%From: "Edward F. Schaefer" <[email protected]>1%Message-Id: <[email protected]>2%To: [email protected], [email protected], [email protected],3% [email protected], [email protected]4%Mime-Version: 1.05%Content-Type: text/plain; charset=X-roman86%Content-Transfer-Encoding: 7bit78%Hi all, below is what I sent to Mathematics of Computation,9%except without the following three %'s:1011% \pagebreak1213%\section{Introduction}14%\label{intro}1516% \normalsize17% \baselineskip=18pt1819%Cheers, Ed20212223\documentclass[12pt]{amsart}24\usepackage{amscd}25\newfont{\cyr}{wncyr10 scaled \magstep1}26\newcommand{\Sh}{\hbox{\cyr Sh}}27\newcommand{\C}{{\mathbf C}}28\newcommand{\Q}{{\mathbf Q}}29\newcommand{\Qbar}{\overline{\Q}}30%\newcommand{\GalQ}{{\Gal}(\Qbar/\Q)}31\newcommand{\CC}{{\mathcal C}}32\newcommand{\Z}{{\mathbf Z}}33\newcommand{\R}{{\mathbf R}}34\newcommand{\F}{{\mathbf F}}35\newcommand{\G}{{\mathbf G}}36\newcommand{\OO}{{\mathcal O}}37\newcommand{\JJ}{{\mathcal J}}38\newcommand{\DD}{{\mathcal D}}39\newcommand{\aaa}{{\mathfrak a}}40\newcommand{\PP}{{\mathbf P}}41\newcommand{\tors}{_{\text{tors}}}42\newcommand{\unr}{^{\text{unr}}}43\newcommand{\nichts}{{\left.\right.}}444546\DeclareMathOperator{\Gal}{Gal}47\DeclareMathOperator{\Norm}{Norm}48\DeclareMathOperator{\Sel}{Sel}49\DeclareMathOperator{\Tr}{Tr}5051\newtheorem{theorem}{Theorem}[section]52\newtheorem{lemma}[theorem]{Lemma}53\newtheorem{cor}[theorem]{Corollary}54\newtheorem{prop}[theorem]{Proposition}5556\theoremstyle{definition}57\newtheorem{question}{Question}58\newtheorem{conj}{Conjecture}5960\theoremstyle{remark}61\newtheorem{rem}{Remark$\!\!$} \renewcommand{\therem}{}62\newtheorem{rems}{Remarks$\!\!$} \renewcommand{\therems}{}6364\topmargin -0.3in65\headsep 0.3in66\oddsidemargin 0in67\evensidemargin 0in68\textwidth 6.5in69\textheight 9in7071%%%\renewcommand{\baselinestretch}{2}7273\begin{document}7475\title[Modular Jacobians]{Empirical evidence for the Birch and76Swinnerton-Dyer conjectures for77modular Jacobians of genus~2 curves}7879\author{E.\ Victor Flynn}80\address{Department of Mathematical Sciences, University of81Liverpool, P.O.Box 147,82Liverpool L69 3BX, England}83\email{evflynn@liverpool.ac.uk}8485\author{Franck Lepr\'{e}vost}86\address{CNRS Equipe d'arithm\'etique, Institut de Math\'ematiques de Paris,87Universit\'e Paris 6,88Tour 46-56, 5\`eme \'etage, Case 247,892-4 place Jussieu, F-75252 Paris cedex 05, France}90\email{leprevot@math.jussieu.fr}9192\author{Edward F.\ Schaefer}93\address{Department of Mathematics and Computer Science \\94Santa Clara University \\ Santa Clara, CA 95053, USA}95\email{eschaefe@math.scu.edu}9697\author{William A.\ Stein}98\address{Department of Mathematics \\ University of California99at Berkeley \\ Berkeley, CA 94720, USA}100\email{was@math.berkeley.edu}101102\author{Michael Stoll}103\address{Mathematisches Institut, Universit\"{a}tsstr.\ 1, D-40225104D\"{u}sseldorf, Germany}105\email{stoll@math.uni-duesseldorf.de}106107\author{Joseph L.\ Wetherell}108\address{Department of Mathematics, University of Southern California,1091042 W.\ 36th Place, Los Angeles, CA 90089-1113, USA}110\email{jlwether@alum.mit.edu}111112\subjclass{Primary 11G40; Secondary 11G10, 11G30, 14H25, 14H40,14H45}113\keywords{Birch and Swinnerton-Dyer conjecture, genus~2, Jacobian, modular114abelian variety}115116\thanks{The first author thanks the Nuffield Foundation117(Grant SCI/180/96/71/G) for financial support.118The second author did some of the research at119the Max-Planck Institut f\"ur Mathematik and120the Technische Universit\"at Berlin.121The third author thanks the National Security Agency (Grant122MDA904-99-1-0013).123The fourth author was supported by a Sarah M. Hallam fellowship.124The fifth author did some of the research at125the Max-Planck-Institut f\"ur Mathematik.126The sixth author thanks the National Science Foundation127(Grant DMS-9705959).128The authors had useful conversations with John Cremona, Qing Liu,129Karl Rubin and130Peter Swinnerton-Dyer and are grateful to Xiangdong Wang and Michael131M\"{u}ller for making data available to them.}132133\date{August 11, 1999}134135\begin{abstract}136This paper provides empirical evidence for the Birch and137Swinnerton-Dyer conjectures for modular Jacobians of genus~2 curves.138The second of these conjectures relates six quantities associated to139a Jacobian over the rational numbers. One of these140six quantities is141the size of the Shafarevich-Tate group.142Unable to compute that, we143computed the five other quantities and solved for the last one. In144all 32~cases, the result is very close to an integer that is a power145of~2. In addition, this power of~2 agrees with the size of the1462-torsion of the Shafarevich-Tate group, which we could compute.147\end{abstract}148149\maketitle150\markboth{FLYNN, LEPR\'{E}VOST, SCHAEFER, STEIN, STOLL, AND WETHERELL}%151{GENUS~2 BIRCH AND SWINNERTON-DYER CONJECTURE}152153% \pagebreak154155156\section{Introduction}157\label{intro}158159% \normalsize160% \baselineskip=18pt161162The conjectures of Birch and Swinnerton-Dyer, originally stated163for elliptic curves over~$\Q$, have been a constant source of164motivation for the study of elliptic curves, with the ultimate165goal being to find a proof.166This has resulted not only in a better167theoretical understanding, but also in the development of better168algorithms for computing the analytic and arithmetic169invariants that are so intriguingly related by them. We now know170that the first and, up to a non-zero rational factor, the171second conjecture hold for modular elliptic curves over~$\Q$172\footnote{It has recently been announced by173Brueil, Conrad, Diamond and Taylor that they have extended Wiles'174results and shown175that all elliptic curves over~$\Q$ are modular.}176in the177analytic rank~0 and~1 cases (see \cite{GZ,Ko,Wal1,Wal2}).178Furthermore,179a number of people have provided numerical evidence for the180conjectures for a large number of elliptic curves; see181for example~\cite{BSD,Ca,Cr}.182183By now, our theoretical and algorithmic knowledge of curves of184genus~2 and their Jacobians has reached a state that makes it185possible to conduct similar investigations. The Birch and186Swinnerton-Dyer conjectures have been generalized to arbitrary187abelian varieties over number fields by Tate~\cite{Ta}. If188$J$ is the Jacobian of a genus~2 curve over $\Q$,189then the first conjecture190states that the order of vanishing of the $L$-series of the Jacobian at191$s=1$ (the {\em analytic rank}) is equal to the Mordell-Weil rank of the192Jacobian. The second conjecture is that193\begin{equation} \label{eqn1}194\lim\limits_{s \to 1} (s-1)^{-r} L(J,s) =195\Omega \cdot {\rm Reg} \cdot \prod\limits_{p} c_{p}196\cdot \#\Sh(J,\Q ) \cdot (\#J(\Q)\tors)^{-2} \,.197\end{equation}198In this equation, $L(J,s)$ is the $L$-series of the Jacobian199$J$, and $r$ is its analytic rank. We use $\Omega$ to denote the200integral over $J(\R)$ of a particular differential 2-form; the201precise choice of this differential is described in202Section~\ref{Omega}. ${\rm Reg}$ is the regulator of $J(\Q)$. For203primes $p$, we use $c_{p}$ to denote the size of $J(\Q_p)/J^0(\Q_p)$,204where $J^0(\Q_p)$ is defined in Section~\ref{Tamagawa}. We let205$\Sh(J,\Q)$ be the Shafarevich-Tate group of $J$ over $\Q$, and we let206$J(\Q)\tors$ denote the torsion subgroup of $J(\Q)$.207208As in the case of elliptic curves, the first conjecture assumes209that the $L$-series can be analytically continued to $s = 1$,210and the second conjecture additionally assumes that the211Shafarevich-Tate group is finite. Neither of these assumptions is212known to hold for arbitrary genus~2 curves. The analytic213continuation of the $L$-series, however, is known to exist for214modular abelian varieties over~$\Q$, where an abelian215variety is called {\em modular} if it is a quotient of the Jacobian~$J_0(N)$216of the modular curve~$X_0(N)$ for some level~$N$. For simplicity,217we will also call a genus~2 curve {\em modular} when its Jacobian is218modular in this sense. So it is certainly a good idea to look219at modular genus~2 curves over~$\Q$, since we then at least know that the220statement of the first conjecture makes sense. Moreover, for many modular221abelian varieties it is also known that the Shafarevich-Tate group222is finite, therefore the statement of the second conjecture also223makes sense. As it turns out, all of our examples belong to this224class. An additional benefit of choosing modular genus~2 curves is225that one can find lists of such curves in the literature.226227In this article, we provide empirical evidence for the Birch and228Swinnerton-Dyer conjectures for such modular genus~2 curves. Since there229is no known effective way of computing the size of the Shafarevich-Tate230group, we computed the other five terms in equation~\eqref{eqn1}231(in two different ways, if possible). This required several different232algorithms, some of which were developed or improved while we were233working on this paper. If one of these algorithms234is already well described in the literature, then we simply cite it.235Otherwise, we describe it here in some detail (in particular,236algorithms for computing $\Omega$ and237$c_p$).238239For modular abelian varieties associated to newforms whose240$L$-series have analytic rank~0 or~1, the first Birch and Swinnerton-Dyer241conjecture has been proven. In such cases, the242Shafarevich-Tate group is also known to be finite and the second conjecture243has been proven, up to a non-zero rational factor. This all244follows {}from results in245\cite{GZ,KL,Wal1,Wal2}.246In our examples, all of the analytic247ranks are either~0 or~1. Thus we already know that the first248conjecture holds. Since the Jacobians we consider are associated to a249quadratic conjugate pair of newforms, the analytic rank of the250Jacobian is twice the analytic rank of either newform (see \cite{GZ}).251252The second Birch and Swinnerton-Dyer conjecture has not been proven253for the cases we consider. In order to verify equation~\eqref{eqn1},254we computed the five terms other than $\#\Sh(J,\Q)$ and solved for255$\#\Sh(J,\Q)$. In each case, the value is an integer to within the256accuracy of our calculations. This number is a power of~2, which257coincides with the independently computed size of the 2-torsion258subgroup of~$\Sh(J,\Q)$. Hence, we have verified the second259Birch and Swinnerton-Dyer conjecture for our curves at least260numerically, if we assume that the Shafarevich-Tate group consists261of 2-torsion only. (This is an ad hoc assumption based only262on the fact that we do not know better.) See Section~\ref{Shah} for263circumstances under which the verification is exact.264265The curves are listed in Table~\ref{table1},266and the numerical results can be found in Table~\ref{table2}.267268269\section{The Curves}270\label{curves}271272Each of the genus~2 curves we consider is related to the Jacobian273$J_0(N)$ of the modular curve $X_0(N)$ for some level $N$. When only274one of these genus~2 curves arises {}from a given level $N$, then we275denote this curve by $C_N$; when there are two curves coming {}from level276$N$ we use the notation $C_{N,A}$, $C_{N,B}$. The relationship277of, say, $C_N$ to $J_0(N)$ depends on the source. Briefly, {}from278Hasegawa \cite{Hs} we obtain quotients of $X_0(N)$ and {}from Wang279\cite{Wan} we obtain curves whose Jacobians are quotients of $J_0(N)$.280In both cases the Jacobian $J_N$ of $C_N$ is isogenous to a2812-dimensional factor of $J_0(N)$. (When not referring to a specific282curve, we will typically drop the subscript $N$ {}from $J$.)283In this way we can also associate284$C_N$ with a 2-dimensional subspace of $S_2(N)$, the space of cusp285forms of weight~2 for $\Gamma_0(N)$.286287We now discuss the precise source of the genus~2 curves we will288consider. Hasegawa \cite{Hs} has provided exact equations for all289genus~2 curves which are quotients of $X_0(N)$ by a subgroup of the290Atkin-Lehner involutions. There are 142 such curves. We are291particularly interested in those where the Jacobian corresponds to a292subspace of $S_2(N)$ spanned by a quadratic conjugate pair of293newforms. There are 21 of these with level $N \leq 200$. For these294curves we will provide evidence for the second conjecture. There are295seven more such curves with $N > 200$. We can classify the other2962-dimensional subspaces into four types. There are2972-dimensional subspaces of oldforms that are irreducible under the298action of the Hecke algebra. There are also 2-dimensional subspaces299that are reducible under the action of the Hecke algebra and are300spanned by two oldforms, two newforms or one of each. The Jacobians301corresponding to the latter three kinds are always isogenous, over302$\Q$, to the product of two elliptic curves. Given the small levels,303these are elliptic curves for which Cremona \cite{Cr} has already304provided evidence for the Birch and Swinnerton-Dyer conjectures. In305Table~\ref{Hasegawa}, we describe the kind of cusp forms spanning the3062-dimensional subspace and the signs of their functional equations307{}from the level at which they are newforms. The analytic and308Mordell-Weil ranks were always the smallest possible given those signs.309310The second set of curves was created by Wang \cite{Wan} and is further311discussed in \cite{FM}. This set consists of 28 curves that were312constructed by considering the spaces $S_2(N)$ with $N \leq 200$.313Whenever a subspace spanned by a pair of quadratic conjugate newforms314was found, these newforms were integrated to produce a quotient315abelian variety~$A$ of $J_0(N)$. These quotients are {\em optimal} in the316sense of \cite{Ma}, in that the kernel of the quotient map is317connected.318319The period matrix for~$A$ was created using certain intersection320numbers. When all of the intersection numbers have the same value,321then the polarization on~$A$ induced {}from the canonical polarization322of~$J_0(N)$ is equivalent to a principal polarization. (Two323polarizations are {\em equivalent} if they differ by an integer multiple.)324Conversely, every 2-dimensional optimal quotient of $J_0(N)$ in which325the induced polarization is equivalent to a principal polarization is326found in this way.327328Using theta functions, numerical approximations were found for the329Igusa invariants of the abelian surfaces. These numbers coincide with330rational numbers of fairly small height within the limits of the331precision used for the computations. Wang then constructed curves332defined over~$\Q$ whose Igusa invariants are the rational numbers333found. (There is one abelian surface at level $N = 177$ for which Wang334was not able to find a curve.) If we assume that these rational335numbers are the true Igusa invariants of the abelian surfaces, then it336follows that Wang's curves have Jacobians isomorphic, over~$\Qbar$, to337the principally polarized abelian surfaces in his list. Since the338classification given by these invariants is only up to isomorphism339over~$\Qbar$, the Jacobians of Wang's curves are not necessarily340isomorphic to, but can be twists of, the optimal quotients341of~$J_0(N)$ over~$\Q$ (see below).342343There are four curves in Hasegawa's list which do not show up in344Wang's list (they are listed in Table~\ref{table1} with an $H$ in the345last column). Their Jacobians are quotients of~$J_0(N)$, but are not346optimal quotients. It is likely that there are modular genus~2 curves347which neither are Atkin-Lehner quotients of~$X_0(N)$ (in Hasegawa's348sense) nor have Jacobians that are optimal quotients. These curves349could be found by looking at the optimal quotient abelian surfaces and350checking whether they are isogenous to a principally polarized abelian351surface over $\Q$.352353For 17 of the curves in Wang's list, the 2-dimensional subspace354spanned by the newforms is the same as that giving one of Hasegawa's355curves. In all of those cases, the curve given by Wang's equation is356isomorphic, over $\Q$, to that given by Hasegawa. This verifies Wang's357equations for these 17 curves. They are listed in Table~\ref{table1}358with $HW$ in the last column.359360The remaining eleven curves (listed in Table~\ref{table1} with a361$W$ in the last column) derive from the other eleven optimal362quotients in Wang's list. These are described in more detail in363Section~\ref{bad11} below.364365With the exception of curves $C_{63}$, $C_{117,A}$ and $C_{189}$, the366Jacobians of all of our curves are absolutely simple, and the367canonically polarized Jacobians have automorphism groups of size two.368We showed that these Jacobians are absolutely simple using an argument369like those in \cite{Le,Sto1}. The automorphism group of the370canonically polarized Jacobian of a hyperelliptic curve is isomorphic371to the automorphism group of the curve (see \cite[Thm.\37212.1]{Mi2}). Each automorphism of a hyperelliptic curve is induced by373a linear fractional transformation on $x$-coordinates (see \cite[p.\3741]{CF}). Each automorphism also permutes the six Weierstrass375points. Once we believed we had found all of the automorphisms, we376were able to show that there are no more by considering all linear377fractional transformations sending three fixed Weierstrass points to378any three Weierstrass points. In each case, we worked with sufficient379accuracy to show that other linear fractional transformations did not380permute the Weierstrass points.381382Let $\zeta_{3}$ denote a primitive third root of unity. The383Jacobians of curves $C_{63}$, $C_{117,A}$ and $C_{189}$ are each384isogenous to the product of two elliptic curves over $\Q(\zeta_3)$,385though not over $\Q$, where they are simple. These genus~2 curves386have automorphism groups of size 12. In the following table we list387the curve at the left. On the right we give one of the elliptic388curves which is a factor of its Jacobian. The second factor is the389conjugate.390\[391\begin{array}{ll}392C_{63}: & y^2 = x(x^2 + (9 - 12\zeta_{3})x - 48\zeta_{3}) \\393C_{117,A}: & y^2 = x(x^2 - (12 + 27\zeta_{3})x - (48 + 48\zeta_{3})) \\394C_{189}: & y^2 = x^3 + (66 - 3\zeta_{3})x^2 + (342 + 81\zeta_{3})x395+ 105 + 21\zeta_{3}396\end{array}397\]398Note that these three Jacobians are examples of abelian varieties399`with extra twist' as discussed in~\cite{Cr2}, where they can be400found in the tables on page~409.401402\subsection{Models for the Wang-only curves}403\label{bad11}404405As we have already noted, a modular genus~2 curve may be found by406either, both, or neither of Wang's and \linebreak407Hasegawa's techniques.408Hasegawa's method allows for the exact determination, over $\Q$, of409the equation of any modular genus~2 curve it has found. On the other410hand, if Wang's technique detects a modular genus~2 curve $C_N$, his411method produces real approximations to a curve $C'_N$ which is defined412over $\Q$ and is isomorphic to $C_N$ over $\Qbar$. We will call413$C'_N$ a {\em twisted modular genus~2 curve}.414415In this section we attempt to determine equations for the eleven416modular genus~2 curves detected by Wang but not by Hasegawa. If we417assume that Wang's equations for the twisted modular genus~2 curves418are correct, we find that we are able to determine the twists. In419turn, this gives us strong evidence that Wang's equations for the420twisted curves were correct. Undoing the twist, we determine probable421equations for the modular genus~2 curves. We end by providing further422evidence for the correctness of these equations.423424In what follows, we will use the notation of~\cite{Cr} and recommend425it as a reference on the general results that we assume here and in426Section~\ref{modular} and the appendix.427Fix a level~$N$ and let428$f(z) \in S_2(N)$. Then $f$ has a Fourier expansion429\[ f(z) = \sum\limits_{n=1}^{\infty} a_{n} e^{2 \pi i n z}\,. \]430For a newform~$f$, we have $a_1 \neq 0$; so we can normalize it by431setting $a_1 = 1$. In our cases, the $a_n$'s are integers in a real432quadratic field. For each prime~$p$ not dividing~$N$, the433corresponding Euler factor of the $L$-series $L(f,s)$ is434$1 - a_p p^{-s} + p^{1-2s}$. Let $N(a_p)$ and $Tr(a_p)$ denote the435norm and trace of~$a_p$. The product of this Euler factor and its436conjugate is437$1 - Tr(a_p)\,p^{-s} + (N(a_p) + 2p)\,p^{-2s}438- p\,Tr(a_p)\,p^{-3s} + p^2\,p^{-4s}$.439Therefore, the characteristic440polynomial of the $p$-Frobenius on the corresponding abelian variety441over $\F_{p}$ is442$x^4 - Tr(a_p)\,x^3 + (N(a_p) + 2p)\,x^2 - p\,Tr(a_p)\,x + p^2$.443Let $C$ be a curve, over $\Q$, whose Jacobian, over $\Q$, comes {}from444the space spanned by $f$ and its conjugate. Then we know that445$p+1 - \#C(\F_{p}) = Tr(a_p)$ and446$\frac{1}{2}(\#C(\F_{p})^{2} + \#C(\F_{p^2})) - (p+1)\# C(\F_{p}) - p =447N(a_p)$ (see \cite[Lemma 3]{MS}).448For the odd primes less than 200, not dividing $N$, we computed449$\# C(\F_{p})$ and $\# C(\F_{p^2})$ for each curve given by one of450Wang's equations. {}From these we could compute the characteristic451polynomials of Frobenius and see if they agreed with those predicted452by the $a_p$'s of the newforms.453454Of the eleven curves, the characteristic polynomials agreed for only455four. In each of the remaining seven cases we found a twist of Wang's456curve whose characteristic polynomials agreed with those predicted by457the newform for all odd primes less than 200 not dividing $N$. Four458of these twists were quadratic and three were of higher degree. It459is these twists that appear in Table~\ref{table1}.460461We can provide further evidence that these equations are correct.462For each curve given in Table~\ref{table1}, it is easy to determine463the primes of singular reduction. In Section~\ref{Tamagawa} we will464provide techniques for determining which of those primes divides the465conductor of its Jacobian. In each case, the primes dividing the466conductor of the Jacobian of the curve are exactly the primes467dividing the level $N$; this is necessary. With the exception of468curve $C_{188}$, all the curves come {}from odd levels. We used Liu's469{\tt genus2reduction} program470({\tt ftp://megrez.math.u-bordeaux.fr/pub/liu}) to compute the471conductor of the curve. In each case (other than curve $C_{188}$),472the conductor is the square of the level; this is also necessary. For473curve $C_{188}$, the odd part of the conductor of the curve is the474square of the odd part of the level.475476In addition, since the Jacobians of the Wang curves are optimal477quotients, we can compute~$k\cdot\Omega$ (where $k$ is the Manin constant,478conjectured to be 1)479using the newforms.480In each case, these agree (to within the accuracy of our computations)481with the $\Omega$'s computed using the equations for the curves.482We can also compute the value of~$c_p$ for optimal quotients from483the newforms, when $p$ exactly divides~$N$ and the eigenvalue of the484$p$th Atkin-Lehner involution is $-1$. When $p$ exactly divides~$N$485and the eigenvalue of the $p$th Atkin-Lehner involution is~$+1$, the486component group is either $0$, $\Z/2\Z$, or~$(\Z/2\Z)^2$. These results487are always in agreement with the values computed using the equations488for the curves. The algorithms based on the newforms are489described in Section~\ref{modular}, those based on the490equations of the curves are described in Section~\ref{algms}.491492Lastly, we were able to compute the Mordell-Weil ranks of the Jacobians493of the curves given by ten of these eleven equations. In494each case it agrees with the analytic rank of the Jacobian,495as deduced {}from the newforms.496497It should be noted that curve~$C_{125,B}$ is the $\sqrt{5}$-twist of498curve~$C_{125,A}$; the corresponding statement holds for the associated4992-dimensional subspaces of~$S_2(125)$. Since curve~$C_{125,A}$ is500a Hasegawa curve, this proves that the equation given in Table~\ref{table1}501for curve~$C_{125,B}$ is correct.502503The $a_p$'s and other information concerning Wang's curves are504currently kept in a database at the Institut f\"{u}r experimentelle505Mathematik in Essen, Germany. Most recently, this database was under506the care of Michael M\"{u}ller. William Stein also keeps a database507of~$a_p$'s for newforms.508509\begin{rem}510For the remainder of this paper we will assume that the equations for511the curves given in Table~\ref{table1} are correct; that is, that512they are equations for the curves whose Jacobians are isogenous513to a factor of~$J_0(N)$ in the way described above.514Some of the quantities can be computed either {}from the newform515or {}from the equation for the curve. We performed both computations516whenever possible, and view this duplicate effort as an attempt to517verify our implementation of the algorithms rather than an attempt518to verify the equations in Table~\ref{table1}. For most quantities,519one method or the other is not guaranteed to produce a value; in this520case, we simply quote the value {}from whichever method did succeed.521The reader who is disturbed by this philosophy should522ignore the Wang-only curves, since the equations for the Hasegawa523curves can be proven to be correct.524\end{rem}525526527\section{Algorithms for genus~2 curves}528\label{algms}529530In this section, we describe the algorithms that are based on the531given models for the curves. We give algorithms that compute all532terms on the right hand side of equation~\eqref{eqn1}, with the533exception of the size of the Shafarevich-Tate group. We describe,534however, how to find the size of its 2-torsion subgroup.535536\subsection{Torsion Subgroup}537\label{torsion}538539The computation of the torsion subgroup of~$J(\Q)$ is straightforward.540We used the technique described in~\cite[pp.~78--82]{CF}.541This technique is not always effective, however. For an algorithm working542in all cases see~\cite{Sto3}.543544\subsection{Mordell-Weil rank and $\Sh(J,\Q)[2]$}545\label{MW}546547The group $J(\Q)$ is a finitely generated abelian group and so is548isomorphic to $\Z^{r} \oplus J(\Q)\tors$ for some $r$ called the549Mordell-Weil rank.550As noted above (see Section~\ref{intro}), we justifiably use551$r$ to denote both the analytic and Mordell-Weil ranks since they552agree for all curves in Table~\ref{table1}.553554We used the algorithm described in \cite{FPS} to compute ${\rm555Sel}^{2}_{\rm fake}(J,\Q)$ (notation {}from \cite{PSc}), which is a556quotient of the 2-Selmer group ${\rm Sel}^{2}(J,\Q)$. More details557on this algorithm can be found in \cite{Sto2}. Theorem 13.2 of558\cite{PSc} explains how to get ${\rm Sel}^{2}(J,\Q)$ {}from ${\rm559Sel}^{2}_{\rm fake}(J,\Q)$. Let $M[2]$ denote the 2-torsion of an560abelian group $M$ and let dim$V$ denote the dimension of an $\F_{2}$561vector space $V$. We have562$\dim {\rm Sel}^{2}(J,\Q) = r + \dim J(\Q)[2] + \dim \Sh(J,\Q)[2]$.563In other words,564\[ \dim\, \Sh (J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2]. \]565566It is interesting to note that in all 30 cases where567$\dim \Sh(J,\Q)[2] \le 1$, we were able to compute the Mordell-Weil rank568independently from the analytic rank.569The570cases where $\dim \Sh(J,\Q)[2] = 1$ are discussed in more571detail in Section~\ref{Shah}.572For both of the remaining cases we have $\dim \Sh(J,\Q)[2]=2$.573One of these cases is574$C_{125,B}$. For this curve we computed575${\rm Sel}^{\sqrt{5}}(J_{125,B},\Q)$576using the technique described in577\cite{Sc}. {}From this, we were able to determine that the Mordell-Weil578rank is 0 independently from the analytic rank.579For the other case,580$C_{133,A}$,581we could show that $r$ had to be either~0582or~2 {}from the equation, but we needed the analytic computation to583show that $r=0$.584585\subsection{Regulator}586\label{reg}587588When the Mordell-Weil rank is~0, then the regulator is~1. When the589Mordell-Weil rank is positive, then to compute the regulator, we590first need to find generators for $J(\Q)/J(\Q)\tors$. The regulator591is the determinant of the canonical height pairing matrix on this set592of generators. An algorithm for computing the generators and593canonical heights is given in~\cite{FS}; it was used to find594generators for $J(\Q)/J(\Q)\tors$ and to compute the regulators. In595that article, the algorithm for computing height constants at the596infinite prime is not clearly explained and there are some errors in597the examples. A clear algorithm for computing infinite height598constants is given in~\cite{Sto3}. In~\cite{Sto4}, some improvements of599the results and algorithms in~\cite{FS} and~\cite{Sto3} are discussed.600The regulators in Table~\ref{table2} have been double-checked using601these improved algorithms.602603\subsection{Tamagawa Numbers}604\label{Tamagawa}605606Let $\OO$ be the integer ring in~$K$ which will be $\Q_{p}$ or607$\Q_{p}\unr$ (the maximal unramified extension of $\Q_{p})$.608Let $\JJ$ be the N\'{e}ron model of~$J$ over~$\OO$.609Define $\JJ^{0}$ to be the open subgroup scheme of~$\JJ$ whose610generic fiber is isomorphic to~$J$ over~$K$ and whose special fiber611is the identity component of the closed fiber of~$\JJ$.612The group $\JJ^{0}(\OO)$ is isomorphic to a subgroup of~$J(K)$ which613we denote $J^{0}(K)$. The group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is614the component group of~$\JJ$ over~$\OO_{\Q_{p}\unr}$. We are615interested in computing $c_p = \#J(\Q_{p})/J^{0}(\Q_{p})$, which is616sometimes called the Tamagawa number.617Since N\'{e}ron models are stable under unramified base extension,618the $\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of619$J^{0}(\Q_{p}\unr)$ is~$J^{0}(\Q_{p})$.620Since $H^1(\Gal(\Q_{p}\unr/\Q_{p}), J^{0}(\Q_{p}\unr))$621is trivial (see~\cite[p.\ 58]{Mi1}) we see that the622$\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of623$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is $ J(\Q_{p})/J^{0}(\Q_{p})$.624625There exist several discussions in the literature on constructing the626group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ starting with an integral627model of the underlying curve. For our purposes, we especially628recommend Silverman's book~\cite{Si}, Chapter~IV, Sections 4 and~7.629For a more detailed treatment, see~\cite[chap.\ 9]{BLR}. In these630books, one can find justifications for what we will do. While631constructing such groups, we ran into a number of difficulties that632we did not find described anywhere. For that reason, we will present633examples of such difficulties that arose, as well as our methods of634resolution. We do not claim that we will describe all situations635that could arise.636637When computing $c_p$ we need a proper, regular model~$\CC$ for~$C$638over~$\Z_p$. Let $\Z_p\unr$ denote the ring of integers of~$\Q_p\unr$639and note that $\Z_p\unr$ is a pro-\'etale Galois extension640of~$\Z_p$ with Galois group641$\Gal(\Z_p\unr/\Z_p) = \Gal(\Q_p\unr/\Q_p)$.642It follows that giving a model for~$C$ over~$\Z_p$ is equivalent to643giving a model for~$C$ over~$\Z_p\unr$ that644is equipped with a Galois action. We have found it convenient to645always work with the latter description. Thus for us, giving a model646over~$\Z_p$ will always mean giving a model over~$\Z_p\unr$ together647with a Galois action.648649In order to find a proper, regular model for~$C$ over~$\Z_p$,650we start with the models in Table~\ref{table1}. Technically, we651consider the curves to be the two affine pieces $y^2+g(x)y=f(x)$ and652$v^2 + u^3 g(1/u)v = u^6 f(1/u)$, glued together by $ux=1$, $v=u^3y$.653We blow them up at all points that are not regular until we have a654regular model. (A point is {\em regular} if the cotangent space there has655two generators.) These curves are all proper, and this is not656affected by blowing up.657658Let $\CC_p$ denote the special fiber of~$\CC$ over~$\Z_p\unr$. The659group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is isomorphic to a quotient660of the degree~0 part of the free group on the irreducible components661of~$\CC_{p}$. Let the irreducible components be denoted $\DD_{i}$ for662$1\leq i\leq n$, and let the multiplicity of~$\DD_{i}$ in~$\CC_p$ be663$d_{i}$. Then the degree~0 part of the free group has the form664\[ L = \{ \sum\limits_{i=1}^{n} \alpha_{i}\DD_{i} \mid665\sum\limits_{i=1}^{n} d_{i}\alpha_{i} = 0 \}\,. \]666667In order to describe the group that we quotient out by, we must668discuss the intersection pairing. For components $\DD_{i}$ and~$\DD_{j}$669of the special fiber, let $\DD_{i} \cdot \DD_{j}$ denote670their intersection pairing. In all of the special fibers that arise671in our examples, distinct components intersect transversally. Thus,672if $i \neq j$, then $\DD_{i} \cdot \DD_{j}$ equals the number of points673at which $\DD_{i}$ and $\DD_{j}$ intersect. The case of674self-intersection ($i=j$) is discussed below.675676The kernel of the map {}from~$L$ to677$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is generated by678divisors of the form679\[ [\DD_j] = \sum\limits_{i=1}^{n} (\DD_{j} \cdot \DD_{i}) \DD_{i} \]680for each component~$\DD_j$. We can deduce $\DD_{j} \cdot \DD_{j}$ by681noting that $[\DD_j]$ must be contained in the group~$L$. This follows682{}from the fact that the intersection pairing of683$\CC_{p} = \sum d_i\DD_{i}$ with any irreducible component is 0.684685\vspace{1mm}686\noindent687{\bf Example 1.} Curve $C_{65,B}$ over $\Z_{2}$.688689An equation for curve~$C_{65,B}$ is690$y^2 = f(x) = -x^6 + 10x^5 - 32x^4 + 20x^3 + 40x^2 + 6x - 1$. The Jacobian691of this curve692is a quotient of the Jacobian of~$X_0(65)$. Though 65 is odd, this693curve has singular reduction at~2. Since the equation for this curve694is conjectural (it is a Wang-only curve), it will be nice to verify695that 2 does not divide the conductor of its Jacobian, i.e.\ that the696Jacobian has good reduction at~2. In addition, we will need a697proper, regular model for this curve in order to find~$\Omega$.698699Note that $f(x)$ has a factor of700$x^2 - 3x - 1$. The special fiber of the arithmetic surface701$y^2 = f(x)$ over~$\Z_{2}\unr$702is given by703$(y + x^3 + 1)^2 = 0 \pmod 2$; this is a genus~0 curve of multiplicity~2704that we denote~$A$. This model is not regular at the two points705$(x-\alpha, y, 2)$, where $\alpha$ is a root of $x^2 - 3x - 1$. It is706regular at infinity so we will blow up only the given affine cover.707The current special fiber is in Figure~\ref{special2} and is labelled708{\it Fiber~1}.709710We fix $\alpha$ and move $(x - \alpha, y, 2)$ to the origin using the711substitution $x_0 = x-\alpha$. We get712\[ y^2 = -x_0^6 + (-6\alpha + 10)x_0^5 + (5\alpha - 47)x_0^4713+ (-28\alpha + 60)x_0^3 + (-11\alpha - 2)x_0^2714+ (-24\alpha - 16)x_0715\]716which we rewrite as the pair of equations717\begin{align*}718g_{1}(x_{0},y,p)719&= -x_0^6 + (-3\alpha + 5) p x_0^5 + (5\alpha - 47) x_0^4720+ (-7\alpha + 15) p^2 x_0^3 \\721& \qquad {} + (-11\alpha - 2) x_0^2 + (-3\alpha - 2) p^3 x_0 - y^2722\\723&= 0,\\724p &= 2.725\end{align*}726To blow up at $(x_0,y,p)$, we introduce projective coordinates727$(x_1,y_1,p_1)$ with $x_{0} y_1 = x_{1} y$, $x_{0} p_{1} = x_{1} p$, and728$y p_1 = y_{1} p$. We look in all three affine covers and check for regularity.729730\begin{description}731\item[$x_{1} = 1$] We have $y = x_{0} y_{1}$, $p = x_{0} p_{1}$. We get732$g_2(x_{0},y_{1},p_{1}) = 0$, $x_{0} p_{1} = 2$, where733\begin{align*}734g_2(x_{0},y_{1},p_{1}) &= x_{0}^{-2}g_{1}(x_{0},x_{0}y_{1},x_{0}p_{1}) \\735&= -x_0^4 + (-3\alpha + 5) p_1 x_0^4 + (5\alpha - 47) x_0^2736+ (-7\alpha + 15) p_1^2 x_0^3 \\737& \qquad{} + (-11\alpha - 2) + (-3\alpha - 2) p_1^3 x_0^2 - y_1^2 \,.738\end{align*}739In the reduction we have either $x_{0} = 0$ or $p_1 = 0$.740\begin{description}741\item[$x_{0} = 0$] $(y_{1} + \alpha + 1)^2 = 0$.742This is a new component which we denote $B$. It has genus~0 and743multiplicity~2. We check regularity along~$B$ at744$(x_{0}, y_{1} + \alpha + 1, p_{1}-t, 2)$, with $t$ in $\Z_2\unr$, and745find that $B$ is nowhere regular.746\item[$p_{1} = 0$]747$(y_{1} + x_{0}^2 + \alpha x_{0} + (\alpha + 1))^2 = 0$.748Using the gluing maps, we see that this is~$A$.749\end{description}750751\item[$y_{1} = 1$] We get no new information {}from this affine cover.752753\item[$p_{1} = 1$] We have $x_{0} = x_{1} p$, $y = y_{1} p$. We get754$g_{3}(x_{1},y_{1},p) = p^{-2} g_{1}(x_{1}p,y_{1}p,p) = 0$, $p = 2$.755In the reduction we have756\begin{description}757\item[$p=0$] $(y_1 + (\alpha+1)x_1)^2 = 0$. Using the gluing maps, we758see that this is~$B$. It is nowhere regular.759\end{description}760\end{description}761762The current special fiber is in763Figure~\ref{special2} and is labelled {\it Fiber~2}. It is not regular764along~$B$ and at the other point on~$A$ which we have not yet blown up.765The component $B$ does not lie entirely in any one affine cover766so we will blow up the affine covers $x_1 = 1$ and $p_1 = 1$ along~$B$.767768To blow up $x_1 = 1$ along~$B$ we make the substitution769$y_2 = y_1 + \alpha + 1$ and replace each factor of~2 in a coefficient770by~$x_0 p_1$. We have $g_{4}(x_0,y_2,p_1) = 0$ and $x_0 p_1 = 2$, and we771want to blow up along the line $(x_0, y_2, 2)$. Blowing up along a line772is similar to blowing up at a point: since we are blowing up at773$(x_0, y_2, 2) = (x_0, y_2)$, we introduce projective774coordinates $x_3, y_3$ together with the relation $x_0 y_3 = x_3 y_2$. We775have two affine covers.776777\begin{description}778\item[$x_3 = 1$] We have $y_2 = y_{3} x_{0}$. We get779$g_{5}(x_{0},y_{3},p_{1}) = x_{0}^{-2} g_{4}(x_{0},y_{3}x_{0},p_1) = 0$780and $x_{0} p_{1} = 2$. In the reduction we have781\begin{description}782\item[$x_{0} = 0$]783$y_{3}^2 + (\alpha + 1) y_{3} p_{1} + \alpha p_{1}^3 + p_{1}^2784+ \alpha + 1 = 0$.785This is~$B$. It is now a non-singular genus~1 curve.786\item[$p_{1} = 0$] $(x_0 + y_3 + \alpha)^2 = 0$. This is~$A$. The point787where $B$ meets~$A$ transversally is regular.788\end{description}789790\item[$y_3 = 1$] We get no new information {}from this affine cover.791\end{description}792793When we blow up $p_1 = 1$ along~$B$ we get essentially the same thing and794all points are again regular.795796The other non-regular point on~$A$ is the conjugate of the one we797blew up. Therefore, after performing the conjugate blow ups, it too798will be a genus~1 component crossing~$A$ transversally. We denote799this component $D$; it is conjugate to~$B$.800801802\begin{figure}803\caption{Special fibers of curve $C_{65,B}$ over $\Z_{2}$;804points not regular are thick}805\label{special2}806\begin{picture}(400,130)807\put(20,5){\begin{picture}(100,125)808\thinlines809\put(20,55){\line(1,0){60}}810\put(85,55){\makebox(0,0){A}}811\put(75,62){\makebox(0,0){2}}812\put(40,55){\circle*{5}}813\put(60,55){\circle*{5}}814\put(50,5){\makebox(0,0){Fiber 1}}815\end{picture}}816\put(145,5){\begin{picture}(100,125)817\thinlines818\put(50,5){\makebox(0,0){Fiber 2}}819\put(20,55){\line(1,0){60}}820\put(85,55){\makebox(0,0){A}}821\put(75,62){\makebox(0,0){2}}822\put(60,55){\circle*{5}}823\put(40,15){\line(0,1){80}}824\put(40.5,15){\line(0,1){80}}825\put(39.5,15){\line(0,1){80}}826\put(39,15){\line(0,1){80}}827\put(41,15){\line(0,1){80}}828\put(40,105){\makebox(0,0){B}}829\put(34,90){\makebox(0,0){2}}830\end{picture}}831\put(270,5){\begin{picture}(100,125)832\thinlines833\put(20,55){\line(1,0){60}}834\put(85,55){\makebox(0,0){A}}835\put(75,62){\makebox(0,0){2}}836\put(40,15){\line(0,1){80}}837\put(40,105){\makebox(0,0){B}}838\put(60,15){\line(0,1){80}}839\put(60,105){\makebox(0,0){D}}840\put(50,5){\makebox(0,0){Fiber 3}}841\end{picture}}842\end{picture}843\end{figure}844845We now have a proper, regular model~$\CC$ of~$C$ over~$\Z_2$.846Let $\CC_2$ be the special fiber of this model; a847diagram of~$\CC_2$ is in Figure~\ref{special2} and is labelled848{\it Fiber~3}. We can use $\CC$ to show that the849N\'eron model $\JJ$ of the Jacobian $J = J_{65,B}$ has good850reduction at~2.851852We know that the reduction of~$\JJ^0$ is the extension of an abelian853variety by a connected linear group. Since $\CC$ is regular and854proper, the abelian variety part of the reduction is the product of855the Jacobians of the normalizations of the components of~$\CC_2$ (see856\cite[9.3/11 and 9.5/4]{BLR}). Thus, the abelian variety part is the857product of the Jacobians of~$B$ and~$D$. Since this is8582-dimensional, the reduction of~$\JJ^0$ is an abelian variety. In859other words, since the sum of the genera of the components of the860special fiber is equal to the dimension of~$J$, the reduction is an861abelian variety. It follows that $\JJ$ has good reduction at~2, that862the conductor of~$J$ is odd, and that $c_2 = 1$. As noted above, this863gives further evidence that the equation given in Table~\ref{table1}864is correct.865866867\vspace{1mm}868\noindent869{\bf Example 2.} Curve $C_{63}$ over $\Z_{3}$.870871The Tamagawa number is often found using the intersection matrix and872sub-determinants. This is not entirely satisfactory for cases where873the special fiber has several components and a non-trivial Galois874action. Here is an example of how to resolve this (see also~\cite{BL}).875876When we blow up curve~$C_{63}$ over~$\Z_{3}\unr$, we get877the special fiber shown in Figure~\ref{special1}.878Elements of $\Gal(\Q_{3}\unr/\Q_{3})$879that do not fix the quadratic unramified extension of~$\Q_{3}$880switch $H$ and~$I$. The other components are defined over~$\Q_{3}$.881All components have genus~0. The group $J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$882is isomorphic to a quotient of883\begin{align*}884L = \{ \alpha A + \beta B + \delta D + \epsilon E + \phi F + \gamma G885&+ \eta H + \iota I \\886&\mid \alpha + \beta + 2\delta + 2\epsilon + 4\phi + 2\gamma887+ 2\eta + 2\iota = 0 \} \,.888\end{align*}889890The kernel is generated by the following divisors.891\begin{center}892\begin{tabular}{*{2}{@{[}c@{]$\;=\;$}r@{\hspace{2cm}}}}893$A$ & $-2A + E$ & $B$ & $-2B + E$ \\894$D$ & $-D + E$ & $E$ & $A + B + D - 4E + F$ \\895$F$ & $E - 2F + G + H + I$ & $G$ & $F - 2G$ \\896$H$ & $F - 2H$ & $I$ & $F - 2I$897\end{tabular}898\end{center}899900\begin{figure}901\caption{Special fiber of curve $C_{63}$ over $\Z_{3}$}902\label{special1}903\begin{picture}(400,130)904\put(100,5){\begin{picture}(200,125)905\thinlines906\put(20,50){\line(1,0){160}}907\put(40,20){\line(0,1){60}}908\put(60,20){\line(0,1){60}}909\put(80,20){\line(0,1){60}}910\put(150,10){\line(0,1){100}}911\put(120,70){\line(1,0){60}}912\put(120,90){\line(1,0){60}}913\put(120,30){\line(1,0){60}}914\put(40,88){\makebox(0,0){G}}915\put(60,88){\makebox(0,0){H}}916\put(80,88){\makebox(0,0){I}}917\put(150,118){\makebox(0,0){E}}918\put(185,50){\makebox(0,0){F}}919\put(185,90){\makebox(0,0){A}}920\put(185,70){\makebox(0,0){B}}921\put(185,30){\makebox(0,0){D}}922\put(35,70){\makebox(0,0){2}}923\put(55,70){\makebox(0,0){2}}924\put(75,70){\makebox(0,0){2}}925\put(165,55){\makebox(0,0){4}}926\put(165,35){\makebox(0,0){2}}927\put(145,104){\makebox(0,0){2}}928\end{picture}}929\end{picture}930\end{figure}931932When we project away {}from~$A$, we find that933$J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$ is isomorphic to934\begin{align*}935\langle B, D, E, F, G, H, I936&\mid E = 0, E = 2B, D = E, 4E = B + D + F, \\937&\quad 2F = E + G + H + I, F = 2G = 2H = 2I \rangle.938\end{align*}939At this point, it is straightforward to simplify the representation by940elimination. Note that we projected away {}from~$A$, which is941Galois-invariant. It is best to continue eliminating Galois-invariant942elements first. We find that this group is isomorphic to943$\langle H, I \mid 2H = 2I = 0 \rangle$ and elements of944$\Gal(\Q_{3}\unr/\Q_{3})$ that do not fix the quadratic unramified945extension of~$\Q_{3}$ switch $H$ and~$I$. Therefore946$J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr) \cong \Z/2\Z \oplus \Z/2\Z$ and947$c_3 = \#J(\Q_{3})/J^{0}(\Q_{3}) = 2$.948949\subsection{Computing $\Omega$}950\label{Omega}951952By an {\em integral differential} (or {\em integral form}) on $J$ we mean the953pullback to $J$ of a global relative differential form on the N\'eron954model of $J$ over $\Z$. The set of integral $n$-forms on $J$ is a955full-rank lattice in the vector space of global holomorphic $n$-forms956on $J$. Since $J$ is an abelian variety of dimension 2, the integral9571-forms are a free $\Z$-module of rank 2 and the integral 2-forms are958a free $\Z$-module of rank 1. Moreover, the wedge of a basis for the959integral 1-forms is a generator for the integral 2-forms. The960quantity $\Omega$ is the integral, over the real points of $J$, of a961generator for the integral 2-forms. (We choose the generator that962leads to a positive integral.)963964We now translate this into a computation on the curve $C$. Let965$\{\omega_1, \omega_2\}$ be a $\Q$-basis for the holomorphic966differentials on $C$ and let $\{\gamma_1, \gamma_2, \gamma_3,967\gamma_4\}$ be a $\Z$-basis for the homology of $C(\C)$. Create a968$2\times 4$ complex matrix $M_{\C} = [ \int_{\gamma_j}\omega_i]$ by969integrating the differentials over the homology and let $M_{\R} =970\Tr_{\C/\R}(M_{\C})$ be the $2\times 4$ real matrix whose entries are971traces {}from the complex matrix. The columns of $M_{\R}$ generate a972lattice $\Lambda$ in $\R^2$. If we make the standard identification973between the holomorphic 1-forms on $J$ and the holomorphic974differentials on $C$ (see \cite{Mi2}), then the notation975$\int_{J(\R)} |\omega_1 \wedge \omega_2|$ makes sense and its value976can be computed as the area of a fundamental domain for $\Lambda$.977978If $\{\omega_1, \omega_2\}$ is a basis for the integral 1-forms on979$J$, then $\int_{J(\R)} |\omega_1 \wedge \omega_2| = \Omega$. On the980other hand, the computation of $M_{\C}$ is simplest if we choose981$\omega_1 = dX/Y$, and $\omega_2=X\,dX/Y$ with respect to a model for982$C$ of the form $Y^2=F(X)$; in this case we obtain $\Omega$ by a983simple change-of-basis calculation. This assumes, of course, that we984know how to express a basis for the integral 1-forms in terms of the985basis $\{\omega_1, \omega_2\}$; this is addressed in more detail986below.987988It is worth mentioning an alternate strategy. Instead of finding a989$\Z$-basis for the homology of $C(\C)$ one could find a $\Z$-basis990$\{\gamma'_1, \gamma'_2\}$ for the subgroup of the homology that is991fixed by complex conjugation (call this the real homology).992Integrating would give us a $2\times 2$ real matrix $M'_{\R}$ and the993determinant of $M'_{\R}$ would equal the integral of $\omega_1994\wedge \omega_2$ over the connected component of $J(\R)$.995In other words, the number of real connected components of $J$ is996equal to the index of the $\C/\R$-traces in the real homology.997998We now come to the question of determining the differentials on $C$999which correspond to the integral 1-forms on $J$. Call these the1000integral differentials on $C$. This computation can be done one1001prime at a time. At each prime $p$ this is equivalent to determining1002a $\Z_p\unr$-basis for the global relative differentials on any1003proper, regular model for $C$ over $\Z_p\unr$. In fact a more1004general class of models can be used; see the discussion of models1005with rational singularities in \cite[\S 6.7]{BLR} and \cite[\S10064.1]{Li}.10071008We start with the model $y^2 + g(x)y=f(x)$ given in1009Table~\ref{table1}. Note that the substitution $X=x$ and $Y=2y+g(x)$1010gives us a model of the form $Y^2=F(X)$. For integration purposes,1011our preferred differentials are $dX/Y=dx/(2y+g(x))$ and1012$X\,dX/Y=x\,dx/(2y+g(x))$. It is not hard to show that at primes of1013non-singular reduction for the $y^2 + g(x)y=f(x)$ model, these1014differentials will generate the integral 1-forms. For each prime $p$1015of singular reduction we give the following algorithm. All steps1016take place over $\Z_p\unr$.10171018\begin{description}1019\item[Step 1]1020Compute explicit equations for a proper, regular model $\CC$.10211022\item[Step 2]1023Diagram the configuration of the special fiber of $\CC$.10241025\item[Step 3] (Optional)1026Identify exceptional components and blow them down in the1027configuration diagram. Repeat step 3 as necessary.10281029\item[Step 4] (Optional)1030Remove components with genus 0 and self-intersection $-2$.1031Since $C$ has genus greater than 1,1032there will be a component that is not of this kind.1033(This1034step corresponds to contracting the given components to create a1035non-proper model with rational singularities. We will not need a1036diagram of the resulting configuration.)10371038\item[Step 5]1039Determine a $\Z_p\unr$-basis for the integral differentials. It1040suffices to check this on a dense open subset of each surviving1041component. Note that we have explicit equations for a dense open1042subset of each of these components {}from the model $\CC$ in step 1. A1043pair of differentials $\{\eta_1, \eta_2\}$ will be a basis for the1044integral differentials (at $p$) if the following three statements are1045true.1046\begin{description}1047\item[a]1048The pair $\{\eta_1, \eta_2\}$ is a basis for the holomorphic1049differentials on $C$.1050\item[b]1051The reductions of $\eta_1$ and $\eta_2$ produce well-defined1052differentials mod $p$ on an open subset of each surviving component.1053\item[c]1054If $a_1\eta_1+a_2\eta_2 = 0 \pmod{p}$ on all surviving components,1055then $p|a_1$ and $p|a_2$.1056\end{description}1057\end{description}10581059Techniques for explicitly computing a proper, regular model are1060discussed in Section~\ref{Tamagawa}. A configuration diagram should1061include the genus, multiplicity and self-intersection number of1062each component and the number and type of intersections between1063components. Note that when an exceptional component is blown down,1064all of the self-intersection numbers of the components intersecting1065it will go up (towards 0). In particular, components which were not1066exceptional before may become exceptional in the new configuration.10671068Steps 3 and 4 are intended to make this algorithm more efficient for1069a human. They are entirely optional. For a computer implementation1070it may be easier to simply check every component than to worry about1071manipulating configurations.10721073The curves in Table~\ref{table1} are given as $y^2 + g(x)y=f(x)$. We1074assumed, at first, that $dx/(2y+g(x))$ and $x\,dx/(2y+g(x))$ generate1075the integral differentials. We integrated these differentials around1076each of the four paths generating the complex homology and found a1077provisional $\Omega$. Then we checked the proper, regular models to1078determine if these differentials really do generate the integral1079differentials and adjusted $\Omega$ when necessary. There were1080three curves where we needed to adjust $\Omega$. We describe the1081adjustment for curve $C_{65,B}$ in the following example. For curve1082$C_{63}$, we used the differentials $3\,dx/(2y+g(x))$ and1083$x\,dx/(2y+g(x))$. For curve $C_{65,A}$, we used the differentials1084$3\,dx/(2y+g(x))$ and $3x\,dx/(2y+g(x))$.10851086\vspace{2mm}1087\noindent1088{\bf Example 3.} Curve $C_{65,B}$.10891090The primes of singular reduction for curve $C_{65,B}$ are 2, 5 and109113. In Example 1 of Section~\ref{Tamagawa}, we found a proper,1092regular model $\CC$ for $C$ over $\Z_2\unr$. The configuration for1093the special fiber of $\CC$ is sketched in Figure~\ref{special2} under1094the label {\it Fiber 3}. Component $A$ is exceptional and can be1095blown down to produce a model in which $B$ and $D$ cross1096transversally. Since $B$ and $D$ both have genus 1, we cannot1097eliminate either of these components. Furthermore, it suffices to1098check $B$, since $D$ is its Galois conjugate.10991100To get {}from the equation of the curve listed in Table~\ref{table1}1101to an affine containing an open subset of $B$ we need to make the1102substitutions $x=x_0 - \alpha$ and $y=x_0 (y_{3}x_0 - \alpha - 1)$.1103We also have $x_{0}p_{1}=2$. Using the substitutions and the1104relation $dx_{0}/x_0 = -dp_{1}/p_1$, we get1105\[ \frac{dx}{2y} = \frac{-dp_1}{2p_1(y_3 x_0 - \alpha - 1)}1106\text{\quad and\quad}1107\frac{x\,dx}{2y}1108= \frac{-(x_0 + \alpha)\,dp_1}{2p_1(y_3 x_0 - \alpha - 1)} \,.1109\]1110Note that $p_1 - t$ is a uniformizer at $p_1 = t$ almost everywhere1111on~$B$. When we multiply each differential by~2, then the1112denominator of each is almost everywhere non-zero; thus, $dx/y$ and1113$x\,dx/y$ are integral at~$2$. Moreover, although the linear1114combination $(x-\alpha)\,dx/y$ is identically zero on~$B$, it is not1115identically zero on~$D$ (its Galois conjugate is not identically zero1116on~$B$). Thus, our new basis is correct at~2. We multiply the1117provisional $\Omega$ by~4 to get a new provisional $\Omega$ which is1118correct at~$2$.11191120Similar (but somewhat simpler) computations at the primes $5$ and~$13$1121show that no adjustment is needed at these primes. Thus, $dx/y$1122and $x\,dx/y$ form a basis for the integral differentials of curve1123$C_{65,B}$, and the correct value of $\Omega$ is 4 times our original1124guess.11251126\section{Modular algorithms}1127\label{modular}11281129In this section, we describe the algorithms that were used to compute1130some of the data from the newforms. This includes the analytic rank1131and leading coefficient of the $L$-series. For optimal quotients,1132the value of~$k\cdot\Omega$ can also be found ($k$ is the Manin constant),1133as well as partial information1134on the Tamagawa numbers~$c_p$ and the size of the torsion subgroup.11351136\subsection{Analytic rank of $L(J,s)$ and leading coefficient at $s=1$}1137\label{l}11381139Fix a Jacobian~$J$ corresponding to the 2-dimensional subspace of1140$S_2(N)$ spanned by quadratic conjugate, normalized newforms~$f$1141and~$\overline{f}$. Let $W_N$ be the Fricke involution. The newforms~$f$1142and~$\overline{f}$ have the same eigenvalue~$\epsilon_N$ with respect1143to~$W_N$, namely $+1$ or~$-1$. In the notation of Section~\ref{curves}, let1144\[ L(f,s) = \sum\limits_{n=1}^{\infty} \frac{a_n}{n^s} \]1145be the $L$-series of~$f$; then $L(\overline{f},s)$ is the Dirichlet1146series whose coefficients are the conjugates of the1147coefficients of~$L(f,s)$. (Recall that the~$a_n$ are integers in some1148real quadratic field.) The order of~$L(f,s)$ at~$s = 1$ is even1149when $\epsilon_N = -1$ and odd when $\epsilon_N = +1$. We have1150$L(J,s) = L(f,s) L(\overline{f},s)$. Thus the analytic rank of $J$ is~01151modulo~4 when $\epsilon_N = -1$ and 2 modulo~4 when $\epsilon_N = +1$.1152We found that the ranks were all 0 or~2. To prove that the analytic1153rank of~$J$ is~0, we need to show $L(f,1) \neq 0$ and1154$L(\overline{f},1) \neq 0$. In the case that $\epsilon_N = +1$, to1155prove that the analytic rank is~2, we need to show that $L'(f,1) \neq 0$1156and $L'(\overline{f},1) \neq 0$. When $\epsilon_N = -1$, we can1157evaluate $L(f,1)$ as in~\cite[\S~2.11]{Cr}. When $\epsilon_N = +1$, we1158can evaluate $L'(f,1)$ as in~\cite[\S~2.13]{Cr}. Each appropriate1159$L(f,1)$ or~$L'(f,1)$ was at least~$0.1$ and the errors in our1160approximations were all less than~$10^{-67}$. In this way we1161determined the analytic ranks, which we denote~$r$. As noted in the1162introduction, the analytic rank equals the Mordell-Weil rank if $r = 0$1163or~$r = 2$. Thus, we can simply call $r$ the rank, without fear of1164ambiguity.11651166To compute the leading coefficient of~$L(J,s)$ at~$s = 1$, we note that1167$\lim_{s \to 1} L(J,s)/(s-1)^r = L^{(r)}(J,1)/r!$.1168In the $r=0$ case, we simply have $L(J,1) = L(f,1)L(\overline{f},1)$.1169In the $r=2$ case, we have1170$L''(J,s)1171= L''(f,s)L(\overline{f},s) + 2L'(f,s)L'(\overline{f},s)1172+ L(f,s)L''(\overline{f},s)$.1173Evaluating both sides1174at $s=1$ we get $\frac{1}{2}L''(J,1) = L'(f,1)L'(\overline{f},1)$.11751176\subsection{Computing $k\cdot\Omega$}\label{modomega}1177Let $J$, $f$ and $\overline{f}$ be as in Section~\ref{l} and1178denote by $V$ the 2-dimensional space spanned by $f$ and1179$\overline{f}$.1180In computing $\Omega$ from an equation for the curve,1181we use a basis of integral1182differentials (see Section~\ref{Omega}) for $J$.1183For optimal quotients we can start with modular symbols and1184use a basis $\{\omega_1,\omega_2\}$1185for the subgroup of $V$ consisting of forms whose1186$q$-expansion coefficients lie in $\Z$, and we will1187obtain the quantity $k\cdot\Omega$. It can be shown1188that $k$ is a rational number. This rational number1189is called the {\em Manin constant}, and1190it is conjectured to equal~$1$.11911192We can compute $k\cdot\Omega$ using a generalization1193to dimension 2~of the algorithm for computing periods1194described in \cite[\S2.10]{Cr}. This is because1195$k\cdot\Omega$ is the volume of the real points1196of the quotient of $\C\times\C$ by the1197lattice of period integrals1198$(\int_\gamma \omega_1, \int_\gamma\omega_2)$1199with $\gamma$ in the integral homology1200$H_1(X_0(N),\Z)$.1201When $L(J,1)\neq 0$ the1202method of \cite[\S2.11]{Cr} coupled with1203Sections~\ref{l} and~\ref{bsdratio} can also be used1204to compute $k\cdot\Omega$.12051206\subsection{Computing $L(J,1)/(k\cdot\Omega)$}\label{bsdratio}1207We compute the rational number $L(J,1)/(k\cdot\Omega)$, for optimal1208quotients,1209using the algorithm in \cite{AS}.1210This algorithm generalizes the algorithm described in1211\cite[\S2.8]{Cr} to dimension greater than 1.12121213\subsection{Tamagawa numbers}1214In this section we assume that $p$ is a prime which1215exactly divides the conductor $N$ of $J$.1216Under these conditions, Grothendieck \cite{Gr} gave a1217description of the component group of $J$ in1218terms of a monodromy pairing on certain character groups.1219(For more details, see Ribet \cite[\S2]{Ri}.)1220If, in addition, $J$ is a new optimal quotient of $J_0(N)$, one1221deduces the following. When1222the eigenvalue for $f$ of the Atkin-Lehner involution $w_p$ is1223$+1$, then the rational component group of $J$ is a subgroup of1224$(\Z/2\Z)^2$. Furthermore, when the eigenvalue of $w_p$ is $-1$,1225the algorithm described in \cite{Ste} can be used to compute1226the value of~$c_p$.12271228\subsection{Torsion subgroup}1229\label{modtors}12301231To compute an integer divisible by the order of the1232torsion subgroup of $J$ we make use of the following two observations.1233First, it is a consequence of the Eichler-Shimura relation1234\cite[\S7.9]{Sh} that if $p$ is a prime not dividing the1235conductor $N$ of $J$ and $f(T)$ is the characteristic polynomial1236of the endomorphism $T_p$1237of $J$, then $\#J(\F_p) = f(p+1)$ (see \cite[\S2.4]{Cr}1238for an algorithm to compute $f(T)$).1239Second, if $p$ is an odd prime at which $J$ has good reduction,1240then the natural map $J(\Q)\tors\rightarrow J(\F_p)$ is injective1241(see \cite[p.\ 70]{CF}). This does not depend on whether $J$ is an1242optimal quotient.1243To obtain a lower bound on the torsion subgroup for optimal quotients,1244we use modular symbols and the Abel-Jacobi theorem \cite[IV.2]{La}1245to compute the order of the image of the rational point1246$(0)-(\infty)\in J_0(N)$.12471248\section{Tables}1249\label{tables}12501251In Table~\ref{table1}, we list the 32 curves described in1252Section~\ref{curves}. We give the level $N$ {}from which each curve1253arose, an integral model for the curve, and list the source(s) {}from1254which it came ($H$ for Hasegawa \cite{Ha}, $W$ for Wang \cite{Wan}).1255Throughout the paper, the curves are denoted $C_N$ (or $C_{N,A}$, $C_{N,B}$).12561257\begin{table}1258\begin{center}1259\begin{tabular}{|l|rcl|c|}1260\hline1261\multicolumn{1}{|c|}{$N$}1262& \multicolumn{3}{|c|}{Equation} & Source\\ \hline\hline126323 & $y^2 + (x^3 + x + 1)y$ & $=$ &1264$-2 x^5 - 3 x^2 + 2 x - 2$ & HW \\126529 & $y^2 + (x^3 + 1)y$ & $=$ &1266$-x^5 - 3 x^4 + 2 x^2 + 2 x - 2$ & HW \\126731 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &1268$-x^5 - 5 x^4 - 5 x^3 + 3 x^2 + 2 x - 3$ & HW \\126935 & $y^2 + (x^3 + x)y$ & $=$ &1270$-x^5 - 8 x^3 - 7 x^2 - 16 x - 19$ & H \\ \hline127139 & $y^2 + (x^3 + 1)y$ & $=$ &1272$-5 x^4 - 2 x^3 + 16 x^2 - 12 x + 2$ & H \\127363 & $y^2 + (x^3 - 1)y$ & $=$ &1274$14 x^3 - 7$ & W \\127565,A & $y^2 + (x^3 + 1)y$ & $=$ &1276$-4 x^6 + 9 x^4 + 7 x^3 + 18 x^2 - 10$ & W \\127765,B & $y^2$ & $=$ &1278$-x^6 + 10 x^5 - 32 x^4 + 20 x^3 + 40 x^2 + 6 x - 1$ & W \\ \hline127967 & $y^2 + (x^3 + x + 1)y$ & $=$ &1280$x^5 - x$ & HW \\128173 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &1282$-x^5 - 2 x^3 + x$ & HW \\128385 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &1284$x^4 + x^3 + 3 x^2 - 2 x + 1$ & H \\128587 & $y^2 + (x^3 + x + 1)y$ & $=$ &1286$-x^4 + x^3 - 3 x^2 + x - 1$ & HW \\ \hline128793 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &1288$-2 x^5 + x^4 + x^3$ & HW \\1289103 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &1290$x^5 + x^4$ & HW \\1291107 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &1292$x^4 - x^2 - x - 1$ & HW \\1293115 & $y^2 + (x^3 + x + 1)y$ & $=$ &1294$2 x^3 + x^2 + x$ & HW \\ \hline1295117,A & $y^2 + (x^3 - 1)y$ & $=$ &1296$3 x^3 - 7$ & W \\1297117,B & $y^2 + (x^3 + 1)y$ & $=$ &1298$-x^6 - 3 x^4 - 5 x^3 - 12 x^2 - 9 x - 7$ & W \\1299125,A & $y^2 + (x^3 + x + 1)y$ & $=$ &1300$x^5 + 2 x^4 + 2 x^3 + x^2 - x - 1$ & HW \\1301125,B & $y^2 + (x^3 + x + 1)y$ & $=$ &1302$x^6 + 5 x^5 + 12 x^4 + 12 x^3 + 6 x^2 - 3 x - 4$ & W \\ \hline1303133,A & $y^2 + (x^3 + x + 1)y$ & $=$ &1304$-2 x^6 + 7 x^5 - 2 x^4 - 19 x^3 + 2 x^2 + 18 x + 7$ & W \\1305133,B & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &1306$-x^5 + x^4 - 2 x^3 + 2 x^2 - 2 x$ & HW \\1307135 & $y^2 + (x^3 + x + 1)y$ & $=$ &1308$x^4 - 3 x^3 + 2 x^2 - 8 x - 3$ & W \\1309147 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &1310$x^5 + 2 x^4 + x^3 + x^2 + 1$ & HW \\ \hline1311161 & $y^2 + (x^3 + x + 1)y$ & $=$ &1312$x^3 + 4 x^2 + 4 x + 1$ & HW \\1313165 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &1314$x^5 + 2 x^4 + 3 x^3 + x^2 - 3 x$ & H \\1315167 & $y^2 + (x^3 + x + 1)y$ & $=$ &1316$-x^5 - x^3 - x^2 - 1$ & HW \\1317175 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &1318$-x^5 - x^4 - 2 x^3 - 4 x^2 - 2 x - 1$ & W \\ \hline1319177 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &1320$x^5 + x^4 + x^3$ & HW \\1321188 & $y^2$ & $=$ &1322$x^5 - x^4 + x^3 + x^2 - 2 x + 1$ & W \\1323189 & $y^2 + (x^3 - 1)y$ & $=$ &1324$x^3 - 7$ & W \\1325191 & $y^2 + (x^3 + x + 1)y$ & $=$ &1326$-x^3 + x^2 + x$ & HW \\ \hline1327\end{tabular}1328\end{center}1329\caption{Levels, integral models and sources for curves}1330\label{table1}1331\end{table}13321333In Table~\ref{table2}, we list the curve~$C_N$ simply by~$N$, the1334level {}from which it arose. Let $r$ denote the rank. We1335list ${\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)$ where $L(J,s)$ is the1336$L$-series for the Jacobian $J$ of~$C_N$ and round off the results to1337five digits. The symbol $\Omega$ was defined in Section~\ref{Omega}1338and is also rounded to five digits. Let Reg denote the regulator,1339also rounded to five digits. We list the $c_{p}$'s by primes of1340increasing order dividing the level~$N$. We denote $J(\Q)\tors = \Phi$1341and list its size. We use $\Sh ?$ to denote the size of1342$({\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)) \cdot1343(\#J(\Q)\tors)^2/(\Omega\cdot {\rm Reg} \cdot \prod c_{p})$,1344rounded to the nearest integer. We will refer to this as the {\em conjectured1345size of} $\Sh(J,\Q)$. The last column gives a bound on the accuracy of the1346computations; all values of $\Sh ?$ were at least this close to the1347nearest integer before rounding.13481349\newcommand{\mcc}[1]{\multicolumn{1}{|c|}{#1}}1350\newcommand{\mcd}[1]{\multicolumn{2}{|c|}{#1}}13511352\begin{table}1353\begin{center}1354\begin{tabular}{|l|c|r@{.}l|r@{.}l|l|l|c|c|l|}1355\hline1356\mcc{$N$} & $r$1357& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}1358& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & $\Phi$ & $\Sh$? & \mcc{error}1359\\ \hline\hline136023 & 0 & 0&24843 & 2&7328 & 1 & 11 & 11 & 1 & $ < 10^{-120} $ \\136129 & 0 & 0&29152 & 2&0407 & 1 & 7 & 7 & 1 & $ < 10^{-50} $ \\136231 & 0 & 0&44929 & 2&2464 & 1 & 5 & 5 & 1 & $ < 10^{-49} $ \\136335 & 0 & 0&37275 & 2&9820 & 1 & 16,2 & 16 & 1 & $ < 10^{-25} $ \\1364\hline136539 & 0 & 0&38204 & 10&697 & 1 & 28,1 & 28 & 1 & $ < 10^{-25} $ \\136663 & 0 & 0&75328 & 4&5197 & 1 & 2,3 & 6 & 1 & $ < 10^{-49} $ \\136765,A & 0 & 0&45207 & 6&3289 & 1 & 7,1 & 14 & 2 & $ < 10^{-48} $ \\136865,B & 0 & 0&91225 & 5&4735 & 1 & 1,3 & 6 & 2 & $ < 10^{-50} $ \\1369\hline137067 & 2 & 0&23410 & 20&465 & 0.011439 & 1 & 1 & 1 & $ < 10^{-50} $ \\137173 & 2 & 0&25812 & 24&093 & 0.010713 & 1 & 1 & 1 & $ < 10^{-49} $ \\137285 & 2 & 0&34334 & 9&1728 & 0.018715 & 4,2 & 2 & 1 & $ < 10^{-26} $ \\137387 & 0 & 1&4323 & 7&1617 & 1 & 5,1 & 5 & 1 & $ < 10^{-49} $ \\1374\hline137593 & 2 & 0&33996 & 18&142 & 0.0046847 & 4,1 & 1 & 1 & $ < 10^{-49} $ \\1376103 & 2 & 0&37585 & 16&855 & 0.022299 & 1 & 1 & 1 & $ < 10^{-49} $ \\1377107 & 2 & 0&53438 & 11&883 & 0.044970 & 1 & 1 & 1 & $ < 10^{-49} $ \\1378115 & 2 & 0&41693 & 10&678 & 0.0097618 & 4,1 & 1 & 1 & $ < 10^{-50} $ \\1379\hline1380117,A & 0 & 1&0985 & 3&2954 & 1 & 4,3 & 6 & 1 & $ < 10^{-49} $ \\1381117,B & 0 & 1&9510 & 1&9510 & 1 & 4,1 & 2 & 1 & $ < 10^{-49} $ \\1382125,A & 2 & 0&62996 & 13&026 & 0.048361 & 1 & 1 & 1 & $ < 10^{-50} $ \\1383125,B & 0 & 2&0842 & 2&6052 & 1 & 5 & 5 & 4 & $ < 10^{-49} $ \\1384\hline1385133,A & 0 & 2&2265 & 2&7832 & 1 & 5,1 & 5 & 4 & $ < 10^{-49} $ \\1386133,B & 2 & 0&43884 & 15&318 & 0.028648 & 1,1 & 1 & 1 & $ < 10^{-49} $ \\1387135 & 0 & 1&5110 & 4&5331 & 1 & 3,1 & 3 & 1 & $ < 10^{-49} $ \\1388147 & 2 & 0&61816 & 13&616 & 0.045400 & 2,2 & 2 & 1 & $ < 10^{-50} $ \\1389\hline1390161 & 2 & 0&82364 & 11&871 & 0.017345 & 4,1 & 1 & 1 & $ < 10^{-47} $ \\1391165 & 2 & 0&68650 & 9&5431 & 0.071936 & 4,2,2 & 4 & 1 & $ < 10^{-26} $ \\1392167 & 2 & 0&91530 & 7&3327 & 0.12482 & 1 & 1 & 1 & $ < 10^{-47} $ \\1393175 & 0 & 0&97209 & 4&8605 & 1 & 1,5 & 5 & 1 & $ < 10^{-44} $ \\1394\hline1395177 & 2 & 0&90451 & 13&742 & 0.065821 & 1,1 & 1 & 1 & $ < 10^{-45} $ \\1396188 & 2 & 1&1708 & 11&519 & 0.011293 & 9,1 & 1 & 1 & $ < 10^{-44} $ \\1397189 & 0 & 1&2982 & 3&8946 & 1 & 1,3 & 3 & 1 & $ < 10^{-43} $ \\1398191 & 2 & 0&95958 & 17&357 & 0.055286 & 1 & 1 & 1 & $ < 10^{-44} $ \\1399\hline1400\end{tabular}1401\end{center}1402\caption{Conjectured sizes of $\Sh (J,\Q)$}1403\label{table2}1404\end{table}14051406In Table~\ref{table3} are generators of $J(\Q)/J(\Q)\tors$ for the1407curves whose Jacobians have Mordell-Weil rank~2. The generators are1408given as divisor classes. Whenever possible, we have chosen1409generators of the form $[P - Q]$ where $P$ and~$Q$ are rational1410points on the curve. Curve~167 is the only example where this is not1411the case, since the degree zero divisors supported on the (known)1412rational points on~$C_{167}$ generate a subgroup of index two in the1413full Mordell-Weil group.1414Affine points are given by their $x$ and $y$ coordinates in the model1415given in Table~\ref{table1}. There are two points at infinity in the1416normalization of the curves described by our equations, with the1417exception of curve~$C_{188}$. These are denoted by $\infty_a$, where1418$a$ is the value of the function $y/x^3$ on the point in question.1419The (only) point at infinity on curve~$C_{188}$ is simply1420denoted~$\infty$.14211422\begin{table}1423\begin{center}1424\begin{tabular}{|l|l|l|}1425\hline1426\mcc{$N$} & \mcd{Generators of $J(\Q)/J(\Q)\tors$} \\ \hline\hline142767 & $ [(0, 0) - \infty_{-1}] $ &1428$ [(0, 0) - (0, -1)] $ \\142973 & $ [(0, -1) - \infty_{-1}] $ &1430$ [(0, 0) - \infty_{-1}] $ \\143185 & $ [(1, 1) - \infty_{-1}]$ &1432$ [(-1, 3) - \infty_{0}] $ \\143393 & $ [(-1, 1) - \infty_{0}] $ &1434$ [(1, -3) - (-1, -2)] $ \\ \hline1435103 & $ [(0, 0) - \infty_{-1}]$ &1436$ [(0, -1) - (0,0)] $ \\1437107 & $ [\infty_{-1} - \infty_{0}]$ &1438$ [(-1, -1) - \infty_{-1}] $ \\1439115 & $ [(1, -4) - \infty_{0}] $ &1440$ [(1, 1) - (-2, 2)] $ \\1441125,A & $ [\infty_{-1} - \infty_{0}] $ &1442$ [(-1, 0) - \infty_{-1}] $ \\ \hline1443133,B & $ [\infty_{-1} - \infty_{0}] $ &1444$ [(0, -1) - \infty_{-1}] $ \\1445147 & $ [\infty_{-1} - \infty_{0}] $ &1446$ [(-1, -1) - \infty_{0}] $ \\1447161 & $ [(1, 2) - (-1, 1)] $ &1448$ [(\frac{1}{2}, -3) - (1, 2)] $ \\1449165 & $ [(1, 1) - \infty_{-1}] $ &1450$ [(0, 0) - \infty_{0} ] $ \\ \hline1451167 & $ [(-1 ,1) - \infty_{0}] $ &1452$ [(i, 0) + (-i, 0) - \infty_{0} - \infty_{-1}] $ \\1453177 & $ [(0, -1) - \infty_{0}] $ &1454$ [(0, 0) - (0, -1)] $ \\1455188 & $ [(0, -1) - \infty] $ &1456$ [(0, 1) - (1, -2)] $ \\1457191 & $ [\infty_{-1} - \infty_{0}]$ &1458$ [(0, -1) - \infty_{0}] $ \\1459\hline1460\end{tabular}1461\end{center}1462\caption{Generators of $J(\Q)/J(\Q)\tors$ in rank 2 cases}1463\label{table3}1464\end{table}14651466In Table~\ref{table4} are the reduction types, {}from the1467classification of~\cite{NU}, of the special fibers of the minimal,1468proper, regular models of the curves for each of the primes of1469singular reduction for the curve. They are the same as the primes1470dividing the level except that curve~$C_{65,A}$ has singular1471reduction at the prime~3 and curve~$C_{65,B}$ has singular reduction1472at the prime~2.14731474\begin{table}1475\begin{center}1476\begin{tabular}{|l|l|l|l|l||l|l|l|l|l|}1477\hline1478\mcc{$N$} & Prime & Type & Prime & Type &1479\mcc{$N$} & Prime & Type & Prime & Type1480\\ \hline\hline148123 & 23 & ${\rm I}_{3-2-1}$ & & &1482117,A & 3 & ${\rm III}-{\rm III}^{\ast}-0$1483& 13 & ${\rm I}_{1-1-1}$ \\148429 & 29 & ${\rm I}_{3-1-1}$ & & &1485117,B & 3 & ${\rm I}_{3-1-1}^{\ast}$1486& 13 & ${\rm I}_{1-1-0}$ \\148731 & 31 & ${\rm I}_{2-1-1}$ & & &1488125,A & 5 & ${\rm VIII}-1$ & & \\148935 & 5 & ${\rm I}_{3-2-2}$1490& 7 & ${\rm I}_{2-1-0}$ &1491125,B & 5 & ${\rm IX}-3$ & & \\ \hline149239 & 3 & ${\rm I}_{6-2-2}$1493& 13 & ${\rm I}_{1-1-0}$ &1494133,A & 7 & ${\rm I}_{2-1-1}$1495& 19 & ${\rm I}_{1-1-0}$ \\149663 & 3 & $2{\rm I}_{0}^{\ast}-0$1497& 7 & ${\rm I}_{1-1-1}$ &1498133,B & 7 & ${\rm I}_{1-1-0}$1499& 19 & ${\rm I}_{1-1-0}$ \\150065,A & 3 & ${\rm I}_{0}-{\rm I}_{0}-1$1501& 5 & ${\rm I}_{3-1-1}$ &1502135 & 3 & III1503& 5 & ${\rm I}_{3-1-0}$ \\150465,A & 13 & ${\rm I}_{1-1-0}$ & & &1505147 & 3 & ${\rm I}_{2-1-0}$1506& 7 & VII \\ \hline150765,B & 2 & ${\rm I}_{0}-{\rm I}_{0}-1$1508& 5 & ${\rm I}_{3-1-0}$ &1509161 & 7 & ${\rm I}_{2-2-0}$1510& 23 & ${\rm I}_{1-1-0}$ \\151165,B & 13 & ${\rm I}_{1-1-1}$ & & &1512165 & 3 & ${\rm I}_{2-2-0}$1513& 5 & ${\rm I}_{2-1-0}$ \\151467 & 67 & ${\rm I}_{1-1-0}$ & & &1515165 & 11 & ${\rm I}_{2-1-0}$ & & \\151673 & 73 & ${\rm I}_{1-1-0}$ & & &1517167 & 167 & ${\rm I}_{1-1-0}$ & & \\ \hline151885 & 5 & ${\rm I}_{2-2-0}$1519& 17 & ${\rm I}_{2-1-0}$ &1520175 & 5 & ${\rm II}-{\rm II}-0$1521& 7 & ${\rm I}_{2-1-1}$ \\152287 & 3 & ${\rm I}_{2-1-1}$1523& 29 & ${\rm I}_{1-1-0}$ &1524177 & 3 & ${\rm I}_{1-1-0}$1525& 59 & ${\rm I}_{1-1-0}$ \\152693 & 3 & ${\rm I}_{2-2-0}$1527& 31 & ${\rm I}_{1-1-0}$ &1528188 & 2 & ${\rm IV}-{\rm IV}-0$1529& 47 & ${\rm I}_{1-1-0}$ \\1530103 & 103 & ${\rm I}_{1-1-0}$ & & &1531189 & 3 & ${\rm II}-{\rm IV}^{\ast}-0$1532& 7 & ${\rm I}_{1-1-1}$ \\ \hline1533107 & 107 & ${\rm I}_{1-1-0}$ & & &1534191 & 191 & ${\rm I}_{1-1-0}$ & & \\1535115 & 5 & ${\rm I}_{2-2-0}$1536& 23 & ${\rm I}_{1-1-0}$ & & & & & \\ \hline1537\end{tabular}1538\end{center}1539\caption{Namikawa and Ueno classification of special fibers}1540\label{table4}1541\end{table}154215431544\section{Discussion of Shafarevich-Tate groups and evidence for the1545second conjecture}1546\label{Shah}15471548{}From Section~\ref{MW} we have1549$\dim \Sh(J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2]$.1550With the exception of curves $C_{65,A}$, $C_{65,B}$, $C_{125,B}$, and1551$C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 0$. Thus we expect1552$\#\Sh(J,\Q)$ to be an odd square. In each case, the conjectured1553size of $\Sh(J,\Q)$ is~1. For curves $C_{65,A}$, $C_{65,B}$,1554$C_{125,B}$ and $C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 1, 1, 2$1555and~2 and the conjectured size of $\Sh(J,\Q) = 2, 2, 4$ and~4,1556respectively. We see that in each case, the (conjectured) size of1557the odd part of $\Sh(J,\Q)$ is~1 and the 2-part is accounted for by1558its 2-torsion.15591560For the optimal quotients, we computed the value of1561the rational number1562$L(J,1)/(k\cdot\Omega)$. Thus we can verify exactly that1563equation~\eqref{eqn1} holds if all of the following1564three conditions are met:1565a) the rank is 0, b) $\Sh(J,\Q) = \Sh(J,\Q)[2]$, and c) the1566Manin constant $k$ is 1 or bounded away from 1. The Manin constants1567are 1 to within the accuracy of our calculations1568(they are defined in Section~\ref{modomega}). Thus, if1569these can be proven to be 1 or bounded away from 1 by some1570amount greater than our degree of accuracy ($10^{-14}$), then1571we have a proof that they are exactly 1.15721573It is also interesting to consider deficient primes. A prime $p$ is1574deficient with respect to a curve $C$ of genus~2, if $C$ has no1575degree 1 rational divisor over~$\Q_{p}$. {}From~\cite{PSt}, the1576number of deficient primes has the same parity as $\dim \Sh(J,\Q)[2]$.1577Curve $C_{65,A}$ has one deficient prime~$3$. Curve1578$C_{65,B}$ has one deficient prime~$2$. Curve $C_{117,B}$ has two1579deficient primes $3$ and~$\infty$. The rest of the curves have no1580deficient primes.15811582Since we have found $r$ (analytic rank) independent points on each1583Jacobian, we have a direct proof that the Mordell-Weil rank must1584equal the analytic rank if $\dim \Sh(J,\Q)[2] = 0$. For1585curves $C_{65,A}$ and $C_{65,B}$, the presence of an odd number of1586deficient primes gives us a1587similar result. For $C_{125,B}$ we used a $\sqrt{5}$-Selmer group1588to get a similar result.1589Thus, we have an independent proof of equality1590between analytic and Mordell-Weil ranks for all curves except1591$C_{133,A}$.15921593The 2-Selmer groups have the same dimensions for the pairs1594$C_{125,A}$, $C_{125,B}$ and $C_{133,A}$, $C_{133,B}$. For each1595pair, the Mordell-Weil rank is~2 for one curve and the 2-torsion of1596the Shafarevich-Tate group has dimension~2 for the other. In1597addition, the two Jacobians, when canonically embedded into~$J_0(N)$,1598intersect in their 2-torsion subgroups, and one can check that their15992-Selmer groups become equal under the identification of1600$H^1(\Q, J_{N,A}[2])$ with $H^1(\Q, J_{N,B}[2])$ induced by the identification1601of the 2-torsion subgroups. Thus these are examples of the principle1602of a `visible part of a Shafarevich-Tate group' as discussed1603in~\cite{CM}.16041605\vspace{5mm}1606\begin{center}1607{\sc Appendix: Other Hasegawa curves}1608\end{center}16091610In Table~\ref{Hasegawa} is data concerning all 142 of Hasegawa's1611curves in the order presented in his paper. Let us explain the1612entries. The first column in each set of three columns gives the1613level, $N$. The second column gives a classification of the cusp1614forms spanning the 2-dimensional subspace of $S_2(N)$ corresponding1615to the Jacobian. When that subspace is irreducible with respect to1616the action of the Hecke algebra and is spanned by two newforms or two1617oldforms, we write $2n$ or $2o$, respectively. When that subspace is1618reducible and is spanned by two oldforms, two newforms or one of1619each, we write $oo$, $nn$ and $on$, respectively. The third column1620contains the sign of the functional equation at the level $M$ at1621which the cusp form is a newform. This is the negative of1622$\epsilon_M$ (described in Section~\ref{l}). The order of the two1623signs in the third column agrees with that of the forms listed in the1624second column. We include this information for those who would like1625to further study these curves. The curves with $N<200$ classified as1626$2n$ appeared already in Table~\ref{table1}.16271628The smallest possible Mordell-Weil ranks corresponding to $++$, $+-$,1629$-+$ and $--$, predicted by the first Birch and Swinnerton-Dyer1630conjecture, are $0$, $1$, $1$ and $2$ respectively. In all cases,1631those were, in fact, the Mordell-Weil ranks. This was determined by1632computing 2-Selmer groups with a computer program based on1633\cite{Sto2}. Of course, these are cases where the first Birch and1634Swinnerton-Dyer conjecture is already known to hold. In the cases1635where the Mordell-Weil rank is positive, the Mordell-Weil group has a1636subgroup of finite index generated by degree zero divisors supported1637on rational points with $x$-coordinates with numerators bounded by 71638(in absolute value) and denominators by 12 with one exception. On1639the second curve with $N=138$, the divisor class1640$[(3+2\sqrt{2},80+56\sqrt{2}) + (3-2\sqrt{2},80-56\sqrt{2})-2\infty]$1641generates a subgroup of finite index in the Mordell-Weil group.16421643\vfill16441645\begin{table}1646\begin{center}1647\begin{tabular}{|c|c|c||c|c|c||c|c|c||c|c|c||c|c|c|}1648\hline164922 & $oo$ & $++$ & 58 & $nn$ & $+-$ & 87 & $2o$ & $++$ & 129 & $on$ & $--$ &1650198 & $2o$ & $+-$ \\165123 & $2n$ & $++$ & 60 & $oo$ & $++$ & 88 & $on$ & $+-$ & 130 & $on$ & $-+$ &1652204 & $2o$ & $+-$ \\165326 & $nn$ & $++$ & 60 & $2o$ & $++$ & 90 & $on$ & $++$ & 132 & $oo$ & $++$ &1654205 & $2n$ & $--$ \\165528 & $oo$ & $++$ & 60 & $2o$ & $++$ & 90 & $oo$ & $++$ & 133 & $2n$ & $--$ &1656206 & $2o$ & $--$ \\165729 & $2n$ & $++$ & 62 & $2o$ & $++$ & 90 & $oo$ & $++$ & 134 & $2o$ & $--$ &1658209 & $2n$ & $--$ \\165930 & $on$ & $++$ & 66 & $nn$ & $++$ & 90 & $oo$ & $++$ & 135 & $on$ & $+-$ &1660210 & $on$ & $+-$ \\166130 & $oo$ & $++$ & 66 & $2o$ & $++$ & 91 & $nn$ & $--$ & 138 & $nn$ & $+-$ &1662213 & $2n$ & $--$ \\166330 & $on$ & $++$ & 66 & $2o$ & $++$ & 93 & $2n$ & $--$ & 138 & $on$ & $+-$ &1664215 & $on$ & $--$ \\166531 & $2n$ & $++$ & 66 & $on$ & $++$ & 98 & $oo$ & $++$ & 140 & $oo$ & $++$ &1666221 & $2n$ & $--$ \\166733 & $on$ & $++$ & 67 & $2n$ & $--$ & 100 & $oo$ & $++$ & 142 & $nn$ & $+-$1668& 230 & $2o$ & $--$ \\ \hline166935 & $2n$ & $++$ & 68 & $oo$ & $++$ & 102 & $on$ & $+-$ & 143 & $on$ & $+-$1670& 255 & $2o$ & $--$ \\167137 & $nn$ & $+-$ & 69 & $2o$ & $++$ & 102 & $on$ & $+-$ & 146 & $2o$ & $--$1672& 266 & $2o$ & $--$ \\167338 & $on$ & $++$ & 70 & $on$ & $++$ & 103 & $2n$ & $--$ & 147 & $2n$ & $--$1674& 276 & $2o$ & $+-$ \\167539 & $2n$ & $++$ & 70 & $2o$ & $++$ & 104 & $2o$ & $++$ & 150 & $on$ & $++$1676& 284 & $2o$ & $+-$ \\167740 & $on$ & $++$ & 70 & $2o$ & $++$ & 106 & $on$ & $--$ & 153 & $on$ & $+-$1678& 285 & $on$ & $--$ \\167940 & $oo$ & $++$ & 70 & $2o$ & $++$ & 107 & $2n$ & $--$ & 154 & $on$ & $--$1680& 286 & $on$ & $--$ \\168142 & $on$ & $++$ & 72 & $on$ & $++$ & 110 & $on$ & $++$ & 156 & $oo$ & $++$1682& 287 & $2n$ & $--$ \\168342 & $oo$ & $++$ & 72 & $oo$ & $++$ & 111 & $oo$ & $+-$ & 158 & $on$ & $--$1684& 299 & $2n$ & $--$ \\168542 & $on$ & $++$ & 73 & $2n$ & $--$ & 112 & $on$ & $+-$ & 161 & $2n$ & $--$1686& 330 & $2o$ & $--$ \\168742 & $oo$ & $++$ & 74 & $oo$ & $+-$ & 114 & $oo$ & $+-$ & 165 & $2n$ & $--$1688& 357 & $2n$ & $--$ \\ \hline168944 & $2o$ & $++$ & 77 & $on$ & $+-$ & 115 & $2n$ & $--$ & 166 & $on$ & $--$1690& 380 & $2o$ & $+-$ \\169146 & $2o$ & $++$ & 78 & $oo$ & $++$ & 116 & $2o$ & $+-$ & 167 & $2n$ & $--$1692& 390 & $on$ & $--$ \\169348 & $on$ & $++$ & 78 & $2o$ & $++$ & 117 & $2o$ & $++$ & 168 & $2o$ & $++$1694& & & \\169548 & $oo$ & $++$ & 80 & $oo$ & $++$ & 120 & $oo$ & $++$ & 170 & $2o$ & $--$1696& & & \\169750 & $nn$ & $++$ & 84 & $oo$ & $++$ & 120 & $on$ & $++$ & 177 & $2n$ & $--$1698& & & \\169952 & $oo$ & $++$ & 84 & $oo$ & $++$ & 121 & $on$ & $+-$ & 180 & $2o$ & $++$1700& & & \\170152 & $oo$ & $++$ & 84 & $oo$ & $++$ & 122 & $on$ & $--$ & 184 & $on$ & $+-$1702& & & \\170354 & $on$ & $++$ & 84 & $oo$ & $++$ & 125 & $2n$ & $--$ & 186 & $2o$ & $--$1704& & & \\170557 & $on$ & $+-$ & 85 & $2n$ & $--$ & 126 & $oo$ & $++$ & 190 & $on$ & $+-$1706& & & \\170757 & $on$ & $+-$ & 87 & $2n$ & $++$ & 126 & $on$ & $++$ & 191 & $2n$ & $--$1708& & & \\1709\hline1710\end{tabular}1711\end{center}1712\caption{Spaces of cusp forms associated to Hasegawa's curves}1713\label{Hasegawa}1714\end{table}17151716\pagebreak1717\begin{thebibliography}{99}17181719\bibitem[AS]{AS}1720A.\ Agash\'{e} and W.A.\ Stein: Some abelian varieties with visible1721Shafarevich-Tate groups. 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