 f(1/u)$, glued together by $ux=1$, $v=u^3y$.
654
We blow them up at all points that are not regular until we have a
655
regular model. (A point is {\em regular} if the cotangent space there has
656
two generators.) These curves are all proper, and this is not
657
affected by blowing up.
658
659
Let $\CC_p$ denote the special fiber of~$\CC$ over~$\Z_p\unr$. The
660
group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is isomorphic to a quotient
661
of the degree~0 part of the free group on the irreducible components
662
of~$\CC_{p}$. Let the irreducible components be denoted $\DD_{i}$ for
663
$1\leq i\leq n$, and let the multiplicity of~$\DD_{i}$ in~$\CC_p$ be
664
$d_{i}$. Then the degree~0 part of the free group has the form
665
\[ L = \{ \sum\limits_{i=1}^{n} \alpha_{i}\DD_{i} \mid
666
\sum\limits_{i=1}^{n} d_{i}\alpha_{i} = 0 \}\,. \]
667
668
In order to describe the group that we quotient out by, we must
669
discuss the intersection pairing. For components $\DD_{i}$ and~$\DD_{j}$
670
of the special fiber, let $\DD_{i} \cdot \DD_{j}$ denote
671
their intersection pairing. In all of the special fibers that arise
672
in our examples, distinct components intersect transversally. Thus,
673
if $i \neq j$, then $\DD_{i} \cdot \DD_{j}$ equals the number of points
674
at which $\DD_{i}$ and $\DD_{j}$ intersect. The case of
675
self-intersection ($i=j$) is discussed below.
676
677
The kernel of the map {}from~$L$ to
678
$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is generated by
679
divisors of the form
680
\[ [\DD_j] = \sum\limits_{i=1}^{n} (\DD_{j} \cdot \DD_{i}) \DD_{i} \]
681
for each component~$\DD_j$. We can deduce $\DD_{j} \cdot \DD_{j}$ by
682
noting that $[\DD_j]$ must be contained in the group~$L$. This follows
683
{}from the fact that the intersection pairing of
684
$\CC_{p} = \sum d_i\DD_{i}$ with any irreducible component is 0.
685
686
\vspace{1mm}
687
\noindent
688
{\bf Example 1.} Curve $C_{65,B}$ over $\Z_{2}$.
689
690
An equation for curve~$C_{65,B}$ is
691
$y^2 = f(x) = -x^6 + 10x^5 - 32x^4 + 20x^3 + 40x^2 + 6x - 1$. The Jacobian
692
of this curve
693
is a quotient of the Jacobian of~$X_0(65)$. Though 65 is odd, this
694
curve has singular reduction at~2. Since the equation for this curve
695
is conjectural (it is a Wang-only curve), it will be nice to verify
696
that 2 does not divide the conductor of its Jacobian, i.e.\ that the
697
Jacobian has good reduction at~2. In addition, we will need a
698
proper, regular model for this curve in order to find~$\Omega$.
699
700
Note that $f(x)$ has a factor of
701
$x^2 - 3x - 1$. The special fiber of the arithmetic surface
702
$y^2 = f(x)$ over~$\Z_{2}\unr$
703
is given by
704
$(y + x^3 + 1)^2 = 0 \pmod 2$; this is a genus~0 curve of multiplicity~2
705
that we denote~$A$. This model is not regular at the two points
706
$(x-\alpha, y, 2)$, where $\alpha$ is a root of $x^2 - 3x - 1$. It is
707
regular at infinity so we will blow up only the given affine cover.
708
The current special fiber is in Figure~\ref{special2} and is labelled
709
{\it Fiber~1}.
710
711
We fix $\alpha$ and move $(x - \alpha, y, 2)$ to the origin using the
712
substitution $x_0 = x-\alpha$. We get
713
\[ y^2 = -x_0^6 + (-6\alpha + 10)x_0^5 + (5\alpha - 47)x_0^4
714
+ (-28\alpha + 60)x_0^3 + (-11\alpha - 2)x_0^2
715
+ (-24\alpha - 16)x_0
716
\]
717
which we rewrite as the pair of equations
718
\begin{align*}
719
g_{1}(x_{0},y,p)
720
&= -x_0^6 + (-3\alpha + 5) p x_0^5 + (5\alpha - 47) x_0^4
721
+ (-7\alpha + 15) p^2 x_0^3 \\
722
& \qquad {} + (-11\alpha - 2) x_0^2 + (-3\alpha - 2) p^3 x_0 - y^2
723
\\
724
&= 0,\\
725
p &= 2.
726
\end{align*}
727
To blow up at $(x_0,y,p)$, we introduce projective coordinates
728
$(x_1,y_1,p_1)$ with $x_{0} y_1 = x_{1} y$, $x_{0} p_{1} = x_{1} p$, and
729
$y p_1 = y_{1} p$. We look in all three affine covers and check for regularity.
730
731
\begin{description}
732
\item[$x_{1} = 1$] We have $y = x_{0} y_{1}$, $p = x_{0} p_{1}$. We get
733
$g_2(x_{0},y_{1},p_{1}) = 0$, $x_{0} p_{1} = 2$, where
734
\begin{align*}
735
g_2(x_{0},y_{1},p_{1}) &= x_{0}^{-2}g_{1}(x_{0},x_{0}y_{1},x_{0}p_{1}) \\
736
&= -x_0^4 + (-3\alpha + 5) p_1 x_0^4 + (5\alpha - 47) x_0^2
737
+ (-7\alpha + 15) p_1^2 x_0^3 \\
738
& \qquad{} + (-11\alpha - 2) + (-3\alpha - 2) p_1^3 x_0^2 - y_1^2 \,.
739
\end{align*}
740
In the reduction we have either $x_{0} = 0$ or $p_1 = 0$.
741
\begin{description}
742
\item[$x_{0} = 0$] $(y_{1} + \alpha + 1)^2 = 0$.
743
This is a new component which we denote $B$. It has genus~0 and
744
multiplicity~2. We check regularity along~$B$ at
745
$(x_{0}, y_{1} + \alpha + 1, p_{1}-t, 2)$, with $t$ in $\Z_2\unr$, and
746
find that $B$ is nowhere regular.
747
\item[$p_{1} = 0$]
748
$(y_{1} + x_{0}^2 + \alpha x_{0} + (\alpha + 1))^2 = 0$.
749
Using the gluing maps, we see that this is~$A$.
750
\end{description}
751
752
\item[$y_{1} = 1$] We get no new information {}from this affine cover.
753
754
\item[$p_{1} = 1$] We have $x_{0} = x_{1} p$, $y = y_{1} p$. We get
755
$g_{3}(x_{1},y_{1},p) = p^{-2} g_{1}(x_{1}p,y_{1}p,p) = 0$, $p = 2$.
756
In the reduction we have
757
\begin{description}
758
\item[$p=0$] $(y_1 + (\alpha+1)x_1)^2 = 0$. Using the gluing maps, we
759
see that this is~$B$. It is nowhere regular.
760
\end{description}
761
\end{description}
762
763
The current special fiber is in
764
Figure~\ref{special2} and is labelled {\it Fiber~2}. It is not regular
765
along~$B$ and at the other point on~$A$ which we have not yet blown up.
766
The component $B$ does not lie entirely in any one affine cover
767
so we will blow up the affine covers $x_1 = 1$ and $p_1 = 1$ along~$B$.
768
769
To blow up $x_1 = 1$ along~$B$ we make the substitution
770
$y_2 = y_1 + \alpha + 1$ and replace each factor of~2 in a coefficient
771
by~$x_0 p_1$. We have $g_{4}(x_0,y_2,p_1) = 0$ and $x_0 p_1 = 2$, and we
772
want to blow up along the line $(x_0, y_2, 2)$. Blowing up along a line
773
is similar to blowing up at a point: since we are blowing up at
774
$(x_0, y_2, 2) = (x_0, y_2)$, we introduce projective
775
coordinates $x_3, y_3$ together with the relation $x_0 y_3 = x_3 y_2$. We
776
have two affine covers.
777
778
\begin{description}
779
\item[$x_3 = 1$] We have $y_2 = y_{3} x_{0}$. We get
780
$g_{5}(x_{0},y_{3},p_{1}) = x_{0}^{-2} g_{4}(x_{0},y_{3}x_{0},p_1) = 0$
781
and $x_{0} p_{1} = 2$. In the reduction we have
782
\begin{description}
783
\item[$x_{0} = 0$]
784
$y_{3}^2 + (\alpha + 1) y_{3} p_{1} + \alpha p_{1}^3 + p_{1}^2
785
+ \alpha + 1 = 0$.
786
This is~$B$. It is now a non-singular genus~1 curve.
787
\item[$p_{1} = 0$] $(x_0 + y_3 + \alpha)^2 = 0$. This is~$A$. The point
788
where $B$ meets~$A$ transversally is regular.
789
\end{description}
790
791
\item[$y_3 = 1$] We get no new information {}from this affine cover.
792
\end{description}
793
794
When we blow up $p_1 = 1$ along~$B$ we get essentially the same thing and
795
all points are again regular.
796
797
The other non-regular point on~$A$ is the conjugate of the one we
798
blew up. Therefore, after performing the conjugate blow ups, it too
799
will be a genus~1 component crossing~$A$ transversally. We denote
800
this component $D$; it is conjugate to~$B$.
801
802
803
\begin{figure}
804
\caption{Special fibers of curve $C_{65,B}$ over $\Z_{2}$;
805
points not regular are thick}
806
\label{special2}
807
\begin{picture}(400,130)
808
\put(20,5){\begin{picture}(100,125)
809
\thinlines
810
\put(20,55){\line(1,0){60}}
811
\put(85,55){\makebox(0,0){A}}
812
\put(75,62){\makebox(0,0){2}}
813
\put(40,55){\circle*{5}}
814
\put(60,55){\circle*{5}}
815
\put(50,5){\makebox(0,0){Fiber 1}}
816
\end{picture}}
817
\put(145,5){\begin{picture}(100,125)
818
\thinlines
819
\put(50,5){\makebox(0,0){Fiber 2}}
820
\put(20,55){\line(1,0){60}}
821
\put(85,55){\makebox(0,0){A}}
822
\put(75,62){\makebox(0,0){2}}
823
\put(60,55){\circle*{5}}
824
\put(40,15){\line(0,1){80}}
825
\put(40.5,15){\line(0,1){80}}
826
\put(39.5,15){\line(0,1){80}}
827
\put(39,15){\line(0,1){80}}
828
\put(41,15){\line(0,1){80}}
829
\put(40,105){\makebox(0,0){B}}
830
\put(34,90){\makebox(0,0){2}}
831
\end{picture}}
832
\put(270,5){\begin{picture}(100,125)
833
\thinlines
834
\put(20,55){\line(1,0){60}}
835
\put(85,55){\makebox(0,0){A}}
836
\put(75,62){\makebox(0,0){2}}
837
\put(40,15){\line(0,1){80}}
838
\put(40,105){\makebox(0,0){B}}
839
\put(60,15){\line(0,1){80}}
840
\put(60,105){\makebox(0,0){D}}
841
\put(50,5){\makebox(0,0){Fiber 3}}
842
\end{picture}}
843
\end{picture}
844
\end{figure}
845
846
We now have a proper, regular model~$\CC$ of~$C$ over~$\Z_2$.
847
Let $\CC_2$ be the special fiber of this model; a
848
diagram of~$\CC_2$ is in Figure~\ref{special2} and is labelled
849
{\it Fiber~3}. We can use $\CC$ to show that the
850
N\'eron model $\JJ$ of the Jacobian $J = J_{65,B}$ has good
851
reduction at~2.
852
853
We know that the reduction of~$\JJ^0$ is the extension of an abelian
854
variety by a connected linear group. Since $\CC$ is regular and
855
proper, the abelian variety part of the reduction is the product of
856
the Jacobians of the normalizations of the components of~$\CC_2$ (see
857
\cite[9.3/11 and 9.5/4]{BLR}). Thus, the abelian variety part is the
858
product of the Jacobians of~$B$ and~$D$. Since this is
859
2-dimensional, the reduction of~$\JJ^0$ is an abelian variety. In
860
other words, since the sum of the genera of the components of the
861
special fiber is equal to the dimension of~$J$, the reduction is an
862
abelian variety. It follows that $\JJ$ has good reduction at~2, that
863
the conductor of~$J$ is odd, and that $c_2 = 1$. As noted above, this
864
gives further evidence that the equation given in Table~\ref{table1}
865
is correct.
866
867
868
\vspace{1mm}
869
\noindent
870
{\bf Example 2.} Curve $C_{63}$ over $\Z_{3}$.
871
872
The Tamagawa number is often found using the intersection matrix and
873
sub-determinants. This is not entirely satisfactory for cases where
874
the special fiber has several components and a non-trivial Galois
875
action. Here is an example of how to resolve this (see also~\cite{BL}).
876
877
When we blow up curve~$C_{63}$ over~$\Z_{3}\unr$, we get
878
the special fiber shown in Figure~\ref{special1}.
879
Elements of $\Gal(\Q_{3}\unr/\Q_{3})$
880
that do not fix the quadratic unramified extension of~$\Q_{3}$
881
switch $H$ and~$I$. The other components are defined over~$\Q_{3}$.
882
All components have genus~0. The group $J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$
883
is isomorphic to a quotient of
884
\begin{align*}
885
L = \{ \alpha A + \beta B + \delta D + \epsilon E + \phi F + \gamma G
886
&+ \eta H + \iota I \\
887
&\mid \alpha + \beta + 2\delta + 2\epsilon + 4\phi + 2\gamma
888
+ 2\eta + 2\iota = 0 \} \,.
889
\end{align*}
890
891
The kernel is generated by the following divisors.
892
\begin{center}
893
\begin{tabular}{*{2}{@{[}c@{]$\;=\;$}r@{\hspace{2cm}}}}
894
$A$ & $-2A + E$ & $B$ & $-2B + E$ \\
895
$D$ & $-D + E$ & $E$ & $A + B + D - 4E + F$ \\
896
$F$ & $E - 2F + G + H + I$ & $G$ & $F - 2G$ \\
897
$H$ & $F - 2H$ & $I$ & $F - 2I$
898
\end{tabular}
899
\end{center}
900
901
\begin{figure}
902
\caption{Special fiber of curve $C_{63}$ over $\Z_{3}$}
903
\label{special1}
904
\begin{picture}(400,130)
905
\put(100,5){\begin{picture}(200,125)
906
\thinlines
907
\put(20,50){\line(1,0){160}}
908
\put(40,20){\line(0,1){60}}
909
\put(60,20){\line(0,1){60}}
910
\put(80,20){\line(0,1){60}}
911
\put(150,10){\line(0,1){100}}
912
\put(120,70){\line(1,0){60}}
913
\put(120,90){\line(1,0){60}}
914
\put(120,30){\line(1,0){60}}
915
\put(40,88){\makebox(0,0){G}}
916
\put(60,88){\makebox(0,0){H}}
917
\put(80,88){\makebox(0,0){I}}
918
\put(150,118){\makebox(0,0){E}}
919
\put(185,50){\makebox(0,0){F}}
920
\put(185,90){\makebox(0,0){A}}
921
\put(185,70){\makebox(0,0){B}}
922
\put(185,30){\makebox(0,0){D}}
923
\put(35,70){\makebox(0,0){2}}
924
\put(55,70){\makebox(0,0){2}}
925
\put(75,70){\makebox(0,0){2}}
926
\put(165,55){\makebox(0,0){4}}
927
\put(165,35){\makebox(0,0){2}}
928
\put(145,104){\makebox(0,0){2}}
929
\end{picture}}
930
\end{picture}
931
\end{figure}
932
933
When we project away {}from~$A$, we find that
934
$J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$ is isomorphic to
935
\begin{align*}
936
\langle B, D, E, F, G, H, I
937
&\mid E = 0, E = 2B, D = E, 4E = B + D + F, \\
938
&\quad 2F = E + G + H + I, F = 2G = 2H = 2I \rangle.
939
\end{align*}
940
At this point, it is straightforward to simplify the representation by
941
elimination. Note that we projected away {}from~$A$, which is
942
Galois-invariant. It is best to continue eliminating Galois-invariant
943
elements first. We find that this group is isomorphic to
944
$\langle H, I \mid 2H = 2I = 0 \rangle$ and elements of
945
$\Gal(\Q_{3}\unr/\Q_{3})$ that do not fix the quadratic unramified
946
extension of~$\Q_{3}$ switch $H$ and~$I$. Therefore
947
$J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr) \cong \Z/2\Z \oplus \Z/2\Z$ and
948
$c_3 = \#J(\Q_{3})/J^{0}(\Q_{3}) = 2$.
949
950
\subsection{Computing $\Omega$}
951
\label{Omega}
952
953
By an {\em integral differential} (or {\em integral form}) on $J$ we mean the
954
pullback to $J$ of a global relative differential form on the N\'eron
955
model of $J$ over $\Z$. The set of integral $n$-forms on $J$ is a
956
full-rank lattice in the vector space of global holomorphic $n$-forms
957
on $J$. Since $J$ is an abelian variety of dimension 2, the integral
958
1-forms are a free $\Z$-module of rank 2 and the integral 2-forms are
959
a free $\Z$-module of rank 1. Moreover, the wedge of a basis for the
960
integral 1-forms is a generator for the integral 2-forms. The
961
quantity $\Omega$ is the integral, over the real points of $J$, of a
962
generator for the integral 2-forms. (We choose the generator that
963
leads to a positive integral.)
964
965
We now translate this into a computation on the curve $C$. Let
966
$\{\omega_1, \omega_2\}$ be a $\Q$-basis for the holomorphic
967
differentials on $C$ and let $\{\gamma_1, \gamma_2, \gamma_3,
968
\gamma_4\}$ be a $\Z$-basis for the homology of $C(\C)$. Create a
969
$2\times 4$ complex matrix $M_{\C} = [ \int_{\gamma_j}\omega_i]$ by
970
integrating the differentials over the homology and let $M_{\R} =
971
\Tr_{\C/\R}(M_{\C})$ be the $2\times 4$ real matrix whose entries are
972
traces {}from the complex matrix. The columns of $M_{\R}$ generate a
973
lattice $\Lambda$ in $\R^2$. If we make the standard identification
974
between the holomorphic 1-forms on $J$ and the holomorphic
975
differentials on $C$ (see \cite{Mi2}), then the notation
976
$\int_{J(\R)} |\omega_1 \wedge \omega_2|$ makes sense and its value
977
can be computed as the area of a fundamental domain for $\Lambda$.
978
979
If $\{\omega_1, \omega_2\}$ is a basis for the integral 1-forms on
980
$J$, then $\int_{J(\R)} |\omega_1 \wedge \omega_2| = \Omega$. On the
981
other hand, the computation of $M_{\C}$ is simplest if we choose
982
$\omega_1 = dX/Y$, and $\omega_2=X\,dX/Y$ with respect to a model for
983
$C$ of the form $Y^2=F(X)$; in this case we obtain $\Omega$ by a
984
simple change-of-basis calculation. This assumes, of course, that we
985
know how to express a basis for the integral 1-forms in terms of the
986
basis $\{\omega_1, \omega_2\}$; this is addressed in more detail
987
below.
988
989
It is worth mentioning an alternate strategy. Instead of finding a
990
$\Z$-basis for the homology of $C(\C)$ one could find a $\Z$-basis
991
$\{\gamma'_1, \gamma'_2\}$ for the subgroup of the homology that is
992
fixed by complex conjugation (call this the real homology).
993
Integrating would give us a $2\times 2$ real matrix $M'_{\R}$ and the
994
determinant of $M'_{\R}$ would equal the integral of $\omega_1
995
\wedge \omega_2$ over the connected component of $J(\R)$.
996
In other words, the number of real connected components of $J$ is
997
equal to the index of the $\C/\R$-traces in the real homology.
998
999
We now come to the question of determining the differentials on $C$
1000
which correspond to the integral 1-forms on $J$. Call these the
1001
integral differentials on $C$. This computation can be done one
1002
prime at a time. At each prime $p$ this is equivalent to determining
1003
a $\Z_p\unr$-basis for the global relative differentials on any
1004
proper, regular model for $C$ over $\Z_p\unr$. In fact a more
1005
general class of models can be used; see the discussion of models
1006
with rational singularities in \cite[\S 6.7]{BLR} and \cite[\S
1007
4.1]{Li}.
1008
1009
We start with the model $y^2 + g(x)y=f(x)$ given in
1010
Table~\ref{table1}. Note that the substitution $X=x$ and $Y=2y+g(x)$
1011
gives us a model of the form $Y^2=F(X)$. For integration purposes,
1012
our preferred differentials are $dX/Y=dx/(2y+g(x))$ and
1013
$X\,dX/Y=x\,dx/(2y+g(x))$. It is not hard to show that at primes of
1014
non-singular reduction for the $y^2 + g(x)y=f(x)$ model, these
1015
differentials will generate the integral 1-forms. For each prime $p$
1016
of singular reduction we give the following algorithm. All steps
1017
take place over $\Z_p\unr$.
1018
1019
\begin{description}
1020
\item[Step 1]
1021
Compute explicit equations for a proper, regular model $\CC$.
1022
1023
\item[Step 2]
1024
Diagram the configuration of the special fiber of $\CC$.
1025
1026
\item[Step 3] (Optional)
1027
Identify exceptional components and blow them down in the
1028
configuration diagram. Repeat step 3 as necessary.
1029
1030
\item[Step 4] (Optional)
1031
Remove components with genus 0 and self-intersection $-2$.
1032
Since $C$ has genus greater than 1,
1033
there will be a component that is not of this kind.
1034
(This
1035
step corresponds to contracting the given components to create a
1036
non-proper model with rational singularities. We will not need a
1037
diagram of the resulting configuration.)
1038
1039
\item[Step 5]
1040
Determine a $\Z_p\unr$-basis for the integral differentials. It
1041
suffices to check this on a dense open subset of each surviving
1042
component. Note that we have explicit equations for a dense open
1043
subset of each of these components {}from the model $\CC$ in step 1. A
1044
pair of differentials $\{\eta_1, \eta_2\}$ will be a basis for the
1045
integral differentials (at $p$) if the following three statements are
1046
true.
1047
\begin{description}
1048
\item[a]
1049
The pair $\{\eta_1, \eta_2\}$ is a basis for the holomorphic
1050
differentials on $C$.
1051
\item[b]
1052
The reductions of $\eta_1$ and $\eta_2$ produce well-defined
1053
differentials mod $p$ on an open subset of each surviving component.
1054
\item[c]
1055
If $a_1\eta_1+a_2\eta_2 = 0 \pmod{p}$ on all surviving components,
1056
then $p|a_1$ and $p|a_2$.
1057
\end{description}
1058
\end{description}
1059
1060
Techniques for explicitly computing a proper, regular model are
1061
discussed in Section~\ref{Tamagawa}. A configuration diagram should
1062
include the genus, multiplicity and self-intersection number of
1063
each component and the number and type of intersections between
1064
components. Note that when an exceptional component is blown down,
1065
all of the self-intersection numbers of the components intersecting
1066
it will go up (towards 0). In particular, components which were not
1067
exceptional before may become exceptional in the new configuration.
1068
1069
Steps 3 and 4 are intended to make this algorithm more efficient for
1070
a human. They are entirely optional. For a computer implementation
1071
it may be easier to simply check every component than to worry about
1072
manipulating configurations.
1073
1074
The curves in Table~\ref{table1} are given as $y^2 + g(x)y=f(x)$. We
1075
assumed, at first, that $dx/(2y+g(x))$ and $x\,dx/(2y+g(x))$ generate
1076
the integral differentials. We integrated these differentials around
1077
each of the four paths generating the complex homology and found a
1078
provisional $\Omega$. Then we checked the proper, regular models to
1079
determine if these differentials really do generate the integral
1080
differentials and adjusted $\Omega$ when necessary. There were
1081
three curves where we needed to adjust $\Omega$. We describe the
1082
adjustment for curve $C_{65,B}$ in the following example. For curve
1083
$C_{63}$, we used the differentials $3\,dx/(2y+g(x))$ and
1084
$x\,dx/(2y+g(x))$. For curve $C_{65,A}$, we used the differentials
1085
$3\,dx/(2y+g(x))$ and $3x\,dx/(2y+g(x))$.
1086
1087
\vspace{2mm}
1088
\noindent
1089
{\bf Example 3.} Curve $C_{65,B}$.
1090
1091
The primes of singular reduction for curve $C_{65,B}$ are 2, 5 and
1092
13. In Example 1 of Section~\ref{Tamagawa}, we found a proper,
1093
regular model $\CC$ for $C$ over $\Z_2\unr$. The configuration for
1094
the special fiber of $\CC$ is sketched in Figure~\ref{special2} under
1095
the label {\it Fiber 3}. Component $A$ is exceptional and can be
1096
blown down to produce a model in which $B$ and $D$ cross
1097
transversally. Since $B$ and $D$ both have genus 1, we cannot
1098
eliminate either of these components. Furthermore, it suffices to
1099
check $B$, since $D$ is its Galois conjugate.
1100
1101
To get {}from the equation of the curve listed in Table~\ref{table1}
1102
to an affine containing an open subset of $B$ we need to make the
1103
substitutions $x=x_0 - \alpha$ and $y=x_0 (y_{3}x_0 - \alpha - 1)$.
1104
We also have $x_{0}p_{1}=2$. Using the substitutions and the
1105
relation $dx_{0}/x_0 = -dp_{1}/p_1$, we get
1106
\[ \frac{dx}{2y} = \frac{-dp_1}{2p_1(y_3 x_0 - \alpha - 1)}
1107
\text{\quad and\quad}
1108
\frac{x\,dx}{2y}
1109
= \frac{-(x_0 + \alpha)\,dp_1}{2p_1(y_3 x_0 - \alpha - 1)} \,.
1110
\]
1111
Note that $p_1 - t$ is a uniformizer at $p_1 = t$ almost everywhere
1112
on~$B$. When we multiply each differential by~2, then the
1113
denominator of each is almost everywhere non-zero; thus, $dx/y$ and
1114
$x\,dx/y$ are integral at~$2$. Moreover, although the linear
1115
combination $(x-\alpha)\,dx/y$ is identically zero on~$B$, it is not
1116
identically zero on~$D$ (its Galois conjugate is not identically zero
1117
on~$B$). Thus, our new basis is correct at~2. We multiply the
1118
provisional $\Omega$ by~4 to get a new provisional $\Omega$ which is
1119
correct at~$2$.
1120
1121
Similar (but somewhat simpler) computations at the primes $5$ and~$13$
1122
show that no adjustment is needed at these primes. Thus, $dx/y$
1123
and $x\,dx/y$ form a basis for the integral differentials of curve
1124
$C_{65,B}$, and the correct value of $\Omega$ is 4 times our original
1125
guess.
1126
1127
\section{Modular algorithms}
1128
\label{modular}
1129
1130
In this section, we describe the algorithms that were used to compute
1131
some of the data from the newforms. This includes the analytic rank
1132
and leading coefficient of the $L$-series. For optimal quotients,
1133
the value of~$k\cdot\Omega$ can also be found ($k$ is the Manin constant),
1134
as well as partial information
1135
on the Tamagawa numbers~$c_p$ and the size of the torsion subgroup.
1136
1137
\subsection{Analytic rank of $L(J,s)$ and leading coefficient at $s=1$}
1138
\label{l}
1139
1140
Fix a Jacobian~$J$ corresponding to the 2-dimensional subspace of
1141
$S_2(N)$ spanned by quadratic conjugate, normalized newforms~$f$
1142
and~$\overline{f}$. Let $W_N$ be the Fricke involution. The newforms~$f$
1143
and~$\overline{f}$ have the same eigenvalue~$\epsilon_N$ with respect
1144
to~$W_N$, namely $+1$ or~$-1$. In the notation of Section~\ref{curves}, let
1145
\[ L(f,s) = \sum\limits_{n=1}^{\infty} \frac{a_n}{n^s} \]
1146
be the $L$-series of~$f$; then $L(\overline{f},s)$ is the Dirichlet
1147
series whose coefficients are the conjugates of the
1148
coefficients of~$L(f,s)$. (Recall that the~$a_n$ are integers in some
1149
real quadratic field.) The order of~$L(f,s)$ at~$s = 1$ is even
1150
when $\epsilon_N = -1$ and odd when $\epsilon_N = +1$. We have
1151
$L(J,s) = L(f,s) L(\overline{f},s)$. Thus the analytic rank of $J$ is~0
1152
modulo~4 when $\epsilon_N = -1$ and 2 modulo~4 when $\epsilon_N = +1$.
1153
We found that the ranks were all 0 or~2. To prove that the analytic
1154
rank of~$J$ is~0, we need to show $L(f,1) \neq 0$ and
1155
$L(\overline{f},1) \neq 0$. In the case that $\epsilon_N = +1$, to
1156
prove that the analytic rank is~2, we need to show that $L'(f,1) \neq 0$
1157
and $L'(\overline{f},1) \neq 0$. When $\epsilon_N = -1$, we can
1158
evaluate $L(f,1)$ as in~\cite[\S~2.11]{Cr}. When $\epsilon_N = +1$, we
1159
can evaluate $L'(f,1)$ as in~\cite[\S~2.13]{Cr}. Each appropriate
1160
$L(f,1)$ or~$L'(f,1)$ was at least~$0.1$ and the errors in our
1161
approximations were all less than~$10^{-67}$. In this way we
1162
determined the analytic ranks, which we denote~$r$. As noted in the
1163
introduction, the analytic rank equals the Mordell-Weil rank if $r = 0$
1164
or~$r = 2$. Thus, we can simply call $r$ the rank, without fear of
1165
ambiguity.
1166
1167
To compute the leading coefficient of~$L(J,s)$ at~$s = 1$, we note that
1168
$\lim_{s \to 1} L(J,s)/(s-1)^r = L^{(r)}(J,1)/r!$.
1169
In the $r=0$ case, we simply have $L(J,1) = L(f,1)L(\overline{f},1)$.
1170
In the $r=2$ case, we have
1171
$L''(J,s)
1172
= L''(f,s)L(\overline{f},s) + 2L'(f,s)L'(\overline{f},s)
1173
+ L(f,s)L''(\overline{f},s)$.
1174
Evaluating both sides
1175
at $s=1$ we get $\frac{1}{2}L''(J,1) = L'(f,1)L'(\overline{f},1)$.
1176
1177
\subsection{Computing $k\cdot\Omega$}\label{modomega}
1178
Let $J$, $f$ and $\overline{f}$ be as in Section~\ref{l} and
1179
denote by $V$ the 2-dimensional space spanned by $f$ and
1180
$\overline{f}$.
1181
In computing $\Omega$ from an equation for the curve,
1182
we use a basis of integral
1183
differentials (see Section~\ref{Omega}) for $J$.
1184
For optimal quotients we can start with modular symbols and
1185
use a basis $\{\omega_1,\omega_2\}$
1186
for the subgroup of $V$ consisting of forms whose
1187
$q$-expansion coefficients lie in $\Z$, and we will
1188
obtain the quantity $k\cdot\Omega$. It can be shown
1189
that $k$ is a rational number. This rational number
1190
is called the {\em Manin constant}, and
1191
it is conjectured to equal~$1$.
1192
1193
We can compute $k\cdot\Omega$ using a generalization
1194
to dimension 2~of the algorithm for computing periods
1195
described in \cite[\S2.10]{Cr}. This is because
1196
$k\cdot\Omega$ is the volume of the real points
1197
of the quotient of $\C\times\C$ by the
1198
lattice of period integrals
1199
$(\int_\gamma \omega_1, \int_\gamma\omega_2)$
1200
with $\gamma$ in the integral homology
1201
$H_1(X_0(N),\Z)$.
1202
When $L(J,1)\neq 0$ the
1203
method of \cite[\S2.11]{Cr} coupled with
1204
Sections~\ref{l} and~\ref{bsdratio} can also be used
1205
to compute $k\cdot\Omega$.
1206
1207
\subsection{Computing $L(J,1)/(k\cdot\Omega)$}\label{bsdratio}
1208
We compute the rational number $L(J,1)/(k\cdot\Omega)$, for optimal
1209
quotients,
1210
using the algorithm in \cite{AS}.
1211
This algorithm generalizes the algorithm described in
1212
\cite[\S2.8]{Cr} to dimension greater than 1.
1213
1214
\subsection{Tamagawa numbers}
1215
In this section we assume that $p$ is a prime which
1216
exactly divides the conductor $N$ of $J$.
1217
Under these conditions, Grothendieck \cite{Gr} gave a
1218
description of the component group of $J$ in
1219
terms of a monodromy pairing on certain character groups.
1220
(For more details, see Ribet \cite[\S2]{Ri}.)
1221
If, in addition, $J$ is a new optimal quotient of $J_0(N)$, one
1222
deduces the following. When
1223
the eigenvalue for $f$ of the Atkin-Lehner involution $w_p$ is
1224
$+1$, then the rational component group of $J$ is a subgroup of
1225
$(\Z/2\Z)^2$. Furthermore, when the eigenvalue of $w_p$ is $-1$,
1226
the algorithm described in \cite{Ste} can be used to compute
1227
the value of~$c_p$.
1228
1229
\subsection{Torsion subgroup}
1230
\label{modtors}
1231
1232
To compute an integer divisible by the order of the
1233
torsion subgroup of $J$ we make use of the following two observations.
1234
First, it is a consequence of the Eichler-Shimura relation
1235
\cite[\S7.9]{Sh} that if $p$ is a prime not dividing the
1236
conductor $N$ of $J$ and $f(T)$ is the characteristic polynomial
1237
of the endomorphism $T_p$
1238
of $J$, then $\#J(\F_p) = f(p+1)$ (see \cite[\S2.4]{Cr}
1239
for an algorithm to compute $f(T)$).
1240
Second, if $p$ is an odd prime at which $J$ has good reduction,
1241
then the natural map $J(\Q)\tors\rightarrow J(\F_p)$ is injective
1242
(see \cite[p.\ 70]{CF}). This does not depend on whether $J$ is an
1243
optimal quotient.
1244
To obtain a lower bound on the torsion subgroup for optimal quotients,
1245
we use modular symbols and the Abel-Jacobi theorem \cite[IV.2]{La}
1246
to compute the order of the image of the rational point
1247
$(0)-(\infty)\in J_0(N)$.
1248
1249
\section{Tables}
1250
\label{tables}
1251
1252
In Table~\ref{table1}, we list the 32 curves described in
1253
Section~\ref{curves}. We give the level $N$ {}from which each curve
1254
arose, an integral model for the curve, and list the source(s) {}from
1255
which it came ($H$ for Hasegawa \cite{Ha}, $W$ for Wang \cite{Wan}).
1256
Throughout the paper, the curves are denoted $C_N$ (or $C_{N,A}$, $C_{N,B}$).
1257
1258
\begin{table}
1259
\begin{center}
1260
\begin{tabular}{|l|rcl|c|}
1261
\hline
1262
\multicolumn{1}{|c|}{$N$}
1263
& \multicolumn{3}{|c|}{Equation} & Source\\ \hline\hline
1264
23 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1265
$-2 x^5 - 3 x^2 + 2 x - 2$ & HW \\
1266
29 & $y^2 + (x^3 + 1)y$ & $=$ &
1267
$-x^5 - 3 x^4 + 2 x^2 + 2 x - 2$ & HW \\
1268
31 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1269
$-x^5 - 5 x^4 - 5 x^3 + 3 x^2 + 2 x - 3$ & HW \\
1270
35 & $y^2 + (x^3 + x)y$ & $=$ &
1271
$-x^5 - 8 x^3 - 7 x^2 - 16 x - 19$ & H \\ \hline
1272
39 & $y^2 + (x^3 + 1)y$ & $=$ &
1273
$-5 x^4 - 2 x^3 + 16 x^2 - 12 x + 2$ & H \\
1274
63 & $y^2 + (x^3 - 1)y$ & $=$ &
1275
$14 x^3 - 7$ & W \\
1276
65,A & $y^2 + (x^3 + 1)y$ & $=$ &
1277
$-4 x^6 + 9 x^4 + 7 x^3 + 18 x^2 - 10$ & W \\
1278
65,B & $y^2$ & $=$ &
1279
$-x^6 + 10 x^5 - 32 x^4 + 20 x^3 + 40 x^2 + 6 x - 1$ & W \\ \hline
1280
67 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1281
$x^5 - x$ & HW \\
1282
73 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1283
$-x^5 - 2 x^3 + x$ & HW \\
1284
85 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1285
$x^4 + x^3 + 3 x^2 - 2 x + 1$ & H \\
1286
87 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1287
$-x^4 + x^3 - 3 x^2 + x - 1$ & HW \\ \hline
1288
93 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1289
$-2 x^5 + x^4 + x^3$ & HW \\
1290
103 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1291
$x^5 + x^4$ & HW \\
1292
107 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1293
$x^4 - x^2 - x - 1$ & HW \\
1294
115 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1295
$2 x^3 + x^2 + x$ & HW \\ \hline
1296
117,A & $y^2 + (x^3 - 1)y$ & $=$ &
1297
$3 x^3 - 7$ & W \\
1298
117,B & $y^2 + (x^3 + 1)y$ & $=$ &
1299
$-x^6 - 3 x^4 - 5 x^3 - 12 x^2 - 9 x - 7$ & W \\
1300
125,A & $y^2 + (x^3 + x + 1)y$ & $=$ &
1301
$x^5 + 2 x^4 + 2 x^3 + x^2 - x - 1$ & HW \\
1302
125,B & $y^2 + (x^3 + x + 1)y$ & $=$ &
1303
$x^6 + 5 x^5 + 12 x^4 + 12 x^3 + 6 x^2 - 3 x - 4$ & W \\ \hline
1304
133,A & $y^2 + (x^3 + x + 1)y$ & $=$ &
1305
$-2 x^6 + 7 x^5 - 2 x^4 - 19 x^3 + 2 x^2 + 18 x + 7$ & W \\
1306
133,B & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1307
$-x^5 + x^4 - 2 x^3 + 2 x^2 - 2 x$ & HW \\
1308
135 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1309
$x^4 - 3 x^3 + 2 x^2 - 8 x - 3$ & W \\
1310
147 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1311
$x^5 + 2 x^4 + x^3 + x^2 + 1$ & HW \\ \hline
1312
161 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1313
$x^3 + 4 x^2 + 4 x + 1$ & HW \\
1314
165 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1315
$x^5 + 2 x^4 + 3 x^3 + x^2 - 3 x$ & H \\
1316
167 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1317
$-x^5 - x^3 - x^2 - 1$ & HW \\
1318
175 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1319
$-x^5 - x^4 - 2 x^3 - 4 x^2 - 2 x - 1$ & W \\ \hline
1320
177 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1321
$x^5 + x^4 + x^3$ & HW \\
1322
188 & $y^2$ & $=$ &
1323
$x^5 - x^4 + x^3 + x^2 - 2 x + 1$ & W \\
1324
189 & $y^2 + (x^3 - 1)y$ & $=$ &
1325
$x^3 - 7$ & W \\
1326
191 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1327
$-x^3 + x^2 + x$ & HW \\ \hline
1328
\end{tabular}
1329
\end{center}
1330
\caption{Levels, integral models and sources for curves}
1331
\label{table1}
1332
\end{table}
1333
1334
In Table~\ref{table2}, we list the curve~$C_N$ simply by~$N$, the
1335
level {}from which it arose. Let $r$ denote the rank. We
1336
list ${\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)$ where $L(J,s)$ is the
1337
$L$-series for the Jacobian $J$ of~$C_N$ and round off the results to
1338
five digits. The symbol $\Omega$ was defined in Section~\ref{Omega}
1339
and is also rounded to five digits. Let Reg denote the regulator,
1340
also rounded to five digits. We list the $c_{p}$'s by primes of
1341
increasing order dividing the level~$N$. We denote $J(\Q)\tors = \Phi$
1342
and list its size. We use $\Sh ?$ to denote the size of
1343
$({\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)) \cdot
1344
(\#J(\Q)\tors)^2/(\Omega\cdot {\rm Reg} \cdot \prod c_{p})$,
1345
rounded to the nearest integer. We will refer to this as the {\em conjectured
1346
size of} $\Sh(J,\Q)$. The last column gives a bound on the accuracy of the
1347
computations; all values of $\Sh ?$ were at least this close to the
1348
nearest integer before rounding.
1349
1350
\newcommand{\mcc}[1]{\multicolumn{1}{|c|}{#1}}
1351
\newcommand{\mcd}[1]{\multicolumn{2}{|c|}{#1}}
1352
1353
\begin{table}
1354
\begin{center}
1355
\begin{tabular}{|l|c|r@{.}l|r@{.}l|l|l|c|c|l|}
1356
\hline
1357
\mcc{$N$} & $r$
1358
& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
1359
& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & $\Phi$ & $\Sh$? & \mcc{error}
1360
\\ \hline\hline
1361
23 & 0 & 0&24843 & 2&7328 & 1 & 11 & 11 & 1 & $ < 10^{-120} $ \\
1362
29 & 0 & 0&29152 & 2&0407 & 1 & 7 & 7 & 1 & $ < 10^{-50} $ \\
1363
31 & 0 & 0&44929 & 2&2464 & 1 & 5 & 5 & 1 & $ < 10^{-49} $ \\
1364
35 & 0 & 0&37275 & 2&9820 & 1 & 16,2 & 16 & 1 & $ < 10^{-25} $ \\
1365
\hline
1366
39 & 0 & 0&38204 & 10&697 & 1 & 28,1 & 28 & 1 & $ < 10^{-25} $ \\
1367
63 & 0 & 0&75328 & 4&5197 & 1 & 2,3 & 6 & 1 & $ < 10^{-49} $ \\
1368
65,A & 0 & 0&45207 & 6&3289 & 1 & 7,1 & 14 & 2 & $ < 10^{-48} $ \\
1369
65,B & 0 & 0&91225 & 5&4735 & 1 & 1,3 & 6 & 2 & $ < 10^{-50} $ \\
1370
\hline
1371
67 & 2 & 0&23410 & 20&465 & 0.011439 & 1 & 1 & 1 & $ < 10^{-50} $ \\
1372
73 & 2 & 0&25812 & 24&093 & 0.010713 & 1 & 1 & 1 & $ < 10^{-49} $ \\
1373
85 & 2 & 0&34334 & 9&1728 & 0.018715 & 4,2 & 2 & 1 & $ < 10^{-26} $ \\
1374
87 & 0 & 1&4323 & 7&1617 & 1 & 5,1 & 5 & 1 & $ < 10^{-49} $ \\
1375
\hline
1376
93 & 2 & 0&33996 & 18&142 & 0.0046847 & 4,1 & 1 & 1 & $ < 10^{-49} $ \\
1377
103 & 2 & 0&37585 & 16&855 & 0.022299 & 1 & 1 & 1 & $ < 10^{-49} $ \\
1378
107 & 2 & 0&53438 & 11&883 & 0.044970 & 1 & 1 & 1 & $ < 10^{-49} $ \\
1379
115 & 2 & 0&41693 & 10&678 & 0.0097618 & 4,1 & 1 & 1 & $ < 10^{-50} $ \\
1380
\hline
1381
117,A & 0 & 1&0985 & 3&2954 & 1 & 4,3 & 6 & 1 & $ < 10^{-49} $ \\
1382
117,B & 0 & 1&9510 & 1&9510 & 1 & 4,1 & 2 & 1 & $ < 10^{-49} $ \\
1383
125,A & 2 & 0&62996 & 13&026 & 0.048361 & 1 & 1 & 1 & $ < 10^{-50} $ \\
1384
125,B & 0 & 2&0842 & 2&6052 & 1 & 5 & 5 & 4 & $ < 10^{-49} $ \\
1385
\hline
1386
133,A & 0 & 2&2265 & 2&7832 & 1 & 5,1 & 5 & 4 & $ < 10^{-49} $ \\
1387
133,B & 2 & 0&43884 & 15&318 & 0.028648 & 1,1 & 1 & 1 & $ < 10^{-49} $ \\
1388
135 & 0 & 1&5110 & 4&5331 & 1 & 3,1 & 3 & 1 & $ < 10^{-49} $ \\
1389
147 & 2 & 0&61816 & 13&616 & 0.045400 & 2,2 & 2 & 1 & $ < 10^{-50} $ \\
1390
\hline
1391
161 & 2 & 0&82364 & 11&871 & 0.017345 & 4,1 & 1 & 1 & $ < 10^{-47} $ \\
1392
165 & 2 & 0&68650 & 9&5431 & 0.071936 & 4,2,2 & 4 & 1 & $ < 10^{-26} $ \\
1393
167 & 2 & 0&91530 & 7&3327 & 0.12482 & 1 & 1 & 1 & $ < 10^{-47} $ \\
1394
175 & 0 & 0&97209 & 4&8605 & 1 & 1,5 & 5 & 1 & $ < 10^{-44} $ \\
1395
\hline
1396
177 & 2 & 0&90451 & 13&742 & 0.065821 & 1,1 & 1 & 1 & $ < 10^{-45} $ \\
1397
188 & 2 & 1&1708 & 11&519 & 0.011293 & 9,1 & 1 & 1 & $ < 10^{-44} $ \\
1398
189 & 0 & 1&2982 & 3&8946 & 1 & 1,3 & 3 & 1 & $ < 10^{-43} $ \\
1399
191 & 2 & 0&95958 & 17&357 & 0.055286 & 1 & 1 & 1 & $ < 10^{-44} $ \\
1400
\hline
1401
\end{tabular}
1402
\end{center}
1403
\caption{Conjectured sizes of $\Sh (J,\Q)$}
1404
\label{table2}
1405
\end{table}
1406
1407
In Table~\ref{table3} are generators of $J(\Q)/J(\Q)\tors$ for the
1408
curves whose Jacobians have Mordell-Weil rank~2. The generators are
1409
given as divisor classes. Whenever possible, we have chosen
1410
generators of the form $[P - Q]$ where $P$ and~$Q$ are rational
1411
points on the curve. Curve~167 is the only example where this is not
1412
the case, since the degree zero divisors supported on the (known)
1413
rational points on~$C_{167}$ generate a subgroup of index two in the
1414
full Mordell-Weil group.
1415
Affine points are given by their $x$ and $y$ coordinates in the model
1416
given in Table~\ref{table1}. There are two points at infinity in the
1417
normalization of the curves described by our equations, with the
1418
exception of curve~$C_{188}$. These are denoted by $\infty_a$, where
1419
$a$ is the value of the function $y/x^3$ on the point in question.
1420
The (only) point at infinity on curve~$C_{188}$ is simply
1421
denoted~$\infty$.
1422
1423
\begin{table}
1424
\begin{center}
1425
\begin{tabular}{|l|l|l|}
1426
\hline
1427
\mcc{$N$} & \mcd{Generators of $J(\Q)/J(\Q)\tors$} \\ \hline\hline
1428
67 & $ [(0, 0) - \infty_{-1}] $ &
1429
$ [(0, 0) - (0, -1)] $ \\
1430
73 & $ [(0, -1) - \infty_{-1}] $ &
1431
$ [(0, 0) - \infty_{-1}] $ \\
1432
85 & $ [(1, 1) - \infty_{-1}]$ &
1433
$ [(-1, 3) - \infty_{0}] $ \\
1434
93 & $ [(-1, 1) - \infty_{0}] $ &
1435
$ [(1, -3) - (-1, -2)] $ \\ \hline
1436
103 & $ [(0, 0) - \infty_{-1}]$ &
1437
$ [(0, -1) - (0,0)] $ \\
1438
107 & $ [\infty_{-1} - \infty_{0}]$ &
1439
$ [(-1, -1) - \infty_{-1}] $ \\
1440
115 & $ [(1, -4) - \infty_{0}] $ &
1441
$ [(1, 1) - (-2, 2)] $ \\
1442
125,A & $ [\infty_{-1} - \infty_{0}] $ &
1443
$ [(-1, 0) - \infty_{-1}] $ \\ \hline
1444
133,B & $ [\infty_{-1} - \infty_{0}] $ &
1445
$ [(0, -1) - \infty_{-1}] $ \\
1446
147 & $ [\infty_{-1} - \infty_{0}] $ &
1447
$ [(-1, -1) - \infty_{0}] $ \\
1448
161 & $ [(1, 2) - (-1, 1)] $ &
1449
$ [(\frac{1}{2}, -3) - (1, 2)] $ \\
1450
165 & $ [(1, 1) - \infty_{-1}] $ &
1451
$ [(0, 0) - \infty_{0} ] $ \\ \hline
1452
167 & $ [(-1 ,1) - \infty_{0}] $ &
1453
$ [(i, 0) + (-i, 0) - \infty_{0} - \infty_{-1}] $ \\
1454
177 & $ [(0, -1) - \infty_{0}] $ &
1455
$ [(0, 0) - (0, -1)] $ \\
1456
188 & $ [(0, -1) - \infty] $ &
1457
$ [(0, 1) - (1, -2)] $ \\
1458
191 & $ [\infty_{-1} - \infty_{0}]$ &
1459
$ [(0, -1) - \infty_{0}] $ \\
1460
\hline
1461
\end{tabular}
1462
\end{center}
1463
\caption{Generators of $J(\Q)/J(\Q)\tors$ in rank 2 cases}
1464
\label{table3}
1465
\end{table}
1466
1467
In Table~\ref{table4} are the reduction types, {}from the
1468
classification of~\cite{NU}, of the special fibers of the minimal,
1469
proper, regular models of the curves for each of the primes of
1470
singular reduction for the curve. They are the same as the primes
1471
dividing the level except that curve~$C_{65,A}$ has singular
1472
reduction at the prime~3 and curve~$C_{65,B}$ has singular reduction
1473
at the prime~2.
1474
1475
\begin{table}
1476
\begin{center}
1477
\begin{tabular}{|l|l|l|l|l||l|l|l|l|l|}
1478
\hline
1479
\mcc{$N$} & Prime & Type & Prime & Type &
1480
\mcc{$N$} & Prime & Type & Prime & Type
1481
\\ \hline\hline
1482
23 & 23 & ${\rm I}_{3-2-1}$ & & &
1483
117,A & 3 & ${\rm III}-{\rm III}^{\ast}-0$
1484
& 13 & ${\rm I}_{1-1-1}$ \\
1485
29 & 29 & ${\rm I}_{3-1-1}$ & & &
1486
117,B & 3 & ${\rm I}_{3-1-1}^{\ast}$
1487
& 13 & ${\rm I}_{1-1-0}$ \\
1488
31 & 31 & ${\rm I}_{2-1-1}$ & & &
1489
125,A & 5 & ${\rm VIII}-1$ & & \\
1490
35 & 5 & ${\rm I}_{3-2-2}$
1491
& 7 & ${\rm I}_{2-1-0}$ &
1492
125,B & 5 & ${\rm IX}-3$ & & \\ \hline
1493
39 & 3 & ${\rm I}_{6-2-2}$
1494
& 13 & ${\rm I}_{1-1-0}$ &
1495
133,A & 7 & ${\rm I}_{2-1-1}$
1496
& 19 & ${\rm I}_{1-1-0}$ \\
1497
63 & 3 & $2{\rm I}_{0}^{\ast}-0$
1498
& 7 & ${\rm I}_{1-1-1}$ &
1499
133,B & 7 & ${\rm I}_{1-1-0}$
1500
& 19 & ${\rm I}_{1-1-0}$ \\
1501
65,A & 3 & ${\rm I}_{0}-{\rm I}_{0}-1$
1502
& 5 & ${\rm I}_{3-1-1}$ &
1503
135 & 3 & III
1504
& 5 & ${\rm I}_{3-1-0}$ \\
1505
65,A & 13 & ${\rm I}_{1-1-0}$ & & &
1506
147 & 3 & ${\rm I}_{2-1-0}$
1507
& 7 & VII \\ \hline
1508
65,B & 2 & ${\rm I}_{0}-{\rm I}_{0}-1$
1509
& 5 & ${\rm I}_{3-1-0}$ &
1510
161 & 7 & ${\rm I}_{2-2-0}$
1511
& 23 & ${\rm I}_{1-1-0}$ \\
1512
65,B & 13 & ${\rm I}_{1-1-1}$ & & &
1513
165 & 3 & ${\rm I}_{2-2-0}$
1514
& 5 & ${\rm I}_{2-1-0}$ \\
1515
67 & 67 & ${\rm I}_{1-1-0}$ & & &
1516
165 & 11 & ${\rm I}_{2-1-0}$ & & \\
1517
73 & 73 & ${\rm I}_{1-1-0}$ & & &
1518
167 & 167 & ${\rm I}_{1-1-0}$ & & \\ \hline
1519
85 & 5 & ${\rm I}_{2-2-0}$
1520
& 17 & ${\rm I}_{2-1-0}$ &
1521
175 & 5 & ${\rm II}-{\rm II}-0$
1522
& 7 & ${\rm I}_{2-1-1}$ \\
1523
87 & 3 & ${\rm I}_{2-1-1}$
1524
& 29 & ${\rm I}_{1-1-0}$ &
1525
177 & 3 & ${\rm I}_{1-1-0}$
1526
& 59 & ${\rm I}_{1-1-0}$ \\
1527
93 & 3 & ${\rm I}_{2-2-0}$
1528
& 31 & ${\rm I}_{1-1-0}$ &
1529
188 & 2 & ${\rm IV}-{\rm IV}-0$
1530
& 47 & ${\rm I}_{1-1-0}$ \\
1531
103 & 103 & ${\rm I}_{1-1-0}$ & & &
1532
189 & 3 & ${\rm II}-{\rm IV}^{\ast}-0$
1533
& 7 & ${\rm I}_{1-1-1}$ \\ \hline
1534
107 & 107 & ${\rm I}_{1-1-0}$ & & &
1535
191 & 191 & ${\rm I}_{1-1-0}$ & & \\
1536
115 & 5 & ${\rm I}_{2-2-0}$
1537
& 23 & ${\rm I}_{1-1-0}$ & & & & & \\ \hline
1538
\end{tabular}
1539
\end{center}
1540
\caption{Namikawa and Ueno classification of special fibers}
1541
\label{table4}
1542
\end{table}
1543
1544
1545
\section{Discussion of Shafarevich-Tate groups and evidence for the
1546
second conjecture}
1547
\label{Shah}
1548
1549
{}From Section~\ref{MW} we have
1550
$\dim \Sh(J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2]$.
1551
With the exception of curves $C_{65,A}$, $C_{65,B}$, $C_{125,B}$, and
1552
$C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 0$. Thus we expect
1553
$\#\Sh(J,\Q)$ to be an odd square. In each case, the conjectured
1554
size of $\Sh(J,\Q)$ is~1. For curves $C_{65,A}$, $C_{65,B}$,
1555
$C_{125,B}$ and $C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 1, 1, 2$
1556
and~2 and the conjectured size of $\Sh(J,\Q) = 2, 2, 4$ and~4,
1557
respectively. We see that in each case, the (conjectured) size of
1558
the odd part of $\Sh(J,\Q)$ is~1 and the 2-part is accounted for by
1559
its 2-torsion.
1560
1561
For the optimal quotients, we computed the value of
1562
the rational number
1563
$L(J,1)/(k\cdot\Omega)$. Thus we can verify exactly that
1564
equation~\eqref{eqn1} holds if all of the following
1565
three conditions are met:
1566
a) the rank is 0, b) $\Sh(J,\Q) = \Sh(J,\Q)[2]$, and c) the
1567
Manin constant $k$ is 1 or bounded away from 1. The Manin constants
1568
are 1 to within the accuracy of our calculations
1569
(they are defined in Section~\ref{modomega}). Thus, if
1570
these can be proven to be 1 or bounded away from 1 by some
1571
amount greater than our degree of accuracy ($10^{-14}$), then
1572
we have a proof that they are exactly 1.
1573
1574
It is also interesting to consider deficient primes. A prime $p$ is
1575
deficient with respect to a curve $C$ of genus~2, if $C$ has no
1576
degree 1 rational divisor over~$\Q_{p}$. {}From~\cite{PSt}, the
1577
number of deficient primes has the same parity as $\dim \Sh(J,\Q)[2]$.
1578
Curve $C_{65,A}$ has one deficient prime~$3$. Curve
1579
$C_{65,B}$ has one deficient prime~$2$. Curve $C_{117,B}$ has two
1580
deficient primes $3$ and~$\infty$. The rest of the curves have no
1581
deficient primes.
1582
1583
Since we have found $r$ (analytic rank) independent points on each
1584
Jacobian, we have a direct proof that the Mordell-Weil rank must
1585
equal the analytic rank if $\dim \Sh(J,\Q)[2] = 0$. For
1586
curves $C_{65,A}$ and $C_{65,B}$, the presence of an odd number of
1587
deficient primes gives us a
1588
similar result. For $C_{125,B}$ we used a $\sqrt{5}$-Selmer group
1589
to get a similar result.
1590
Thus, we have an independent proof of equality
1591
between analytic and Mordell-Weil ranks for all curves except
1592
$C_{133,A}$.
1593
1594
The 2-Selmer groups have the same dimensions for the pairs
1595
$C_{125,A}$, $C_{125,B}$ and $C_{133,A}$, $C_{133,B}$. For each
1596
pair, the Mordell-Weil rank is~2 for one curve and the 2-torsion of
1597
the Shafarevich-Tate group has dimension~2 for the other. In
1598
addition, the two Jacobians, when canonically embedded into~$J_0(N)$,
1599
intersect in their 2-torsion subgroups, and one can check that their
1600
2-Selmer groups become equal under the identification of
1601
$H^1(\Q, J_{N,A}[2])$ with $H^1(\Q, J_{N,B}[2])$ induced by the identification
1602
of the 2-torsion subgroups. Thus these are examples of the principle
1603
of a `visible part of a Shafarevich-Tate group' as discussed
1604
in~\cite{CM}.
1605
1606
\vspace{5mm}
1607
\begin{center}
1608
{\sc Appendix: Other Hasegawa curves}
1609
\end{center}
1610
1611
In Table~\ref{Hasegawa} is data concerning all 142 of Hasegawa's
1612
curves in the order presented in his paper. Let us explain the
1613
entries. The first column in each set of three columns gives the
1614
level, $N$. The second column gives a classification of the cusp
1615
forms spanning the 2-dimensional subspace of $S_2(N)$ corresponding
1616
to the Jacobian. When that subspace is irreducible with respect to
1617
the action of the Hecke algebra and is spanned by two newforms or two
1618
oldforms, we write $2n$ or $2o$, respectively. When that subspace is
1619
reducible and is spanned by two oldforms, two newforms or one of
1620
each, we write $oo$, $nn$ and $on$, respectively. The third column
1621
contains the sign of the functional equation at the level $M$ at
1622
which the cusp form is a newform. This is the negative of
1623
$\epsilon_M$ (described in Section~\ref{l}). The order of the two
1624
signs in the third column agrees with that of the forms listed in the
1625
second column. We include this information for those who would like
1626
to further study these curves. The curves with $N<200$ classified as
1627
$2n$ appeared already in Table~\ref{table1}.
1628
1629
The smallest possible Mordell-Weil ranks corresponding to $++$, $+-$,
1630
$-+$ and $--$, predicted by the first Birch and Swinnerton-Dyer
1631
conjecture, are $0$, $1$, $1$ and $2$ respectively. In all cases,
1632
those were, in fact, the Mordell-Weil ranks. This was determined by
1633
computing 2-Selmer groups with a computer program based on
1634
\cite{Sto2}. Of course, these are cases where the first Birch and
1635
Swinnerton-Dyer conjecture is already known to hold. In the cases
1636
where the Mordell-Weil rank is positive, the Mordell-Weil group has a
1637
subgroup of finite index generated by degree zero divisors supported
1638
on rational points with $x$-coordinates with numerators bounded by 7
1639
(in absolute value) and denominators by 12 with one exception. On
1640
the second curve with $N=138$, the divisor class
1641
$[(3+2\sqrt{2},80+56\sqrt{2}) + (3-2\sqrt{2},80-56\sqrt{2})-2\infty]$
1642
generates a subgroup of finite index in the Mordell-Weil group.
1643
1644
\vfill
1645
1646
\begin{table}
1647
\begin{center}
1648
\begin{tabular}{|c|c|c||c|c|c||c|c|c||c|c|c||c|c|c|}
1649
\hline
1650
22 & $oo$ & $++$ & 58 & $nn$ & $+-$ & 87 & $2o$ & $++$ & 129 & $on$ & $--$ &
1651
198 & $2o$ & $+-$ \\
1652
23 & $2n$ & $++$ & 60 & $oo$ & $++$ & 88 & $on$ & $+-$ & 130 & $on$ & $-+$ &
1653
204 & $2o$ & $+-$ \\
1654
26 & $nn$ & $++$ & 60 & $2o$ & $++$ & 90 & $on$ & $++$ & 132 & $oo$ & $++$ &
1655
205 & $2n$ & $--$ \\
1656
28 & $oo$ & $++$ & 60 & $2o$ & $++$ & 90 & $oo$ & $++$ & 133 & $2n$ & $--$ &
1657
206 & $2o$ & $--$ \\
1658
29 & $2n$ & $++$ & 62 & $2o$ & $++$ & 90 & $oo$ & $++$ & 134 & $2o$ & $--$ &
1659
209 & $2n$ & $--$ \\
1660
30 & $on$ & $++$ & 66 & $nn$ & $++$ & 90 & $oo$ & $++$ & 135 & $on$ & $+-$ &
1661
210 & $on$ & $+-$ \\
1662
30 & $oo$ & $++$ & 66 & $2o$ & $++$ & 91 & $nn$ & $--$ & 138 & $nn$ & $+-$ &
1663
213 & $2n$ & $--$ \\
1664
30 & $on$ & $++$ & 66 & $2o$ & $++$ & 93 & $2n$ & $--$ & 138 & $on$ & $+-$ &
1665
215 & $on$ & $--$ \\
1666
31 & $2n$ & $++$ & 66 & $on$ & $++$ & 98 & $oo$ & $++$ & 140 & $oo$ & $++$ &
1667
221 & $2n$ & $--$ \\
1668
33 & $on$ & $++$ & 67 & $2n$ & $--$ & 100 & $oo$ & $++$ & 142 & $nn$ & $+-$
1669
& 230 & $2o$ & $--$ \\ \hline
1670
35 & $2n$ & $++$ & 68 & $oo$ & $++$ & 102 & $on$ & $+-$ & 143 & $on$ & $+-$
1671
& 255 & $2o$ & $--$ \\
1672
37 & $nn$ & $+-$ & 69 & $2o$ & $++$ & 102 & $on$ & $+-$ & 146 & $2o$ & $--$
1673
& 266 & $2o$ & $--$ \\
1674
38 & $on$ & $++$ & 70 & $on$ & $++$ & 103 & $2n$ & $--$ & 147 & $2n$ & $--$
1675
& 276 & $2o$ & $+-$ \\
1676
39 & $2n$ & $++$ & 70 & $2o$ & $++$ & 104 & $2o$ & $++$ & 150 & $on$ & $++$
1677
& 284 & $2o$ & $+-$ \\
1678
40 & $on$ & $++$ & 70 & $2o$ & $++$ & 106 & $on$ & $--$ & 153 & $on$ & $+-$
1679
& 285 & $on$ & $--$ \\
1680
40 & $oo$ & $++$ & 70 & $2o$ & $++$ & 107 & $2n$ & $--$ & 154 & $on$ & $--$
1681
& 286 & $on$ & $--$ \\
1682
42 & $on$ & $++$ & 72 & $on$ & $++$ & 110 & $on$ & $++$ & 156 & $oo$ & $++$
1683
& 287 & $2n$ & $--$ \\
1684
42 & $oo$ & $++$ & 72 & $oo$ & $++$ & 111 & $oo$ & $+-$ & 158 & $on$ & $--$
1685
& 299 & $2n$ & $--$ \\
1686
42 & $on$ & $++$ & 73 & $2n$ & $--$ & 112 & $on$ & $+-$ & 161 & $2n$ & $--$
1687
& 330 & $2o$ & $--$ \\
1688
42 & $oo$ & $++$ & 74 & $oo$ & $+-$ & 114 & $oo$ & $+-$ & 165 & $2n$ & $--$
1689
& 357 & $2n$ & $--$ \\ \hline
1690
44 & $2o$ & $++$ & 77 & $on$ & $+-$ & 115 & $2n$ & $--$ & 166 & $on$ & $--$
1691
& 380 & $2o$ & $+-$ \\
1692
46 & $2o$ & $++$ & 78 & $oo$ & $++$ & 116 & $2o$ & $+-$ & 167 & $2n$ & $--$
1693
& 390 & $on$ & $--$ \\
1694
48 & $on$ & $++$ & 78 & $2o$ & $++$ & 117 & $2o$ & $++$ & 168 & $2o$ & $++$
1695
& & & \\
1696
48 & $oo$ & $++$ & 80 & $oo$ & $++$ & 120 & $oo$ & $++$ & 170 & $2o$ & $--$
1697
& & & \\
1698
50 & $nn$ & $++$ & 84 & $oo$ & $++$ & 120 & $on$ & $++$ & 177 & $2n$ & $--$
1699
& & & \\
1700
52 & $oo$ & $++$ & 84 & $oo$ & $++$ & 121 & $on$ & $+-$ & 180 & $2o$ & $++$
1701
& & & \\
1702
52 & $oo$ & $++$ & 84 & $oo$ & $++$ & 122 & $on$ & $--$ & 184 & $on$ & $+-$
1703
& & & \\
1704
54 & $on$ & $++$ & 84 & $oo$ & $++$ & 125 & $2n$ & $--$ & 186 & $2o$ & $--$
1705
& & & \\
1706
57 & $on$ & $+-$ & 85 & $2n$ & $--$ & 126 & $oo$ & $++$ & 190 & $on$ & $+-$
1707
& & & \\
1708
57 & $on$ & $+-$ & 87 & $2n$ & $++$ & 126 & $on$ & $++$ & 191 & $2n$ & $--$
1709
& & & \\
1710
\hline
1711
\end{tabular}
1712
\end{center}
1713
\caption{Spaces of cusp forms associated to Hasegawa's curves}
1714
\label{Hasegawa}
1715
\end{table}
1716
1717
\pagebreak
1718
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\end{document}
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