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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{Universit\'{e} Grenoble I, Institut Fourier, BP 74, F-38402 Saint
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Martin d'H\`{e}res Cedex, 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, Harvard University, One Oxford Street,
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Cambridge, MA 02138, USA}
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\email{was@math.berkeley.edu}
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\author{Michael Stoll}
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\address{Mathematisches Institut der Heinrich-Heine-Universit\"{a}t,
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Universit\"{a}tsstr.~1, 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{June 5, 2000}
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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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Breuil, 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 (see \cite{BCDT}).}
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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{BGZ,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.
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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 induces
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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
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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
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quotients, we can compute~$k\cdot\Omega$ (where $k$ is the Manin constant,
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conjectured to be 1)
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using the newforms.
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In each case, these agree (to within the accuracy of our computations)
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with the $\Omega$'s computed using the equations for the curves.
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We can also compute the value of~$c_p$ for optimal quotients from
474
the newforms, when $p$ exactly divides~$N$ and the eigenvalue of the
475
$p$th Atkin-Lehner involution is $-1$. When $p$ exactly divides~$N$
476
and the eigenvalue of the $p$th Atkin-Lehner involution is~$+1$, the
477
component group is either $0$, $\Z/2\Z$, or~$(\Z/2\Z)^2$. These results
478
are always in agreement with the values computed using the equations
479
for the curves. The algorithms based on the newforms are
480
described in Section~\ref{modular}, those based on the
481
equations of the curves are described in Section~\ref{algms}.
482
483
Lastly, we were able to compute the Mordell-Weil ranks of the Jacobians
484
of the curves given by ten of these eleven equations. In
485
each case it agrees with the analytic rank of the Jacobian,
486
as deduced {}from the newforms.
487
488
It should be noted that curve~$C_{125,B}$ is the $\sqrt{5}$-twist of
489
curve~$C_{125,A}$; the corresponding statement holds for the associated
490
2-dimensional subspaces of~$S_2(125)$. Since curve~$C_{125,A}$ is
491
a Hasegawa curve, this proves that the equation given in Table~\ref{table1}
492
for curve~$C_{125,B}$ is correct.
493
494
The $a_p$'s and other information concerning Wang's curves are
495
currently kept in a database at the Institut f\"{u}r experimentelle
496
Mathematik in Essen, Germany. Most recently, this database was under
497
the care of Michael M\"{u}ller. William Stein also keeps a database
498
of~$a_p$'s for newforms.
499
500
\begin{remark}
501
For the remainder of this paper we will assume that the equations for
502
the curves given in Table~\ref{table1} are correct; that is, that
503
they are equations for the curves whose Jacobians are isogenous
504
to a factor of~$J_0(N)$ in the way described above.
505
Some of the quantities can be computed either {}from the newform
506
or {}from the equation for the curve. We performed both computations
507
whenever possible, and view this duplicate effort as an attempt to
508
verify our implementation of the algorithms rather than an attempt
509
to verify the equations in Table~\ref{table1}. For most quantities,
510
one method or the other is not guaranteed to produce a value; in this
511
case, we simply quote the value {}from whichever method did succeed.
512
The reader who is disturbed by this philosophy should
513
ignore the Wang-only curves, since the equations for the Hasegawa
514
curves can be proven to be correct.
515
\end{remark}
516
517
518
\section{Algorithms for genus~2 curves}
519
\label{algms}
520
521
In this section, we describe the algorithms that are based on the
522
given models for the curves. We give algorithms that compute all
523
terms on the right hand side of equation~\eqref{eqn1}, with the
524
exception of the size of the Shafarevich-Tate group. We describe,
525
however, how to find the size of its 2-torsion subgroup. Note that these
526
algorithms are for general genus 2 curves and do not depend on modularity.
527
528
\subsection{Torsion Subgroup}
529
\label{torsion}
530
531
The computation of the torsion subgroup of~$J(\Q)$ is straightforward.
532
We used the technique described in~\cite[pp.~78--82]{CF}.
533
This technique is not always effective, however. For an algorithm working
534
in all cases see~\cite{Sto3}.
535
536
\subsection{Mordell-Weil rank and $\Sh(J,\Q)[2]$}
537
\label{MW}
538
539
The group $J(\Q)$ is a finitely generated abelian group and so is
540
isomorphic to $\Z^{r} \oplus J(\Q)\tors$ for some $r$ called the
541
Mordell-Weil rank.
542
As noted above (see Section~\ref{intro}), we justifiably use
543
$r$ to denote both the analytic and Mordell-Weil ranks since they
544
agree for all curves in Table~\ref{table1}.
545
546
We used the algorithm described in \cite{FPS} to compute ${\rm
547
Sel}^{2}_{\rm fake}(J,\Q)$ (notation {}from \cite{PSc}), which is a
548
quotient of the 2-Selmer group ${\rm Sel}^{2}(J,\Q)$. More details
549
on this algorithm can be found in \cite{Sto2}. Theorem 13.2 of
550
\cite{PSc} explains how to get ${\rm Sel}^{2}(J,\Q)$ {}from ${\rm
551
Sel}^{2}_{\rm fake}(J,\Q)$. Let $M[2]$ denote the 2-torsion of an
552
abelian group $M$ and let dim$V$ denote the dimension of an $\F_{2}$
553
vector space $V$. We have
554
$\dim {\rm Sel}^{2}(J,\Q) = r + \dim J(\Q)[2] + \dim \Sh(J,\Q)[2]$.
555
In other words,
556
\[ \dim\, \Sh (J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2]. \]
557
558
It is interesting to note that in all 30 cases where
559
$\dim \Sh(J,\Q)[2] \le 1$, we were able to compute the Mordell-Weil rank
560
independently from the analytic rank.
561
The
562
cases where $\dim \Sh(J,\Q)[2] = 1$ are discussed in more
563
detail in Section~\ref{Shah}.
564
For both of the remaining cases we have $\dim \Sh(J,\Q)[2]=2$.
565
One of these cases is
566
$C_{125,B}$. For this curve we computed
567
${\rm Sel}^{\sqrt{5}}(J_{125,B},\Q)$
568
using the technique described in
569
\cite{Sc}. {}From this, we were able to determine that the Mordell-Weil
570
rank is 0 independently from the analytic rank.
571
For the other case,
572
$C_{133,A}$,
573
we could show that $r$ had to be either~0
574
or~2 {}from the equation, but we needed the analytic computation to
575
show that $r=0$.
576
577
\subsection{Regulator}
578
\label{reg}
579
580
When the Mordell-Weil rank is~0, then the regulator is~1. When the
581
Mordell-Weil rank is positive, then to compute the regulator, we
582
first need to find generators for $J(\Q)/J(\Q)\tors$. The regulator
583
is the determinant of the canonical height pairing matrix on this set
584
of generators. An algorithm for computing the generators and
585
canonical heights is given in~\cite{FS}; it was used to find
586
generators for $J(\Q)/J(\Q)\tors$ and to compute the regulators. In
587
that article, the algorithm for computing height constants at the
588
infinite prime is not clearly explained and there are some errors in
589
the examples. A clear algorithm for computing infinite height
590
constants is given in~\cite{Sto3}. In~\cite{Sto4}, some improvements of
591
the results and algorithms in~\cite{FS} and~\cite{Sto3} are discussed.
592
The regulators in Table~\ref{table2} have been double-checked using
593
these improved algorithms.
594
595
\subsection{Tamagawa Numbers}
596
\label{Tamagawa}
597
598
Let $\OO$ be the integer ring in~$K$ which will be $\Q_{p}$ or
599
$\Q_{p}\unr$ (the maximal unramified extension of $\Q_{p})$.
600
Let $\JJ$ be the N\'{e}ron model of~$J$ over~$\OO$.
601
Define $\JJ^{0}$ to be the open subgroup scheme of~$\JJ$ whose
602
generic fiber is isomorphic to~$J$ over~$K$ and whose special fiber
603
is the identity component of the closed fiber of~$\JJ$.
604
The group $\JJ^{0}(\OO)$ is isomorphic to a subgroup of~$J(K)$ which
605
we denote $J^{0}(K)$. The group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is
606
the component group of~$\JJ$ over~$\OO_{\Q_{p}\unr}$. We are
607
interested in computing $c_p = \#J(\Q_{p})/J^{0}(\Q_{p})$, which is
608
sometimes called the Tamagawa number.
609
Since N\'{e}ron models are stable under unramified base extension,
610
the $\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of
611
$J^{0}(\Q_{p}\unr)$ is~$J^{0}(\Q_{p})$.
612
Since $H^1(\Gal(\Q_{p}\unr/\Q_{p}), J^{0}(\Q_{p}\unr))$
613
is trivial (see~\cite[p.\ 58]{Mi1}) we see that the
614
$\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of
615
$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is $ J(\Q_{p})/J^{0}(\Q_{p})$.
616
617
There exist several discussions in the literature on constructing the
618
group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ starting with an integral
619
model of the underlying curve. For our purposes, we especially
620
recommend Silverman's book~\cite{Si}, Chapter~IV, Sections 4 and~7.
621
For a more detailed treatment, see~\cite[chap.\ 9]{BLR} and~\cite[\S 2]{Ed2}.
622
One can find justifications for what we will do in these sources. While
623
constructing such groups, we ran into a number of difficulties that
624
we did not find described anywhere. For that reason, we will present
625
examples of such difficulties that arose, as well as our methods of
626
resolution. We do not claim that we will describe all situations
627
that could arise.
628
629
When computing $c_p$ we need a proper, regular model~$\CC$ for~$C$
630
over~$\Z_p$. Let $\Z_p\unr$ denote the ring of integers of~$\Q_p\unr$
631
and note that $\Z_p\unr$ is a pro-\'etale Galois extension
632
of~$\Z_p$ with Galois group
633
$\Gal(\Z_p\unr/\Z_p) = \Gal(\Q_p\unr/\Q_p)$.
634
It follows that giving a model for~$C$ over~$\Z_p$ is equivalent to
635
giving a model for~$C$ over~$\Z_p\unr$ that
636
is equipped with a Galois action. We have found it convenient to
637
always work with the latter description. Thus for us, giving a model
638
over~$\Z_p$ will always mean giving a model over~$\Z_p\unr$ together
639
with a Galois action.
640
641
In order to find a proper, regular model for~$C$ over~$\Z_p$,
642
we start with the models in Table~\ref{table1}. Technically, we
643
consider the curves to be the two affine pieces $y^2+g(x)y=f(x)$ and
644
$v^2 + u^3 g(1/u)v = u^6 f(1/u)$, glued together by $ux=1$, $v=u^3y$.
645
We blow them up at all points that are not regular until we have a
646
regular model. (A point is {\em regular} if the cotangent space there has
647
two generators.) These curves are all proper, and this is not
648
affected by blowing up.
649
650
Let $\CC_p$ denote the special fiber of~$\CC$ over~$\Z_p\unr$. The
651
group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is isomorphic to a quotient
652
of the degree~0 part of the free group on the irreducible components
653
of~$\CC_{p}$. Let the irreducible components be denoted $\DD_{i}$ for
654
$1\leq i\leq n$, and let the multiplicity of~$\DD_{i}$ in~$\CC_p$ be
655
$d_{i}$. Then the degree~0 part of the free group has the form
656
\[ L = \{ \sum\limits_{i=1}^{n} \alpha_{i}\DD_{i} \mid
657
\sum\limits_{i=1}^{n} d_{i}\alpha_{i} = 0 \}\,. \]
658
659
In order to describe the group that we quotient out by, we must
660
discuss the intersection pairing. For components $\DD_{i}$ and~$\DD_{j}$
661
of the special fiber, let $\DD_{i} \cdot \DD_{j}$ denote
662
their intersection pairing. In all of the special fibers that arise
663
in our examples, distinct components intersect transversally. Thus,
664
if $i \neq j$, then $\DD_{i} \cdot \DD_{j}$ equals the number of points
665
at which $\DD_{i}$ and $\DD_{j}$ intersect. The case of
666
self-intersection ($i=j$) is discussed below.
667
668
The kernel of the map {}from~$L$ to
669
$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is generated by
670
divisors of the form
671
\[ [\DD_j] = \sum\limits_{i=1}^{n} (\DD_{j} \cdot \DD_{i}) \DD_{i} \]
672
for each component~$\DD_j$. We can deduce $\DD_{j} \cdot \DD_{j}$ by
673
noting that $[\DD_j]$ must be contained in the group~$L$. This follows
674
{}from the fact that the intersection pairing of
675
$\CC_{p} = \sum d_i\DD_{i}$ with any irreducible component is 0.
676
677
\vspace{1mm}
678
\noindent
679
{\bf Example 1.} Curve $C_{65,B}$ over $\Z_{2}$.
680
681
The Jacobian
682
of $C_{65,B}$
683
is a quotient of the Jacobian of~$X_0(65)$.
684
Since 65 is odd, $J_0(65)$ has good reducation at~2; however,
685
$C_{65,B}$ has singular
686
reduction at~2. Since the equation for this curve
687
is conjectural (it is a Wang-only curve), it will be nice to verify
688
that 2 does not divide the conductor of its Jacobian, i.e.\ that the
689
Jacobian has good reduction at~2. In addition, we will need a
690
proper, regular model for this curve in order to find~$\Omega$.
691
692
We start with the arithmetic surface over~$\Z_{2}\unr$ given by the
693
two pieces
694
$y^2 = f(x) = -x^6 + 10x^5 - 32x^4 + 20x^3 + 40x^2 + 6x - 1$ and
695
$v^2 = u^6 f(1/u)$. (Here and in the following we will not specify the
696
gluing maps.) This arithmetic surface is regular at $u=0$ so we
697
focus our attention on the first affine piece. The special fiber of
698
$y^2 = f(x)$ over~$\Z_{2}\unr$
699
is given by
700
$(y + x^3 + 1)^2 = 0 \pmod 2$; this is a genus~0 curve of multiplicity~2
701
that we denote~$A$. This model is not regular at the two points
702
$(x-\alpha, y, 2)$, where $\alpha$ is a root of $x^2 - 3x - 1$.
703
The current special fiber is in Figure~\ref{special2} and is labelled
704
{\it Fiber~1}.
705
706
We fix $\alpha$ and move $(x - \alpha, y, 2)$ to the origin using the
707
substitution $x_0 = x-\alpha$. We get
708
\[ y^2 = -x_0^6 + (-6\alpha + 10)x_0^5 + (5\alpha - 47)x_0^4
709
+ (-28\alpha + 60)x_0^3 + (-11\alpha - 2)x_0^2
710
+ (-24\alpha - 16)x_0
711
\]
712
which we rewrite as the pair of equations
713
\begin{align*}
714
g_{1}(x_{0},y,p)
715
&= -x_0^6 + (-3\alpha + 5) p x_0^5 + (5\alpha - 47) x_0^4
716
+ (-7\alpha + 15) p^2 x_0^3 \\
717
& \qquad {} + (-11\alpha - 2) x_0^2 + (-3\alpha - 2) p^3 x_0 - y^2
718
\\
719
&= 0,\\
720
p &= 2.
721
\end{align*}
722
To blow up at $(x_0,y,p)$, we introduce projective coordinates
723
$(x_1,y_1,p_1)$ with $x_{0} y_1 = x_{1} y$, $x_{0} p_{1} = x_{1} p$, and
724
$y p_1 = y_{1} p$. We look in three affine pieces that cover the blow-up
725
of $g_1(x_{0},y,p)=0,$ $p=2$
726
and check for regularity.
727
728
\begin{description}
729
\item[$x_{1} = 1$] We have $y = x_{0} y_{1}$, $p = x_{0} p_{1}$. We get
730
$g_2(x_{0},y_{1},p_{1}) = 0$, $x_{0} p_{1} = 2$, where
731
\begin{align*}
732
g_2(x_{0},y_{1},p_{1}) &= x_{0}^{-2}g_{1}(x_{0},x_{0}y_{1},x_{0}p_{1}) \\
733
&= -x_0^4 + (-3\alpha + 5) p_1 x_0^4 + (5\alpha - 47) x_0^2
734
+ (-7\alpha + 15) p_1^2 x_0^3 \\
735
& \qquad{} + (-11\alpha - 2) + (-3\alpha - 2) p_1^3 x_0^2 - y_1^2 \,.
736
\end{align*}
737
In the reduction we have either $x_{0} = 0$ or $p_1 = 0$.
738
\begin{description}
739
\item[$x_{0} = 0$] $(y_{1} + \alpha + 1)^2 = 0$.
740
This is a new component which we denote $B$. It has genus~0 and
741
multiplicity~2. We check regularity along~$B$ at
742
$(x_{0}, y_{1} + \alpha + 1, p_{1}-t, 2)$, with $t$ in $\Z_2\unr$, and
743
find that $B$ is nowhere regular.
744
\item[$p_{1} = 0$]
745
$(y_{1} + x_{0}^2 + \alpha x_{0} + (\alpha + 1))^2 = 0$.
746
Using the gluing maps, we see that this is~$A$.
747
\end{description}
748
749
\item[$y_{1} = 1$] We get no new information {}from this affine piece.
750
751
\item[$p_{1} = 1$] We have $x_{0} = x_{1} p$, $y = y_{1} p$. We get
752
$g_{3}(x_{1},y_{1},p) = p^{-2} g_{1}(x_{1}p,y_{1}p,p) = 0$, $p = 2$.
753
In the reduction we have
754
\begin{description}
755
\item[$p=0$] $(y_1 + (\alpha+1)x_1)^2 = 0$. Using the gluing maps, we
756
see that this is~$B$. It is nowhere regular.
757
\end{description}
758
\end{description}
759
760
The current special fiber is in
761
Figure~\ref{special2} and is labelled {\it Fiber~2}. It is not regular
762
along~$B$ and at the other point on~$A$ which we have not yet blown up.
763
The component $B$ does not lie entirely in any one affine piece
764
so we will blow up the affine pieces $x_1 = 1$ and $p_1 = 1$ along~$B$.
765
766
To blow up $x_1 = 1$ along~$B$ we make the substitution
767
$y_2 = y_1 + \alpha + 1$ and replace each factor of~2 in a coefficient
768
by~$x_0 p_1$. We have $g_{4}(x_0,y_2,p_1) = 0$ and $x_0 p_1 = 2$, and we
769
want to blow up along the line $(x_0, y_2, 2)$. Blowing up along a line
770
is similar to blowing up at a point: since we are blowing up at
771
$(x_0, y_2, 2) = (x_0, y_2)$, we introduce projective
772
coordinates $x_3, y_3$ together with the relation $x_0 y_3 = x_3 y_2$. We
773
consider two affine pieces that cover the blow-up of $x_1 = 1$.
774
775
\begin{description}
776
\item[$x_3 = 1$] We have $y_2 = y_{3} x_{0}$. We get
777
$g_{5}(x_{0},y_{3},p_{1}) = x_{0}^{-2} g_{4}(x_{0},y_{3}x_{0},p_1) = 0$
778
and $x_{0} p_{1} = 2$. In the reduction we have
779
\begin{description}
780
\item[$x_{0} = 0$]
781
$y_{3}^2 + (\alpha + 1) y_{3} p_{1} + \alpha p_{1}^3 + p_{1}^2
782
+ \alpha + 1 = 0$.
783
This is~$B$. It is now a non-singular genus~1 curve.
784
\item[$p_{1} = 0$] $(x_0 + y_3 + \alpha)^2 = 0$. This is~$A$. The point
785
where $B$ meets~$A$ transversally is regular.
786
\end{description}
787
788
\item[$y_3 = 1$] We get no new information {}from this affine piece.
789
\end{description}
790
791
When we blow up $p_1 = 1$ along~$B$ we get essentially the same thing and
792
all points are again regular.
793
794
The other non-regular point on~$A$ is the conjugate of the one we
795
blew up. Therefore, after performing the conjugate blow ups, it too
796
will be a genus~1 component crossing~$A$ transversally. We denote
797
this component $D$; it is conjugate to~$B$.
798
799
800
\begin{figure}
801
\caption{Special fibers of curve $C_{65,B}$ over $\Z_{2}$;
802
points not regular are thick}
803
\label{special2}
804
\begin{picture}(400,130)
805
\put(20,5){\begin{picture}(100,125)
806
\thinlines
807
\put(20,55){\line(1,0){60}}
808
\put(85,55){\makebox(0,0){A}}
809
\put(75,62){\makebox(0,0){2}}
810
\put(40,55){\circle*{5}}
811
\put(60,55){\circle*{5}}
812
\put(50,5){\makebox(0,0){Fiber 1}}
813
\end{picture}}
814
\put(145,5){\begin{picture}(100,125)
815
\thinlines
816
\put(50,5){\makebox(0,0){Fiber 2}}
817
\put(20,55){\line(1,0){60}}
818
\put(85,55){\makebox(0,0){A}}
819
\put(75,62){\makebox(0,0){2}}
820
\put(60,55){\circle*{5}}
821
\put(40,15){\line(0,1){80}}
822
\put(40.5,15){\line(0,1){80}}
823
\put(39.5,15){\line(0,1){80}}
824
\put(39,15){\line(0,1){80}}
825
\put(41,15){\line(0,1){80}}
826
\put(40,105){\makebox(0,0){B}}
827
\put(34,90){\makebox(0,0){2}}
828
\end{picture}}
829
\put(270,5){\begin{picture}(100,125)
830
\thinlines
831
\put(20,55){\line(1,0){60}}
832
\put(85,55){\makebox(0,0){A}}
833
\put(75,62){\makebox(0,0){2}}
834
\put(40,15){\line(0,1){80}}
835
\put(40,105){\makebox(0,0){B}}
836
\put(60,15){\line(0,1){80}}
837
\put(60,105){\makebox(0,0){D}}
838
\put(50,5){\makebox(0,0){Fiber 3}}
839
\end{picture}}
840
\end{picture}
841
\end{figure}
842
843
We now have a proper, regular model~$\CC$ of~$C$ over~$\Z_2$.
844
Let $\CC_2$ be the special fiber of this model; a
845
diagram of~$\CC_2$ is in Figure~\ref{special2} and is labelled
846
{\it Fiber~3}. We can use $\CC$ to show that the
847
N\'eron model $\JJ$ of the Jacobian $J = J_{65,B}$ has good
848
reduction at~2.
849
850
We know that the reduction of~$\JJ^0$ is the extension of an abelian
851
variety by a connected linear group. Since $\CC$ is regular and
852
proper, the abelian variety part of the reduction is the product of
853
the Jacobians of the normalizations of the components of~$\CC_2$ (see
854
\cite[9.3/11 and 9.5/4]{BLR}). Thus, the abelian variety part is the
855
product of the Jacobians of~$B$ and~$D$. Since this is
856
2-dimensional, the reduction of~$\JJ^0$ is an abelian variety. In
857
other words, since the sum of the genera of the components of the
858
special fiber is equal to the dimension of~$J$, the reduction is an
859
abelian variety. It follows that $\JJ$ has good reduction at~2, that
860
the conductor of~$J$ is odd, and that $c_2 = 1$. As noted above, this
861
gives further evidence that the equation given in Table~\ref{table1}
862
is correct.
863
864
865
\vspace{1mm}
866
\noindent
867
{\bf Example 2.} Curve $C_{63}$ over $\Z_{3}$.
868
869
The Tamagawa number is often found using the intersection matrix and
870
sub-determinants. This is not entirely satisfactory for cases where
871
the special fiber has several components and a non-trivial Galois
872
action. Here is an example of how to resolve this (see also~\cite{BL}).
873
874
When we blow up curve~$C_{63}$ over~$\Z_{3}\unr$, we get
875
the special fiber shown in Figure~\ref{special1}.
876
Elements of $\Gal(\Q_{3}\unr/\Q_{3})$
877
that do not fix the quadratic unramified extension of~$\Q_{3}$
878
switch $H$ and~$I$. The other components are defined over~$\Q_{3}$.
879
All components have genus~0. The group $J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$
880
is isomorphic to a quotient of
881
%\begin{align*}
882
% L = \{ \alpha A + \beta B + \delta D + \epsilon E + \phi F + \gamma G
883
% &+ \eta H + \iota I \\
884
% &\mid \alpha + \beta + 2\delta + 2\epsilon + 4\phi + 2\gamma
885
% + 2\eta + 2\iota = 0 \} \,.
886
%\end{align*}
887
888
\[
889
L = \{ \alpha A + \beta B + \delta D + \epsilon E + \phi F + \gamma G
890
+ \eta H + \iota I
891
\mid \alpha + \beta + 2\delta + 2\epsilon + 4\phi + 2\gamma
892
+ 2\eta + 2\iota = 0 \} \,.\]
893
894
895
The kernel is generated by the following divisors.
896
\begin{center}
897
\begin{tabular}{*{2}{@{[}c@{]$\;=\;$}r@{\hspace{2cm}}}}
898
$A$ & $-2A + E$ & $B$ & $-2B + E$ \\
899
$D$ & $-D + E$ & $E$ & $A + B + D - 4E + F$ \\
900
$F$ & $E - 2F + G + H + I$ & $G$ & $F - 2G$ \\
901
$H$ & $F - 2H$ & $I$ & $F - 2I$
902
\end{tabular}
903
\end{center}
904
905
\begin{figure}
906
\caption{Special fiber of curve $C_{63}$ over $\Z_{3}$}
907
\label{special1}
908
\begin{picture}(400,130)
909
\put(100,5){\begin{picture}(200,125)
910
\thinlines
911
\put(20,50){\line(1,0){160}}
912
\put(40,20){\line(0,1){60}}
913
\put(60,20){\line(0,1){60}}
914
\put(80,20){\line(0,1){60}}
915
\put(150,10){\line(0,1){100}}
916
\put(120,70){\line(1,0){60}}
917
\put(120,90){\line(1,0){60}}
918
\put(120,30){\line(1,0){60}}
919
\put(40,88){\makebox(0,0){G}}
920
\put(60,88){\makebox(0,0){H}}
921
\put(80,88){\makebox(0,0){I}}
922
\put(150,118){\makebox(0,0){E}}
923
\put(185,50){\makebox(0,0){F}}
924
\put(185,90){\makebox(0,0){A}}
925
\put(185,70){\makebox(0,0){B}}
926
\put(185,30){\makebox(0,0){D}}
927
\put(35,70){\makebox(0,0){2}}
928
\put(55,70){\makebox(0,0){2}}
929
\put(75,70){\makebox(0,0){2}}
930
\put(165,55){\makebox(0,0){4}}
931
\put(165,35){\makebox(0,0){2}}
932
\put(145,104){\makebox(0,0){2}}
933
\end{picture}}
934
\end{picture}
935
\end{figure}
936
937
When we project away {}from~$A$, we find that
938
$J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$ is isomorphic to
939
\begin{align*}
940
\langle B, D, E, F, G, H, I
941
&\mid E = 0, E = 2B, D = E, 4E = B + D + F, \\
942
&\quad 2F = E + G + H + I, F = 2G = 2H = 2I \rangle.
943
\end{align*}
944
At this point, it is straightforward to simplify the representation by
945
elimination. Note that we projected away {}from~$A$, which is
946
Galois-invariant. It is best to continue eliminating Galois-invariant
947
elements first. We find that this group is isomorphic to
948
$\langle H, I \mid 2H = 2I = 0 \rangle$ and elements of
949
$\Gal(\Q_{3}\unr/\Q_{3})$ that do not fix the quadratic unramified
950
extension of~$\Q_{3}$ switch $H$ and~$I$. Therefore
951
$J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr) \cong \Z/2\Z \oplus \Z/2\Z$ and
952
$c_3 = \#J(\Q_{3})/J^{0}(\Q_{3}) = 2$.
953
954
\subsection{Computing $\Omega$}
955
\label{Omega}
956
957
By an {\em integral differential} (or {\em integral form}) on $J$ we mean the
958
pullback to $J$ of a global relative differential form on the N\'eron
959
model of $J$ over $\Z$. The set of integral $n$-forms on $J$ is a
960
full-rank lattice in the $\Q$-vector space of global holomorphic $n$-forms
961
on $J$. Since $J$ is an abelian variety of dimension 2, the integral
962
1-forms are a free $\Z$-module of rank 2 and the integral 2-forms are
963
a free $\Z$-module of rank 1. Moreover, the wedge of a basis for the
964
integral 1-forms is a generator for the integral 2-forms. The
965
quantity $\Omega$ is the integral, over the real points of $J$, of a
966
generator for the integral 2-forms. (We choose the generator that
967
leads to a positive integral.)
968
969
We now translate this into a computation on the curve $C$. Let
970
$\{\omega_1, \omega_2\}$ be a $\Q$-basis for the holomorphic
971
differentials on $C$ and let $\{\gamma_1, \gamma_2, \gamma_3,
972
\gamma_4\}$ be a $\Z$-basis for the homology of $C(\C)$. Create a
973
$2\times 4$ complex matrix $M_{\C} = [ \int_{\gamma_j}\omega_i]$ by
974
integrating the differentials over the homology and let $M_{\R} =
975
\Tr_{\C/\R}(M_{\C})$ be the $2\times 4$ real matrix whose entries are
976
traces {}from the complex matrix. The columns of $M_{\R}$ generate a
977
lattice $\Lambda$ in $\R^2$. If we make the standard identification
978
between the holomorphic 1-forms on $J$ and the holomorphic
979
differentials on $C$ (see \cite{Mi2}), then the notation
980
$\int_{J(\R)} |\omega_1 \wedge \omega_2|$ makes sense and its value
981
can be computed as the area of a fundamental domain for $\Lambda$.
982
983
If $\{\omega_1, \omega_2\}$ is a basis for the integral 1-forms on
984
$J$, then $\int_{J(\R)} |\omega_1 \wedge \omega_2| = \Omega$. On the
985
other hand, the computation of $M_{\C}$ is simplest if we choose
986
$\omega_1 = dX/Y$, and $\omega_2=X\,dX/Y$ with respect to a model for
987
$C$ of the form $Y^2=F(X)$; in this case we obtain $\Omega$ by a
988
simple change-of-basis calculation. This assumes, of course, that we
989
know how to express a basis for the integral 1-forms in terms of the
990
basis $\{\omega_1, \omega_2\}$; this is addressed in more detail
991
below.
992
993
It is worth mentioning an alternate strategy. Instead of finding a
994
$\Z$-basis for the homology of $C(\C)$ one could find a $\Z$-basis
995
$\{\gamma'_1, \gamma'_2\}$ for the subgroup of the homology that is
996
fixed by complex conjugation (call this the real homology).
997
Integrating would give us a $2\times 2$ real matrix $M'_{\R}$ and the
998
determinant of $M'_{\R}$ would equal the integral of $\omega_1
999
\wedge \omega_2$ over the connected component of $J(\R)$.
1000
In other words, the number of real connected components of $J$ is
1001
equal to the index of the $\C/\R$-traces in the real homology.
1002
1003
We now come to the question of determining the differentials on $C$
1004
which correspond to the integral 1-forms on $J$. Call these the
1005
integral differentials on $C$. This computation can be done one
1006
prime at a time. At each prime $p$ this is equivalent to determining
1007
a $\Z_p\unr$-basis for the global relative differentials on any
1008
proper, regular model for $C$ over $\Z_p\unr$. In fact a more
1009
general class of models can be used; see the discussion of models
1010
with rational singularities in \cite[\S 6.7]{BLR} and \cite[\S
1011
4.1]{Li}.
1012
1013
We start with the model $y^2 + g(x)y=f(x)$ given in
1014
Table~\ref{table1}. Note that the substitution $X=x$ and $Y=2y+g(x)$
1015
gives us a model of the form $Y^2=F(X)$. For integration purposes,
1016
our preferred differentials are $dX/Y=dx/(2y+g(x))$ and
1017
$X\,dX/Y=x\,dx/(2y+g(x))$. It is not hard to show that at primes of
1018
non-singular reduction for the $y^2 + g(x)y=f(x)$ model, these
1019
differentials will generate the integral 1-forms. For each prime $p$
1020
of singular reduction we give the following algorithm. All steps
1021
take place over $\Z_p\unr$.
1022
1023
\begin{description}
1024
\item[Step 1]
1025
Compute explicit equations for a proper, regular model $\CC$.
1026
1027
\item[Step 2]
1028
Diagram the configuration of the special fiber of $\CC$.
1029
1030
\item[Step 3] (Optional)
1031
Identify exceptional components and blow them down in the
1032
configuration diagram. Repeat step 3 as necessary.
1033
1034
\item[Step 4] (Optional)
1035
Remove components with genus 0 and self-intersection $-2$.
1036
Since $C$ has genus greater than 1,
1037
there will be a component that is not of this kind.
1038
1039
(This step corresponds to contracting the given components. The model
1040
obtained would no longer be regular; it would, however, be a proper model
1041
with rational singularities. We will not need a
1042
diagram of the resulting configuration.)
1043
1044
\item[Step 5]
1045
Determine a $\Z_p\unr$-basis for the integral differentials. It
1046
suffices to check this on a dense open subset of each surviving
1047
component. Note that we have explicit equations for a dense open
1048
subset of each of these components {}from the model $\CC$ in step 1. A
1049
pair of differentials $\{\eta_1, \eta_2\}$ will be a basis for the
1050
integral differentials (at $p$) if the following three statements are
1051
true.
1052
\begin{description}
1053
\item[a]
1054
The pair $\{\eta_1, \eta_2\}$ is a basis for the holomorphic
1055
differentials on $C$.
1056
\item[b]
1057
The reductions of $\eta_1$ and $\eta_2$ produce well-defined
1058
differentials mod $p$ on an open subset of each surviving component.
1059
\item[c]
1060
If $a_1\eta_1+a_2\eta_2 = 0 \pmod{p}$ on all surviving components,
1061
then $p|a_1$ and $p|a_2$.
1062
\end{description}
1063
\end{description}
1064
1065
Techniques for explicitly computing a proper, regular model are
1066
discussed in Section~\ref{Tamagawa}. A configuration diagram should
1067
include the genus, multiplicity and self-intersection number of
1068
each component and the number and type of intersections between
1069
components. Note that when an exceptional component is blown down,
1070
all of the self-intersection numbers of the components intersecting
1071
it will go up (towards 0). In particular, components which were not
1072
exceptional before may become exceptional in the new configuration.
1073
1074
Steps 3 and 4 are intended to make this algorithm more efficient for
1075
a human. They are entirely optional. For a computer implementation
1076
it may be easier to simply check every component than to worry about
1077
manipulating configurations.
1078
1079
The curves in Table~\ref{table1} are given as $y^2 + g(x)y=f(x)$. We
1080
assumed, at first, that $dx/(2y+g(x))$ and $x\,dx/(2y+g(x))$ generate
1081
the integral differentials. We integrated these differentials around
1082
each of the four paths generating the complex homology and found a
1083
provisional $\Omega$. Then we checked the proper, regular models to
1084
determine if these differentials really do generate the integral
1085
differentials and adjusted $\Omega$ when necessary. There were
1086
three curves where we needed to adjust $\Omega$. We describe the
1087
adjustment for curve $C_{65,B}$ in the following example. For curve
1088
$C_{63}$, we used the differentials $3\,dx/(2y+g(x))$ and
1089
$x\,dx/(2y+g(x))$. For curve $C_{65,A}$, we used the differentials
1090
$3\,dx/(2y+g(x))$ and $3x\,dx/(2y+g(x))$.
1091
1092
\vspace{2mm}
1093
\noindent
1094
{\bf Example 3.} Curve $C_{65,B}$.
1095
1096
The primes of singular reduction for curve $C_{65,B}$ are 2, 5 and
1097
13. In Example 1 of Section~\ref{Tamagawa}, we found a proper,
1098
regular model $\CC$ for $C$ over $\Z_2\unr$. The configuration for
1099
the special fiber of $\CC$ is sketched in Figure~\ref{special2} under
1100
the label {\it Fiber 3}. Component $A$ is exceptional and can be
1101
blown down to produce a model in which $B$ and $D$ cross
1102
transversally. Since $B$ and $D$ both have genus 1, we cannot
1103
eliminate either of these components. Furthermore, it suffices to
1104
check $B$, since $D$ is its Galois conjugate.
1105
1106
To get {}from the equation of the curve listed in Table~\ref{table1}
1107
to an affine containing an open subset of $B$ we need to make the
1108
substitutions $x=x_0 - \alpha$ and $y=x_0 (y_{3}x_0 - \alpha - 1)$.
1109
We also have $x_{0}p_{1}=2$. Using the substitutions and the
1110
relation $dx_{0}/x_0 = -dp_{1}/p_1$, we get
1111
\[ \frac{dx}{2y} = \frac{-dp_1}{2p_1(y_3 x_0 - \alpha - 1)}
1112
\text{\quad and\quad}
1113
\frac{x\,dx}{2y}
1114
= \frac{-(x_0 + \alpha)\,dp_1}{2p_1(y_3 x_0 - \alpha - 1)} \,.
1115
\]
1116
Note that $p_1 - t$ is a uniformizer at $p_1 = t$ almost everywhere
1117
on~$B$. When we multiply each differential by~2, then the
1118
denominator of each is almost everywhere non-zero; thus, $dx/y$ and
1119
$x\,dx/y$ are integral at~$2$. Moreover, although the linear
1120
combination $(x-\alpha)\,dx/y$ is identically zero on~$B$, it is not
1121
identically zero on~$D$ (its Galois conjugate is not identically zero
1122
on~$B$). Thus, our new basis is correct at~2. We multiply the
1123
provisional $\Omega$ by~4 to get a new provisional $\Omega$ which is
1124
correct at~$2$.
1125
1126
Similar (but somewhat simpler) computations at the primes $5$ and~$13$
1127
show that no adjustment is needed at these primes. Thus, $dx/y$
1128
and $x\,dx/y$ form a basis for the integral differentials of curve
1129
$C_{65,B}$, and the correct value of $\Omega$ is 4 times our original
1130
guess.
1131
1132
\section{Modular algorithms}
1133
\label{modular}
1134
1135
In this section, we describe the algorithms that were used to compute
1136
some of the data from the newforms. This includes the analytic rank
1137
and leading coefficient of the $L$-series. For optimal quotients,
1138
the value of~$k\cdot\Omega$ can also be found ($k$ is the Manin constant),
1139
as well as partial information
1140
on the Tamagawa numbers~$c_p$ and the size of the torsion subgroup.
1141
1142
\subsection{Analytic rank of $L(J,s)$ and leading coefficient at $s=1$}
1143
\label{l}
1144
1145
Fix a Jacobian~$J$ corresponding to the 2-dimensional subspace of
1146
$S_2(N)$ spanned by quadratic conjugate, normalized newforms~$f$
1147
and~$\overline{f}$. Let $W_N$ be the Fricke involution. The newforms~$f$
1148
and~$\overline{f}$ have the same eigenvalue~$\epsilon_N$ with respect
1149
to~$W_N$, namely $+1$ or~$-1$. In the notation of Section~\ref{curves}, let
1150
\[ L(f,s) = \sum\limits_{n=1}^{\infty} \frac{a_n}{n^s} \]
1151
be the $L$-series of~$f$; then $L(\overline{f},s)$ is the Dirichlet
1152
series whose coefficients are the conjugates of the
1153
coefficients of~$L(f,s)$. (Recall that the~$a_n$ are integers in some
1154
real quadratic field.) The order of~$L(f,s)$ at~$s = 1$ is even
1155
when $\epsilon_N = -1$ and odd when $\epsilon_N = +1$. We have
1156
$L(J,s) = L(f,s) L(\overline{f},s)$. Thus the analytic rank of $J$ is~0
1157
modulo~4 when $\epsilon_N = -1$ and 2 modulo~4 when $\epsilon_N = +1$.
1158
We found that the ranks were all 0 or~2. To prove that the analytic
1159
rank of~$J$ is~0, we need to show $L(f,1) \neq 0$ and
1160
$L(\overline{f},1) \neq 0$. In the case that $\epsilon_N = +1$, to
1161
prove that the analytic rank is~2, we need to show that $L'(f,1) \neq 0$
1162
and $L'(\overline{f},1) \neq 0$. When $\epsilon_N = -1$, we can
1163
evaluate $L(f,1)$ as in~\cite[\S 2.11]{Cr}. When $\epsilon_N = +1$, we
1164
can evaluate $L'(f,1)$ as in~\cite[\S 2.13]{Cr}. Each appropriate
1165
$L(f,1)$ or~$L'(f,1)$ was at least~$0.1$ and the errors in our
1166
approximations were all less than~$10^{-67}$. In this way we
1167
determined the analytic ranks, which we denote~$r$. As noted in the
1168
introduction, the analytic rank equals the Mordell-Weil rank if $r = 0$
1169
or~$r = 2$. Thus, we can simply call $r$ the rank, without fear of
1170
ambiguity.
1171
1172
To compute the leading coefficient of~$L(J,s)$ at~$s = 1$, we note that
1173
$\lim_{s \to 1} L(J,s)/(s-1)^r = L^{(r)}(J,1)/r!$.
1174
In the $r=0$ case, we simply have $L(J,1) = L(f,1)L(\overline{f},1)$.
1175
In the $r=2$ case, we have
1176
$L''(J,s)
1177
= L''(f,s)L(\overline{f},s) + 2L'(f,s)L'(\overline{f},s)
1178
+ L(f,s)L''(\overline{f},s)$.
1179
Evaluating both sides
1180
at $s=1$ we get $\frac{1}{2}L''(J,1) = L'(f,1)L'(\overline{f},1)$.
1181
1182
\subsection{Computing $k\cdot\Omega$}\label{modomega}
1183
Let $J$, $f$ and $\overline{f}$ be as in Section~\ref{l} and assume
1184
$J$ is an optimal quotient. Let $V$ be the 2-dimensional space
1185
spanned by $f$ and $\overline{f}$. Choose a basis
1186
$\{\omega_1,\omega_2\}$ for the subgroup of $V$ consisting of forms
1187
whose $q$-expansion coefficients lie in $\Z$. Let $k\cdot\Omega$ be
1188
the volume of the real points of the quotient of $\C\times\C$ by the
1189
lattice of period integrals $(\int_\gamma \omega_1,
1190
\int_\gamma\omega_2)$ with $\gamma$ in the integral homology
1191
$H_1(X_0(N),\Z)$.
1192
The rational number $k$
1193
is called
1194
the {\em Manin constant}. In practice we compute $k\cdot\Omega$
1195
using modular symbols and a generalization to dimension 2~of the
1196
algorithm for computing periods described in \cite[\S2.10]{Cr}. When
1197
$L(J,1)\neq 0$ the method of \cite[\S2.11]{Cr} coupled with
1198
Sections~\ref{l} and~\ref{bsdratio} can also be used to compute
1199
$k\cdot\Omega$.
1200
1201
A slight generalization of the argument of
1202
Proposition 2 of \cite{Ed1} proves that $k$ is, in fact, an integer.
1203
This generalization can be found in \cite{AS2}, where
1204
one also finds a conjecture that~$k$ must equal~$1$ for all optimal quotients of
1205
Jacobians of modular curves, which generalizes the longstanding conjecture of Manin
1206
that~$k$ equals~$1$ for all optimal elliptic curves. In unpublished work, Edixhoven
1207
has partially proven Manin's conjecture.
1208
1209
The computations of the present paper verify that $k$ equals~$1$ for the
1210
optimal quotients that we are considering. Specifically, we computed
1211
$k\cdot\Omega$ as above and $\Omega$ as described in Section~\ref{Omega}.
1212
The quotient of the two values was always well within $0.5$ of $1$.
1213
1214
\subsection{Computing $L(J,1)/(k\cdot\Omega)$}\label{bsdratio}
1215
We compute the rational number $L(J,1)/(k\cdot\Omega)$, for optimal
1216
quotients,
1217
using the algorithm in \cite{AS1}.
1218
This algorithm generalizes the algorithm described in
1219
\cite[\S2.8]{Cr} to dimension greater than 1.
1220
1221
\subsection{Tamagawa numbers}
1222
In this section we assume that $p$ is a prime which
1223
exactly divides the conductor $N$ of $J$.
1224
Under these conditions, Grothendieck \cite{Gr} gave a
1225
description of the component group of $J$ in
1226
terms of a monodromy pairing on certain character groups.
1227
(For more details, see Ribet \cite[\S2]{Ri}.)
1228
If, in addition, $J$ is a new optimal quotient of $J_0(N)$, one
1229
deduces the following. When
1230
the eigenvalue for $f$ of the Atkin-Lehner involution $W_p$ is
1231
$+1$, then the rational component group of $J$ is a subgroup of
1232
$(\Z/2\Z)^2$. Furthermore, when the eigenvalue of $W_p$ is $-1$,
1233
the algorithm described in \cite{Ste} can be used to compute
1234
the value of~$c_p$.
1235
1236
\subsection{Torsion subgroup}
1237
\label{modtors}
1238
1239
To compute an integer divisible by the order of the
1240
torsion subgroup of $J$ we make use of the following two observations.
1241
First, it is a consequence of the Eichler-Shimura relation
1242
\cite[\S7.9]{Sh} that if $p$ is a prime not dividing the
1243
conductor $N$ of $J$ and $f(T)$ is the characteristic polynomial
1244
of the endomorphism $T_p$
1245
of $J$, then $\#J(\F_p) = f(p+1)$ (see \cite[\S2.4]{Cr}
1246
for an algorithm to compute $f(T)$).
1247
Second, if $p$ is an odd prime at which $J$ has good reduction,
1248
then the natural map $J(\Q)\tors\rightarrow J(\F_p)$ is injective
1249
(see \cite[p.\ 70]{CF}). This does not depend on whether $J$ is an
1250
optimal quotient.
1251
To obtain a lower bound on the torsion subgroup for optimal quotients,
1252
we use modular symbols and the Abel-Jacobi theorem \cite[IV.2]{La}
1253
to compute the order of the image of the rational point
1254
$(0)-(\infty)\in J_0(N)$.
1255
1256
\section{Tables}
1257
\label{tables}
1258
1259
In Table~\ref{table1}, we list the 32 curves described in
1260
Section~\ref{curves}. We give the level $N$ {}from which each curve
1261
arose, an integral model for the curve, and list the source(s) {}from
1262
which it came ($H$ for Hasegawa \cite{Ha}, $W$ for Wang \cite{Wan}).
1263
Throughout the paper, the curves are denoted $C_N$ (or $C_{N,A}$, $C_{N,B}$).
1264
1265
\begin{table}
1266
\begin{center}
1267
\begin{tabular}{|l|rcl|c|}
1268
\hline
1269
\multicolumn{1}{|c|}{$N$}
1270
& \multicolumn{3}{|c|}{Equation} & Source\\ \hline\hline
1271
23 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1272
$-2 x^5 - 3 x^2 + 2 x - 2$ & HW \\
1273
29 & $y^2 + (x^3 + 1)y$ & $=$ &
1274
$-x^5 - 3 x^4 + 2 x^2 + 2 x - 2$ & HW \\
1275
31 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1276
$-x^5 - 5 x^4 - 5 x^3 + 3 x^2 + 2 x - 3$ & HW \\
1277
35 & $y^2 + (x^3 + x)y$ & $=$ &
1278
$-x^5 - 8 x^3 - 7 x^2 - 16 x - 19$ & H \\ \hline
1279
39 & $y^2 + (x^3 + 1)y$ & $=$ &
1280
$-5 x^4 - 2 x^3 + 16 x^2 - 12 x + 2$ & H \\
1281
63 & $y^2 + (x^3 - 1)y$ & $=$ &
1282
$14 x^3 - 7$ & W \\
1283
65,A & $y^2 + (x^3 + 1)y$ & $=$ &
1284
$-4 x^6 + 9 x^4 + 7 x^3 + 18 x^2 - 10$ & W \\
1285
65,B & $y^2$ & $=$ &
1286
$-x^6 + 10 x^5 - 32 x^4 + 20 x^3 + 40 x^2 + 6 x - 1$ & W \\ \hline
1287
67 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1288
$x^5 - x$ & HW \\
1289
73 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1290
$-x^5 - 2 x^3 + x$ & HW \\
1291
85 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1292
$x^4 + x^3 + 3 x^2 - 2 x + 1$ & H \\
1293
87 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1294
$-x^4 + x^3 - 3 x^2 + x - 1$ & HW \\ \hline
1295
93 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1296
$-2 x^5 + x^4 + x^3$ & HW \\
1297
103 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1298
$x^5 + x^4$ & HW \\
1299
107 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1300
$x^4 - x^2 - x - 1$ & HW \\
1301
115 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1302
$2 x^3 + x^2 + x$ & HW \\ \hline
1303
117,A & $y^2 + (x^3 - 1)y$ & $=$ &
1304
$3 x^3 - 7$ & W \\
1305
117,B & $y^2 + (x^3 + 1)y$ & $=$ &
1306
$-x^6 - 3 x^4 - 5 x^3 - 12 x^2 - 9 x - 7$ & W \\
1307
125,A & $y^2 + (x^3 + x + 1)y$ & $=$ &
1308
$x^5 + 2 x^4 + 2 x^3 + x^2 - x - 1$ & HW \\
1309
125,B & $y^2 + (x^3 + x + 1)y$ & $=$ &
1310
$x^6 + 5 x^5 + 12 x^4 + 12 x^3 + 6 x^2 - 3 x - 4$ & W \\ \hline
1311
133,A & $y^2 + (x^3 + x + 1)y$ & $=$ &
1312
$-2 x^6 + 7 x^5 - 2 x^4 - 19 x^3 + 2 x^2 + 18 x + 7$ & W \\
1313
133,B & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1314
$-x^5 + x^4 - 2 x^3 + 2 x^2 - 2 x$ & HW \\
1315
135 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1316
$x^4 - 3 x^3 + 2 x^2 - 8 x - 3$ & W \\
1317
147 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1318
$x^5 + 2 x^4 + x^3 + x^2 + 1$ & HW \\ \hline
1319
161 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1320
$x^3 + 4 x^2 + 4 x + 1$ & HW \\
1321
165 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1322
$x^5 + 2 x^4 + 3 x^3 + x^2 - 3 x$ & H \\
1323
167 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1324
$-x^5 - x^3 - x^2 - 1$ & HW \\
1325
175 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1326
$-x^5 - x^4 - 2 x^3 - 4 x^2 - 2 x - 1$ & W \\ \hline
1327
177 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1328
$x^5 + x^4 + x^3$ & HW \\
1329
188 & $y^2$ & $=$ &
1330
$x^5 - x^4 + x^3 + x^2 - 2 x + 1$ & W \\
1331
189 & $y^2 + (x^3 - 1)y$ & $=$ &
1332
$x^3 - 7$ & W \\
1333
191 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1334
$-x^3 + x^2 + x$ & HW \\ \hline
1335
\end{tabular}
1336
\end{center}
1337
\caption{Levels, integral models and sources for curves}
1338
\label{table1}
1339
\end{table}
1340
1341
In Table~\ref{table2}, we list the curve~$C_N$ simply by~$N$, the
1342
level {}from which it arose. Let $r$ denote the rank. We
1343
list ${\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)$ where $L(J,s)$ is the
1344
$L$-series for the Jacobian $J$ of~$C_N$ and round off the results to
1345
five digits. The symbol $\Omega$ was defined in Section~\ref{Omega}
1346
and is also rounded to five digits. Let Reg denote the regulator,
1347
also rounded to five digits. We list the $c_{p}$'s by primes of
1348
increasing order dividing the level~$N$. We denote $J(\Q)\tors = \Phi$
1349
and list its size. We use $\Sh ?$ to denote the size of
1350
$({\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)) \cdot
1351
(\#J(\Q)\tors)^2/(\Omega\cdot {\rm Reg} \cdot \prod c_{p})$,
1352
rounded to the nearest integer. We will refer to this as the {\em conjectured
1353
size of} $\Sh(J,\Q)$.
1354
For rank~$0$ optimal quotients this integer equals the (a priori)
1355
rational number $(L(J,1)/(k\cdot\Omega))\cdot((\#J(\Q)\tors)^2/\prod c_{p})$;
1356
of course there is no rounding error in this computation. For all other cases
1357
the last column gives a bound on the accuracy of the
1358
computations; all values of $\Sh ?$ were at least this close to the
1359
nearest integer before rounding.
1360
1361
\newcommand{\mcc}[1]{\multicolumn{1}{|c|}{#1}}
1362
\newcommand{\mcd}[1]{\multicolumn{2}{|c|}{#1}}
1363
1364
\begin{table}
1365
\begin{center}
1366
\begin{tabular}{|l|c|r@{.}l|r@{.}l|l|l|c|c|l|}
1367
\hline
1368
\mcc{$N$} & $r$
1369
& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
1370
& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & $\Phi$ & $\Sh$? & \mcc{error}
1371
\\ \hline\hline
1372
23 & 0 & 0&24843 & 2&7328 & 1 & 11 & 11 & 1 & $ $ \\
1373
29 & 0 & 0&29152 & 2&0407 & 1 & 7 & 7 & 1 & $ $ \\
1374
31 & 0 & 0&44929 & 2&2464 & 1 & 5 & 5 & 1 & $ $ \\
1375
35 & 0 & 0&37275 & 2&9820 & 1 & 16,2 & 16 & 1 & $ < 10^{-25} $ \\
1376
\hline
1377
39 & 0 & 0&38204 & 10&697 & 1 & 28,1 & 28 & 1 & $ < 10^{-25} $ \\
1378
63 & 0 & 0&75328 & 4&5197 & 1 & 2,3 & 6 & 1 & $ $ \\
1379
65,A & 0 & 0&45207 & 6&3289 & 1 & 7,1 & 14 & 2 & $ $ \\
1380
65,B & 0 & 0&91225 & 5&4735 & 1 & 1,3 & 6 & 2 & $ $ \\
1381
\hline
1382
67 & 2 & 0&23410 & 20&465 & 0.011439 & 1 & 1 & 1 & $ < 10^{-50} $ \\
1383
73 & 2 & 0&25812 & 24&093 & 0.010713 & 1 & 1 & 1 & $ < 10^{-49} $ \\
1384
85 & 2 & 0&34334 & 9&1728 & 0.018715 & 4,2 & 2 & 1 & $ < 10^{-26} $ \\
1385
87 & 0 & 1&4323 & 7&1617 & 1 & 5,1 & 5 & 1 & $ $ \\
1386
\hline
1387
93 & 2 & 0&33996 & 18&142 & 0.0046847 & 4,1 & 1 & 1 & $ < 10^{-49} $ \\
1388
103 & 2 & 0&37585 & 16&855 & 0.022299 & 1 & 1 & 1 & $ < 10^{-49} $ \\
1389
107 & 2 & 0&53438 & 11&883 & 0.044970 & 1 & 1 & 1 & $ < 10^{-49} $ \\
1390
115 & 2 & 0&41693 & 10&678 & 0.0097618 & 4,1 & 1 & 1 & $ < 10^{-50} $ \\
1391
\hline
1392
117,A & 0 & 1&0985 & 3&2954 & 1 & 4,3 & 6 & 1 & $ $ \\
1393
117,B & 0 & 1&9510 & 1&9510 & 1 & 4,1 & 2 & 1 & $ $ \\
1394
125,A & 2 & 0&62996 & 13&026 & 0.048361 & 1 & 1 & 1 & $ < 10^{-50} $ \\
1395
125,B & 0 & 2&0842 & 2&6052 & 1 & 5 & 5 & 4 & $ $ \\
1396
\hline
1397
133,A & 0 & 2&2265 & 2&7832 & 1 & 5,1 & 5 & 4 & $ $ \\
1398
133,B & 2 & 0&43884 & 15&318 & 0.028648 & 1,1 & 1 & 1 & $ < 10^{-49} $ \\
1399
135 & 0 & 1&5110 & 4&5331 & 1 & 3,1 & 3 & 1 & $ $ \\
1400
147 & 2 & 0&61816 & 13&616 & 0.045400 & 2,2 & 2 & 1 & $ < 10^{-50} $ \\
1401
\hline
1402
161 & 2 & 0&82364 & 11&871 & 0.017345 & 4,1 & 1 & 1 & $ < 10^{-47} $ \\
1403
165 & 2 & 0&68650 & 9&5431 & 0.071936 & 4,2,2 & 4 & 1 & $ < 10^{-26} $ \\
1404
167 & 2 & 0&91530 & 7&3327 & 0.12482 & 1 & 1 & 1 & $ < 10^{-47} $ \\
1405
175 & 0 & 0&97209 & 4&8605 & 1 & 1,5 & 5 & 1 & $ $ \\
1406
\hline
1407
177 & 2 & 0&90451 & 13&742 & 0.065821 & 1,1 & 1 & 1 & $ < 10^{-45} $ \\
1408
188 & 2 & 1&1708 & 11&519 & 0.011293 & 9,1 & 1 & 1 & $ < 10^{-44} $ \\
1409
189 & 0 & 1&2982 & 3&8946 & 1 & 1,3 & 3 & 1 & $ $ \\
1410
191 & 2 & 0&95958 & 17&357 & 0.055286 & 1 & 1 & 1 & $ < 10^{-44} $ \\
1411
\hline
1412
\end{tabular}
1413
\end{center}
1414
\caption{Conjectured sizes of $\Sh (J,\Q)$}
1415
\label{table2}
1416
\end{table}
1417
1418
In Table~\ref{table3} are generators of $J(\Q)/J(\Q)\tors$ for the
1419
curves whose Jacobians have Mordell-Weil rank~2. The generators are
1420
given as divisor classes. Whenever possible, we have chosen
1421
generators of the form $[P - Q]$ where $P$ and~$Q$ are rational
1422
points on the curve. Curve~167 is the only example where this is not
1423
the case, since the degree zero divisors supported on the (known)
1424
rational points on~$C_{167}$ generate a subgroup of index two in the
1425
full Mordell-Weil group.
1426
Affine points are given by their $x$ and $y$ coordinates in the model
1427
given in Table~\ref{table1}. There are two points at infinity in the
1428
normalization of the curves described by our equations, with the
1429
exception of curve~$C_{188}$. These are denoted by $\infty_a$, where
1430
$a$ is the value of the function $y/x^3$ on the point in question.
1431
The (only) point at infinity on curve~$C_{188}$ is simply
1432
denoted~$\infty$.
1433
1434
\begin{table}
1435
\begin{center}
1436
\begin{tabular}{|l|l|l|}
1437
\hline
1438
\mcc{$N$} & \mcd{Generators of $J(\Q)/J(\Q)\tors$} \\ \hline\hline
1439
67 & $ [(0, 0) - \infty_{-1}] $ &
1440
$ [(0, 0) - (0, -1)] $ \\
1441
73 & $ [(0, -1) - \infty_{-1}] $ &
1442
$ [(0, 0) - \infty_{-1}] $ \\
1443
85 & $ [(1, 1) - \infty_{-1}]$ &
1444
$ [(-1, 3) - \infty_{0}] $ \\
1445
93 & $ [(-1, 1) - \infty_{0}] $ &
1446
$ [(1, -3) - (-1, -2)] $ \\ \hline
1447
103 & $ [(0, 0) - \infty_{-1}]$ &
1448
$ [(0, -1) - (0,0)] $ \\
1449
107 & $ [\infty_{-1} - \infty_{0}]$ &
1450
$ [(-1, -1) - \infty_{-1}] $ \\
1451
115 & $ [(1, -4) - \infty_{0}] $ &
1452
$ [(1, 1) - (-2, 2)] $ \\
1453
125,A & $ [\infty_{-1} - \infty_{0}] $ &
1454
$ [(-1, 0) - \infty_{-1}] $ \\ \hline
1455
133,B & $ [\infty_{-1} - \infty_{0}] $ &
1456
$ [(0, -1) - \infty_{-1}] $ \\
1457
147 & $ [\infty_{-1} - \infty_{0}] $ &
1458
$ [(-1, -1) - \infty_{0}] $ \\
1459
161 & $ [(1, 2) - (-1, 1)] $ &
1460
$ [(\frac{1}{2}, -3) - (1, 2)] $ \\
1461
165 & $ [(1, 1) - \infty_{-1}] $ &
1462
$ [(0, 0) - \infty_{0} ] $ \\ \hline
1463
167 & $ [(-1 ,1) - \infty_{0}] $ &
1464
$ [(i, 0) + (-i, 0) - \infty_{0} - \infty_{-1}] $ \\
1465
177 & $ [(0, -1) - \infty_{0}] $ &
1466
$ [(0, 0) - (0, -1)] $ \\
1467
188 & $ [(0, -1) - \infty] $ &
1468
$ [(0, 1) - (1, -2)] $ \\
1469
191 & $ [\infty_{-1} - \infty_{0}]$ &
1470
$ [(0, -1) - \infty_{0}] $ \\
1471
\hline
1472
\end{tabular}
1473
\end{center}
1474
\caption{Generators of $J(\Q)/J(\Q)\tors$ in rank 2 cases}
1475
\label{table3}
1476
\end{table}
1477
1478
In Table~\ref{table4} are the reduction types, {}from the
1479
classification of~\cite{NU}, of the special fibers of the minimal,
1480
proper, regular models of the curves for each of the primes of
1481
singular reduction for the curve. They are the same as the primes
1482
dividing the level except that curve~$C_{65,A}$ has singular
1483
reduction at the prime~3 and curve~$C_{65,B}$ has singular reduction
1484
at the prime~2.
1485
1486
\begin{table}
1487
\begin{center}
1488
\begin{tabular}{|l|l|l|l|l||l|l|l|l|l|}
1489
\hline
1490
\mcc{$N$} & Prime & Type & Prime & Type &
1491
\mcc{$N$} & Prime & Type & Prime & Type
1492
\\ \hline\hline
1493
23 & 23 & ${\rm I}_{3-2-1}$ & & &
1494
117,A & 3 & ${\rm III}-{\rm III}^{\ast}-0$
1495
& 13 & ${\rm I}_{1-1-1}$ \\
1496
29 & 29 & ${\rm I}_{3-1-1}$ & & &
1497
117,B & 3 & ${\rm I}_{3-1-1}^{\ast}$
1498
& 13 & ${\rm I}_{1-1-0}$ \\
1499
31 & 31 & ${\rm I}_{2-1-1}$ & & &
1500
125,A & 5 & ${\rm VIII}-1$ & & \\
1501
35 & 5 & ${\rm I}_{3-2-2}$
1502
& 7 & ${\rm I}_{2-1-0}$ &
1503
125,B & 5 & ${\rm IX}-3$ & & \\ \hline
1504
39 & 3 & ${\rm I}_{6-2-2}$
1505
& 13 & ${\rm I}_{1-1-0}$ &
1506
133,A & 7 & ${\rm I}_{2-1-1}$
1507
& 19 & ${\rm I}_{1-1-0}$ \\
1508
63 & 3 & $2{\rm I}_{0}^{\ast}-0$
1509
& 7 & ${\rm I}_{1-1-1}$ &
1510
133,B & 7 & ${\rm I}_{1-1-0}$
1511
& 19 & ${\rm I}_{1-1-0}$ \\
1512
65,A & 3 & ${\rm I}_{0}-{\rm I}_{0}-1$
1513
& 5 & ${\rm I}_{3-1-1}$ &
1514
135 & 3 & III
1515
& 5 & ${\rm I}_{3-1-0}$ \\
1516
65,A & 13 & ${\rm I}_{1-1-0}$ & & &
1517
147 & 3 & ${\rm I}_{2-1-0}$
1518
& 7 & VII \\ \hline
1519
65,B & 2 & ${\rm I}_{0}-{\rm I}_{0}-1$
1520
& 5 & ${\rm I}_{3-1-0}$ &
1521
161 & 7 & ${\rm I}_{2-2-0}$
1522
& 23 & ${\rm I}_{1-1-0}$ \\
1523
65,B & 13 & ${\rm I}_{1-1-1}$ & & &
1524
165 & 3 & ${\rm I}_{2-2-0}$
1525
& 5 & ${\rm I}_{2-1-0}$ \\
1526
67 & 67 & ${\rm I}_{1-1-0}$ & & &
1527
165 & 11 & ${\rm I}_{2-1-0}$ & & \\
1528
73 & 73 & ${\rm I}_{1-1-0}$ & & &
1529
167 & 167 & ${\rm I}_{1-1-0}$ & & \\ \hline
1530
85 & 5 & ${\rm I}_{2-2-0}$
1531
& 17 & ${\rm I}_{2-1-0}$ &
1532
175 & 5 & ${\rm II}-{\rm II}-0$
1533
& 7 & ${\rm I}_{2-1-1}$ \\
1534
87 & 3 & ${\rm I}_{2-1-1}$
1535
& 29 & ${\rm I}_{1-1-0}$ &
1536
177 & 3 & ${\rm I}_{1-1-0}$
1537
& 59 & ${\rm I}_{1-1-0}$ \\
1538
93 & 3 & ${\rm I}_{2-2-0}$
1539
& 31 & ${\rm I}_{1-1-0}$ &
1540
188 & 2 & ${\rm IV}-{\rm IV}-0$
1541
& 47 & ${\rm I}_{1-1-0}$ \\
1542
103 & 103 & ${\rm I}_{1-1-0}$ & & &
1543
189 & 3 & ${\rm II}-{\rm IV}^{\ast}-0$
1544
& 7 & ${\rm I}_{1-1-1}$ \\ \hline
1545
107 & 107 & ${\rm I}_{1-1-0}$ & & &
1546
191 & 191 & ${\rm I}_{1-1-0}$ & & \\
1547
115 & 5 & ${\rm I}_{2-2-0}$
1548
& 23 & ${\rm I}_{1-1-0}$ & & & & & \\ \hline
1549
\end{tabular}
1550
\end{center}
1551
\caption{Namikawa and Ueno classification of special fibers}
1552
\label{table4}
1553
\end{table}
1554
1555
1556
\section{Discussion of Shafarevich-Tate groups and evidence for the
1557
second conjecture}
1558
\label{Shah}
1559
1560
{}From Section~\ref{MW} we have
1561
$\dim \Sh(J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2]$.
1562
With the exception of curves $C_{65,A}$, $C_{65,B}$, $C_{125,B}$, and
1563
$C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 0$. Thus we expect
1564
$\#\Sh(J,\Q)$ to be an odd square. In each case, the conjectured
1565
size of $\Sh(J,\Q)$ is~1. For curves $C_{65,A}$, $C_{65,B}$,
1566
$C_{125,B}$ and $C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 1, 1, 2$
1567
and~2 and the conjectured size of $\Sh(J,\Q) = 2, 2, 4$ and~4,
1568
respectively. We see that in each case, the (conjectured) size of
1569
the odd part of $\Sh(J,\Q)$ is~1 and the 2-part is accounted for by
1570
its 2-torsion.
1571
1572
Recall that for rank 0 optimal quotients we are able to exactly
1573
determine the value which the second Birch and Swinnerton-Dyer
1574
conjecture predicts for $\Sh(J,\Q)$. From the previous paragraph,
1575
we then see that equation~\eqref{eqn1} holds if and only if
1576
$\Sh(J,\Q)$ is killed by $2$.
1577
1578
It is also interesting to consider deficient primes. A prime $p$ is
1579
{\em deficient} with respect to a curve $C$ of genus~2, if $C$ has no
1580
degree 1 rational divisor over~$\Q_{p}$. {}From~\cite{PSt}, the
1581
number of deficient primes has the same parity as $\dim \Sh(J,\Q)[2]$.
1582
Curve $C_{65,A}$ has one deficient prime~$3$. Curve
1583
$C_{65,B}$ has one deficient prime~$2$. Curve $C_{117,B}$ has two
1584
deficient primes $3$ and~$\infty$. The rest of the curves have no
1585
deficient primes.
1586
1587
Since we have found $r$ (analytic rank) independent points on each
1588
Jacobian, we have a direct proof that the Mordell-Weil rank must
1589
equal the analytic rank if $\dim \Sh(J,\Q)[2] = 0$. For
1590
curves $C_{65,A}$ and $C_{65,B}$, the presence of an odd number of
1591
deficient primes gives us a
1592
similar result. For $C_{125,B}$ we used a $\sqrt{5}$-Selmer group
1593
to get a similar result.
1594
Thus, we have an independent proof of equality
1595
between analytic and Mordell-Weil ranks for all curves except
1596
$C_{133,A}$.
1597
1598
The 2-Selmer groups have the same dimensions for the pairs
1599
$C_{125,A}$, $C_{125,B}$ and $C_{133,A}$, $C_{133,B}$. For each
1600
pair, the Mordell-Weil rank is~2 for one curve and the 2-torsion of
1601
the Shafarevich-Tate group has dimension~2 for the other. In
1602
addition, the two Jacobians, when canonically embedded into~$J_0(N)$,
1603
intersect in their 2-torsion subgroups, and one can check that their
1604
2-Selmer groups become equal under the identification of
1605
$H^1(\Q, J_{N,A}[2])$ with $H^1(\Q, J_{N,B}[2])$ induced by the identification
1606
of the 2-torsion subgroups. Thus these are examples of the principle
1607
of a `visible part of a Shafarevich-Tate group' as discussed
1608
in~\cite{CM}.
1609
1610
\vspace{5mm}
1611
\begin{center}
1612
{\sc Appendix: Other Hasegawa curves}
1613
\end{center}
1614
1615
In Table~\ref{Hasegawa} is data concerning all 142 of Hasegawa's
1616
curves in the order presented in his paper. Let us explain the
1617
entries. The first column in each set of three columns gives the
1618
level, $N$. The second column gives a classification of the cusp
1619
forms spanning the 2-dimensional subspace of $S_2(N)$ corresponding
1620
to the Jacobian. When that subspace is irreducible with respect to
1621
the action of the Hecke algebra and is spanned by two newforms or two
1622
oldforms, we write $2n$ or $2o$, respectively. When that subspace is
1623
reducible and is spanned by two oldforms, two newforms or one of
1624
each, we write $oo$, $nn$ and $on$, respectively. The third column
1625
contains the sign of the functional equation at the level $M$ at
1626
which the cusp form is a newform. This is the negative of
1627
$\epsilon_M$ (described in Section~\ref{l}). The order of the two
1628
signs in the third column agrees with that of the forms listed in the
1629
second column. We include this information for those who would like
1630
to further study these curves. The curves with $N<200$ classified as
1631
$2n$ appeared already in Table~\ref{table1}.
1632
1633
The smallest possible Mordell-Weil ranks corresponding to $++$, $+-$,
1634
$-+$ and $--$, predicted by the first Birch and Swinnerton-Dyer
1635
conjecture, are $0$, $1$, $1$ and $2$ respectively. In all cases,
1636
those were, in fact, the Mordell-Weil ranks. This was determined by
1637
computing 2-Selmer groups with a computer program based on
1638
\cite{Sto2}. Of course, these are cases where the first Birch and
1639
Swinnerton-Dyer conjecture is already known to hold. In the cases
1640
where the Mordell-Weil rank is positive, the Mordell-Weil group has a
1641
subgroup of finite index generated by degree zero divisors supported
1642
on rational points with $x$-coordinates with numerators bounded by 7
1643
(in absolute value) and denominators by 12 with one exception. On
1644
the second curve with $N=138$, the divisor class
1645
$[(3+2\sqrt{2},80+56\sqrt{2}) + (3-2\sqrt{2},80-56\sqrt{2})-2\infty]$
1646
generates a subgroup of finite index in the Mordell-Weil group.
1647
1648
\vfill
1649
1650
\begin{table}
1651
\begin{center}
1652
\begin{tabular}{|c|c|c||c|c|c||c|c|c||c|c|c||c|c|c|}
1653
\hline
1654
22 & $oo$ & $++$ & 58 & $nn$ & $+-$ & 87 & $2o$ & $++$ & 129 & $on$ & $--$ &
1655
198 & $2o$ & $+-$ \\
1656
23 & $2n$ & $++$ & 60 & $oo$ & $++$ & 88 & $on$ & $+-$ & 130 & $on$ & $-+$ &
1657
204 & $2o$ & $+-$ \\
1658
26 & $nn$ & $++$ & 60 & $2o$ & $++$ & 90 & $on$ & $++$ & 132 & $oo$ & $++$ &
1659
205 & $2n$ & $--$ \\
1660
28 & $oo$ & $++$ & 60 & $2o$ & $++$ & 90 & $oo$ & $++$ & 133 & $2n$ & $--$ &
1661
206 & $2o$ & $--$ \\
1662
29 & $2n$ & $++$ & 62 & $2o$ & $++$ & 90 & $oo$ & $++$ & 134 & $2o$ & $--$ &
1663
209 & $2n$ & $--$ \\
1664
30 & $on$ & $++$ & 66 & $nn$ & $++$ & 90 & $oo$ & $++$ & 135 & $on$ & $+-$ &
1665
210 & $on$ & $+-$ \\
1666
30 & $oo$ & $++$ & 66 & $2o$ & $++$ & 91 & $nn$ & $--$ & 138 & $nn$ & $+-$ &
1667
213 & $2n$ & $--$ \\
1668
30 & $on$ & $++$ & 66 & $2o$ & $++$ & 93 & $2n$ & $--$ & 138 & $on$ & $+-$ &
1669
215 & $on$ & $--$ \\
1670
31 & $2n$ & $++$ & 66 & $on$ & $++$ & 98 & $oo$ & $++$ & 140 & $oo$ & $++$ &
1671
221 & $2n$ & $--$ \\
1672
33 & $on$ & $++$ & 67 & $2n$ & $--$ & 100 & $oo$ & $++$ & 142 & $nn$ & $+-$
1673
& 230 & $2o$ & $--$ \\ \hline
1674
35 & $2n$ & $++$ & 68 & $oo$ & $++$ & 102 & $on$ & $+-$ & 143 & $on$ & $+-$
1675
& 255 & $2o$ & $--$ \\
1676
37 & $nn$ & $+-$ & 69 & $2o$ & $++$ & 102 & $on$ & $+-$ & 146 & $2o$ & $--$
1677
& 266 & $2o$ & $--$ \\
1678
38 & $on$ & $++$ & 70 & $on$ & $++$ & 103 & $2n$ & $--$ & 147 & $2n$ & $--$
1679
& 276 & $2o$ & $+-$ \\
1680
39 & $2n$ & $++$ & 70 & $2o$ & $++$ & 104 & $2o$ & $++$ & 150 & $on$ & $++$
1681
& 284 & $2o$ & $+-$ \\
1682
40 & $on$ & $++$ & 70 & $2o$ & $++$ & 106 & $on$ & $--$ & 153 & $on$ & $+-$
1683
& 285 & $on$ & $--$ \\
1684
40 & $oo$ & $++$ & 70 & $2o$ & $++$ & 107 & $2n$ & $--$ & 154 & $on$ & $--$
1685
& 286 & $on$ & $--$ \\
1686
42 & $on$ & $++$ & 72 & $on$ & $++$ & 110 & $on$ & $++$ & 156 & $oo$ & $++$
1687
& 287 & $2n$ & $--$ \\
1688
42 & $oo$ & $++$ & 72 & $oo$ & $++$ & 111 & $oo$ & $+-$ & 158 & $on$ & $--$
1689
& 299 & $2n$ & $--$ \\
1690
42 & $on$ & $++$ & 73 & $2n$ & $--$ & 112 & $on$ & $+-$ & 161 & $2n$ & $--$
1691
& 330 & $2o$ & $--$ \\
1692
42 & $oo$ & $++$ & 74 & $oo$ & $+-$ & 114 & $oo$ & $+-$ & 165 & $2n$ & $--$
1693
& 357 & $2n$ & $--$ \\ \hline
1694
44 & $2o$ & $++$ & 77 & $on$ & $+-$ & 115 & $2n$ & $--$ & 166 & $on$ & $--$
1695
& 380 & $2o$ & $+-$ \\
1696
46 & $2o$ & $++$ & 78 & $oo$ & $++$ & 116 & $2o$ & $+-$ & 167 & $2n$ & $--$
1697
& 390 & $on$ & $--$ \\
1698
48 & $on$ & $++$ & 78 & $2o$ & $++$ & 117 & $2o$ & $++$ & 168 & $2o$ & $++$
1699
& & & \\
1700
48 & $oo$ & $++$ & 80 & $oo$ & $++$ & 120 & $oo$ & $++$ & 170 & $2o$ & $--$
1701
& & & \\
1702
50 & $nn$ & $++$ & 84 & $oo$ & $++$ & 120 & $on$ & $++$ & 177 & $2n$ & $--$
1703
& & & \\
1704
52 & $oo$ & $++$ & 84 & $oo$ & $++$ & 121 & $on$ & $+-$ & 180 & $2o$ & $++$
1705
& & & \\
1706
52 & $oo$ & $++$ & 84 & $oo$ & $++$ & 122 & $on$ & $--$ & 184 & $on$ & $+-$
1707
& & & \\
1708
54 & $on$ & $++$ & 84 & $oo$ & $++$ & 125 & $2n$ & $--$ & 186 & $2o$ & $--$
1709
& & & \\
1710
57 & $on$ & $+-$ & 85 & $2n$ & $--$ & 126 & $oo$ & $++$ & 190 & $on$ & $+-$
1711
& & & \\
1712
57 & $on$ & $+-$ & 87 & $2n$ & $++$ & 126 & $on$ & $++$ & 191 & $2n$ & $--$
1713
& & & \\
1714
\hline
1715
\end{tabular}
1716
\end{center}
1717
\caption{Spaces of cusp forms associated to Hasegawa's curves}
1718
\label{Hasegawa}
1719
\end{table}
1720
1721
\pagebreak
1722
\begin{thebibliography}{99}
1723
1724
\bibitem[AS1]{AS1}
1725
A.\ Agash\'{e} and W.A.\ Stein, \textit{Some abelian varieties with visible
1726
Shafarevich-Tate groups}, preprint, 2000.
1727
\bibitem[AS2]{AS2}
1728
A.\ Agash\'{e}, and W.A.\ Stein, \textit{The
1729
generalized Manin constant, congruence
1730
primes, and the modular degree}, in preparation, 2000.
1731
\bibitem[BSD]{BSD}
1732
B.\ Birch and H.P.F.\ Swinnerton-Dyer, \textit{Notes on elliptic curves.
1733
II}, J. reine angew. Math., \textbf{ 218} (1965), 79--108. MR 31
1734
\#3419
1735
\bibitem[BL]{BL}
1736
S.\ Bosch and Q.\ Liu, \textit{Rational points of the group of components
1737
of a N\'{e}ron model}, Manuscripta Math, \textbf{ 98} (1999), 275--293.
1738
\bibitem[BLR]{BLR}
1739
S.\ Bosch, W.\ L\"{u}tkebohmert and M.\ Raynaud, \textit{N\'{e}ron models},
1740
Springer-Verlag, Berlin, 1990. MR \textbf{ 91i}:14034
1741
\bibitem[BCDT]{BCDT}
1742
C.\ Breuil, B.\ Conrad, F.\ Diamond and R.\ Taylor \textit{On the modularity
1743
of elliptic curves over $\Q$: Wild 3-adic exercises}.
1744
\texttt{http://abel.math.harvard.edu/HTML/Individuals/Richard\_Taylor.html}
1745
(2000).
1746
\bibitem[BGZ]{BGZ}
1747
J.\ Buhler, B.H.\ Gross and D.B.\ Zagier, \textit{On the conjecture of
1748
Birch and Swinnerton-Dyer for an elliptic curve of rank $3$}.
1749
Math. Comp., \textbf{ 44} (1985), 473--481. MR \textbf{ 86g}:11037
1750
\bibitem[Ca]{Ca}
1751
J.W.S.\ Cassels, \textit{Arithmetic on curves of genus 1. VIII. On conjectures
1752
of Birch and Swinnerton-Dyer.},
1753
J. reine angew. Math., \textbf{ 217} (1965), 180--199.
1754
MR 31 \#3420
1755
\bibitem[CF]{CF}
1756
J.W.S.\ Cassels and E.V.\ Flynn, \textit{Prolegomena to a middlebrow
1757
arithmetic of curves of genus~2}, London Math. Soc., Lecture Note Series
1758
230,
1759
Cambridge Univ. Press, Cambridge, 1996. MR \textbf{ 97i}:11071
1760
\bibitem[Cr1]{Cr2}
1761
J.E.\ Cremona, \textit{Abelian varieties with extra twist, cusp forms, and
1762
elliptic curves over imaginary quadratic fields},
1763
J. London Math.\ Soc.\ (2), \textbf{ 45} (1992), 404--416.
1764
MR \textbf{ 93h}:11056
1765
\bibitem[Cr2]{Cr}
1766
J.E.\ Cremona, \textit{Algorithms for modular elliptic curves. 2nd edition},
1767
Cambridge Univ. Press, Cambridge, 1997. MR \textbf{ 93m}:11053
1768
\bibitem[CM]{CM}
1769
J.E.\ Cremona and B.\ Mazur,
1770
\textit{Visualizing elements in the Shafarevich-Tate group},
1771
to appear in {\it Experiment.\ Math.}
1772
\bibitem[Ed1]{Ed1}
1773
B.\ Edixhoven, \textit{On
1774
the Manin constants of modular elliptic curves}, Arithmetic
1775
algebraic geometry (Texel, 1989), Progr. Math., 89, Birkhauser Boston,
1776
Boston, MA, 1991, pp.\ 25--39.
1777
\bibitem[Ed2]{Ed2}
1778
B.\ Edixhoven, \textit{L'action de l'alg\`{e}bre de Hecke sur les groupes de
1779
composantes des jacobiennes des courbes modulaires est ``Eisenstein''},
1780
Ast\'{e}risque, No.\ 196--197 (1992), 159--170. MR \textbf{ 92k}:11059
1781
\bibitem[FPS]{FPS}
1782
E.V.\ Flynn, B.\ Poonen and E.F.\ Schaefer, \textit{Cycles of quadratic
1783
polynomials and rational points on a genus-two curve}, Duke Math.\ J.,
1784
\textbf{ 90} (1997), 435--463. MR \textbf{ 98j}:11048
1785
\bibitem[FS]{FS}
1786
E.V.\ Flynn and N.P.\ Smart, \textit{Canonical heights on the Jacobians
1787
of curves of genus~2 and the infinite descent}, Acta Arith.,
1788
\textbf{ 79} (1997), 333--352. MR \textbf{ 98f}:11066
1789
\bibitem[FM]{FM}
1790
G.\ Frey and M.\ M\"{u}ller, \textit{Arithmetic
1791
of modular curves and applications},
1792
in {\it Algorithmic algebra and number theory}, Ed.\ Matzat et al.,
1793
Springer-Verlag, Berlin, 1999, pp.\ 11--48. MR \textbf{ 00a}:11095
1794
\bibitem[GZ]{GZ}
1795
B.H.\ Gross and D.B.\ Zagier,
1796
\textit{Heegner points and derivatives of $L$-series},
1797
Invent. Math.,
1798
\textbf{ 84} (1986), 225--320. MR \textbf{ 87j}:11057
1799
\bibitem[Gr]{Gr}
1800
A.\ Grothendieck, \textit{Groupes de monodromie en g\'eom\'etrie alg\'ebrique},
1801
SGA 7 I, Expos\'{e} IX, Lecture Notes in Math.\ vol.\
1802
288, Springer, Berlin--Heidelberg--New York, 1972, pp.\ 313--523.
1803
MR 50 \#7134
1804
\bibitem[Ha]{Ha}
1805
R.\ Hartshorne, \textit{Algebraic geometry}, Grad.\ Texts in Math.\ 52,
1806
Springer-Verlag, New York, 1977.
1807
MR 57 \#3116
1808
\bibitem[Hs]{Hs}
1809
Y.\ Hasegawa, \textit{Table of quotient curves of modular curves $X_0(N)$
1810
with genus~2}, Proc.\ Japan.\ Acad., \textbf{ 71} (1995), 235--239.
1811
MR \textbf{ 97e}:11071
1812
\bibitem[Ko]{Ko}
1813
V.A.\ Kolyvagin, \textit{Finiteness of $E(\Q)$ and $\Sh (E,\Q)$ for a subclass
1814
of Weil curves}, Izv.\ Akad.\ Nauk SSSR Ser.\ Mat., \textbf{ 52} (1988),
1815
522--540. MR \textbf{ 89m}:11056
1816
\bibitem[KL]{KL}
1817
V.A.\ Kolyvagin and D.Y.\ Logachev, \textit{Finiteness of the Shafarevich-Tate
1818
group and the group of rational points for some modular abelian varieties},
1819
Leningrad Math J., \textbf{ 1} (1990), 1229--1253. MR \textbf{ 91c}:11032
1820
\bibitem[La]{La}
1821
S.\ Lang, \textit{Introduction to modular forms}, Springer-Verlag, Berlin, 1976.
1822
MR 55 \#2751
1823
\bibitem[Le]{Le}
1824
F.\ Lepr\'{e}vost, \textit{Jacobiennes de certaines courbes de genre 2:
1825
torsion et simplicit\'e}, J. Th\'eor. Nombres Bordeaux, \textbf{ 7} (1995),
1826
283--306. MR \textbf{ 98a}:11078
1827
\bibitem[Li]{Li}
1828
Q.\ Liu, \textit{Conducteur et discriminant minimal de courbes de genre 2},
1829
Compos.\ Math., \textbf{ 94} (1994), 51--79. MR \textbf{ 96b}:14038
1830
\bibitem[Ma]{Ma}
1831
B.\ Mazur, \textit{Rational isogenies of prime degree (with an appendix by D.
1832
Goldfeld)}, Invent.\ Math., ~\textbf{ 44} (1978), 129--162.
1833
MR \textbf{ 80h}:14022
1834
\bibitem[MS]{MS}
1835
J.R.\ Merriman and N.P.\ Smart, \textit{Curves of genus~2 with good reduction
1836
away {}from 2 with a rational Weierstrass point}, Math.\ Proc.\ Cambridge
1837
Philos.\ Soc., \textbf{114} (1993), 203--214. MR \textbf{ 94h}:14031
1838
\bibitem[Mi1]{Mi1}
1839
J.S.\ Milne, \textit{Arithmetic duality theorems}, Academic Press, Boston, 1986.
1840
MR \textbf{ 88e}:14028
1841
\bibitem[Mi2]{Mi2}
1842
J.S.\ Milne, \textit{Jacobian varieties}, in: {\it Arithmetic geometry},
1843
Ed.\ G.\ Cornell, G. and J.H.\ Silverman, Springer-Verlag, New York, 1986,
1844
pp.\ 167--212. MR \textbf{ 89b}:14029
1845
\bibitem[NU]{NU}
1846
Y.\ Namikawa and K.\ Ueno, \textit{The complete classification of fibres in
1847
pencils of curves of genus two}, Manuscripta Math., \textbf{ 9} (1973),
1848
143--186. MR 51 \#5595
1849
\bibitem[PSc]{PSc}
1850
B.\ Poonen and E.F.\ Schaefer, \textit{Explicit descent for Jacobians of
1851
cyclic covers of the projective line}, J. reine angew. Math.,
1852
\textbf{ 488} (1997), 141--188. MR \textbf{ 98k}:11087
1853
\bibitem[PSt]{PSt}
1854
B.\ Poonen and M.\ Stoll, \textit{The Cassels-Tate pairing on polarized
1855
abelian varieties}, Ann.\ of Math. (2), \textbf{150} (1999), 1109--1149.
1856
\bibitem[Ri]{Ri}
1857
K.\ Ribet, \textit{On modular representations of $\Gal(\Qbar/\Q)$
1858
arising from modular forms}, Invent. math.,
1859
\textbf{100} (1990), 431--476. MR \textbf{ 91g}:11066
1860
\bibitem[Sc]{Sc}
1861
E.F.\ Schaefer, \textit{Computing a Selmer group of a Jacobian using functions
1862
on the curve}, Math.\ Ann., \textbf{310} (1998), 447-471.
1863
MR \textbf{ 99h}:11063
1864
\bibitem[Sh]{Sh}
1865
G.\ Shimura, \textit{Introduction to the arithmetic theory of
1866
automorphic functions}, Princeton University Press, 1994.
1867
MR \textbf{ 95e}:11048
1868
\bibitem[Si]{Si}
1869
J.H.\ Silverman, \textit{Advanced topics in the arithmetic of elliptic curves},
1870
Grad.\ Texts in Math.\ 151, Springer-Verlag, New York,
1871
1994. MR \textbf{ 96b}:11074
1872
\bibitem[Ste]{Ste}
1873
W.A.\ Stein, \textit{Component groups of optimal quotients of Jacobians},
1874
preprint, 2000.
1875
\bibitem[Sto1]{Sto1}
1876
M.\ Stoll, \textit{Two simple 2-dimensional abelian varieties
1877
defined over~$\Q$ with Mordell-Weil rank at least~$19$},
1878
C. R. Acad. Sci. Paris, S\'erie I, \textbf{321} (1995), 1341--1344.
1879
MR \textbf{ 96j}:11084
1880
\bibitem[Sto2