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9%Hi all, below is what I sent to Mathematics of Computation,
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12% \pagebreak
13
14%\section{Introduction}
15%\label{intro}
16
17% \normalsize
18% \baselineskip=18pt
19
20%Cheers, Ed
21
22
23
24\documentclass[12pt]{amsart}
25\usepackage{amscd}
26\newfont{\cyr}{wncyr10 scaled \magstep1}
27\newcommand{\Sh}{\hbox{\cyr Sh}}
28\newcommand{\C}{{\mathbf C}}
29\newcommand{\Q}{{\mathbf Q}}
30\newcommand{\Qbar}{\overline{\Q}}
31%\newcommand{\GalQ}{{\Gal}(\Qbar/\Q)}
32\newcommand{\CC}{{\mathcal C}}
33\newcommand{\Z}{{\mathbf Z}}
34\newcommand{\R}{{\mathbf R}}
35\newcommand{\F}{{\mathbf F}}
36\newcommand{\G}{{\mathbf G}}
37\newcommand{\OO}{{\mathcal O}}
38\newcommand{\JJ}{{\mathcal J}}
39\newcommand{\DD}{{\mathcal D}}
40\newcommand{\aaa}{{\mathfrak a}}
41\newcommand{\PP}{{\mathbf P}}
42\newcommand{\tors}{_{\text{tors}}}
43\newcommand{\unr}{^{\text{unr}}}
44\newcommand{\nichts}{{\left.\right.}}
45
46
47\DeclareMathOperator{\Gal}{Gal}
48\DeclareMathOperator{\Norm}{Norm}
49\DeclareMathOperator{\Sel}{Sel}
50\DeclareMathOperator{\Tr}{Tr}
51
52\newtheorem{theorem}{Theorem}[section]
53\newtheorem{lemma}[theorem]{Lemma}
54\newtheorem{cor}[theorem]{Corollary}
55\newtheorem{prop}[theorem]{Proposition}
56
57\theoremstyle{definition}
58\newtheorem{question}{Question}
59\newtheorem{conj}{Conjecture}
60
61\theoremstyle{remark}
62\newtheorem{rem}{Remark$\!\!$}          \renewcommand{\therem}{}
63\newtheorem{rems}{Remarks$\!\!$}        \renewcommand{\therems}{}
64
65\topmargin -0.3in
67\oddsidemargin 0in
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69\textwidth 6.5in
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72%%%\renewcommand{\baselinestretch}{2}
73
74\begin{document}
75
76\title[Modular Jacobians]{Empirical evidence for the Birch and
77Swinnerton-Dyer conjectures for
78modular Jacobians of genus~2 curves}
79
80\author{E.\ Victor Flynn}
81\address{Department of Mathematical Sciences, University of
82Liverpool, P.O.Box 147,
83Liverpool L69 3BX, England}
84\email{evflynn@liverpool.ac.uk}
85
86\author{Franck Lepr\'{e}vost}
87\address{CNRS Equipe d'arithm\'etique, Institut de Math\'ematiques de Paris,
88Universit\'e Paris 6,
89Tour 46-56, 5\eme \'etage, Case 247,
902-4 place Jussieu, F-75252 Paris cedex 05, France}
91\email{leprevot@math.jussieu.fr}
92
93\author{Edward F.\ Schaefer}
94\address{Department of Mathematics and Computer Science \\
95Santa Clara University \\ Santa Clara, CA 95053, USA}
96\email{eschaefe@math.scu.edu}
97
98\author{William A.\ Stein}
99\address{Department of Mathematics \\ University of California
100at Berkeley \\ Berkeley, CA  94720, USA}
101\email{was@math.berkeley.edu}
102
103\author{Michael Stoll}
105D\"{u}sseldorf, Germany}
106\email{stoll@math.uni-duesseldorf.de}
107
108\author{Joseph L.\ Wetherell}
109\address{Department of Mathematics, University of Southern California,
1101042 W.\ 36th Place, Los Angeles, CA  90089-1113, USA}
111\email{jlwether@alum.mit.edu}
112
113\subjclass{Primary 11G40; Secondary 11G10, 11G30, 14H25, 14H40,14H45}
114\keywords{Birch and Swinnerton-Dyer conjecture, genus~2, Jacobian, modular
115abelian variety}
116
117\thanks{The first author thanks the Nuffield Foundation
118(Grant SCI/180/96/71/G) for financial support.
119The second author did some of the research at
120the Max-Planck Institut f\"ur Mathematik and
121the Technische Universit\"at Berlin.
122The third author thanks the National Security Agency (Grant
123MDA904-99-1-0013).
124The fourth author was supported by a Sarah M. Hallam fellowship.
125The fifth author did some of the research at
126the Max-Planck-Institut f\"ur Mathematik.
127The sixth author thanks the National Science Foundation
128(Grant DMS-9705959).
129The authors had useful conversations with John Cremona, Qing Liu,
130Karl Rubin and
131Peter Swinnerton-Dyer and are grateful to Xiangdong Wang and Michael
132M\"{u}ller for making data available to them.}
133
134\date{August 11, 1999}
135
136\begin{abstract}
137This paper provides empirical evidence for the Birch and
138Swinnerton-Dyer conjectures for modular Jacobians of genus~2 curves.
139The second of these conjectures relates six quantities associated to
140a Jacobian over the rational numbers.  One of these
141six quantities is
142the size of the Shafarevich-Tate group.
143Unable to compute that, we
144computed the five other quantities and solved for the last one.  In
145all 32~cases, the result is very close to an integer that is a power
146of~2.  In addition, this power of~2 agrees with the size of the
1472-torsion of the Shafarevich-Tate group, which we could compute.
148\end{abstract}
149
150\maketitle
151\markboth{FLYNN, LEPR\'{E}VOST, SCHAEFER, STEIN, STOLL, AND WETHERELL}%
152         {GENUS~2 BIRCH AND SWINNERTON-DYER CONJECTURE}
153
154% \pagebreak
155
156
157\section{Introduction}
158\label{intro}
159
160% \normalsize
161% \baselineskip=18pt
162
163The conjectures of Birch and Swinnerton-Dyer, originally stated
164for elliptic curves over~$\Q$, have been a constant source of
165motivation for the study of elliptic curves, with the ultimate
166goal being to find a proof.
167This has resulted not only in a better
168theoretical understanding, but also in the development of better
169algorithms for computing the analytic and arithmetic
170invariants that are so intriguingly related by them. We now know
171that the first and, up to a non-zero rational factor, the
172second conjecture hold for modular elliptic curves over~$\Q$
173\footnote{It has recently been announced by
174Brueil, Conrad, Diamond and Taylor that they have extended Wiles'
175results and shown
176that all elliptic curves over~$\Q$ are modular.}
177in the
178analytic rank~0 and~1 cases (see \cite{GZ,Ko,Wal1,Wal2}).
179Furthermore,
180a number of people have provided numerical evidence for the
181conjectures for a large number of elliptic curves; see
182for example~\cite{BSD,Ca,Cr}.
183
184By now, our theoretical and algorithmic knowledge of curves of
185genus~2 and their Jacobians has reached a state that makes it
186possible to conduct similar investigations. The Birch and
187Swinnerton-Dyer conjectures have been generalized to arbitrary
188abelian varieties over number fields by Tate~\cite{Ta}. If
189$J$ is the Jacobian of a genus~2 curve over $\Q$,
190then the first conjecture
191states that the order of vanishing of the $L$-series of the Jacobian at
192$s=1$ (the {\em analytic rank}) is equal to the Mordell-Weil rank of the
193Jacobian. The second conjecture is that
194\begin{equation} \label{eqn1}
195  \lim\limits_{s \to 1} (s-1)^{-r} L(J,s) =
196     \Omega \cdot {\rm Reg} \cdot \prod\limits_{p} c_{p}
197        \cdot \#\Sh(J,\Q ) \cdot (\#J(\Q)\tors)^{-2} \,.
198\end{equation}
199In this equation, $L(J,s)$ is the $L$-series of the Jacobian
200$J$, and $r$ is its analytic rank.  We use $\Omega$ to denote the
201integral over $J(\R)$ of a particular differential 2-form; the
202precise choice of this differential is described in
203Section~\ref{Omega}.  ${\rm Reg}$ is the regulator of $J(\Q)$.  For
204primes $p$, we use $c_{p}$ to denote the size of $J(\Q_p)/J^0(\Q_p)$,
205where $J^0(\Q_p)$ is defined in Section~\ref{Tamagawa}.  We let
206$\Sh(J,\Q)$ be the Shafarevich-Tate group of $J$ over $\Q$, and we let
207$J(\Q)\tors$ denote the torsion subgroup of $J(\Q)$.
208
209As in the case of elliptic curves, the first conjecture assumes
210that the $L$-series can be analytically continued to $s = 1$,
211and the second conjecture additionally assumes that the
212Shafarevich-Tate group is finite. Neither of these assumptions is
213known to hold for arbitrary genus~2 curves. The analytic
214continuation of the $L$-series, however, is known to exist for
215modular abelian varieties over~$\Q$, where an abelian
216variety is called {\em modular} if it is a quotient of the Jacobian~$J_0(N)$
217of the modular curve~$X_0(N)$ for some level~$N$. For simplicity,
218we will also call a genus~2 curve {\em modular} when its Jacobian is
219modular in this sense. So it is certainly a good idea to look
220at modular genus~2 curves over~$\Q$, since we then at least know that the
221statement of the first conjecture makes sense. Moreover, for many modular
222abelian varieties it is also known that the Shafarevich-Tate group
223is finite, therefore the statement of the second conjecture also
224makes sense. As it turns out, all of our examples belong to this
225class. An additional benefit of choosing modular genus~2 curves is
226that one can find lists of such curves in the literature.
227
229Swinnerton-Dyer conjectures for such modular genus~2 curves. Since there
230is no known effective way of computing the size of the Shafarevich-Tate
231group, we computed the other five terms in equation~\eqref{eqn1}
232(in two different ways, if possible). This required several different
233algorithms, some of which were developed or improved while we were
234working on this paper. If one of these algorithms
235is already well described in the literature, then we simply cite it.
236Otherwise, we describe it here in some detail (in particular,
237algorithms for computing $\Omega$ and
238$c_p$).
239
240For modular abelian varieties associated to newforms whose
241$L$-series have analytic rank~0 or~1, the first Birch and Swinnerton-Dyer
242conjecture has been proven. In such cases, the
243Shafarevich-Tate group is also known to be finite and the second conjecture
244has been proven, up to a non-zero rational factor. This all
245follows {}from results in
246\cite{GZ,KL,Wal1,Wal2}.
247In our examples, all of the analytic
248ranks are either~0 or~1.  Thus we already know that the first
249conjecture holds.  Since the Jacobians we consider are associated to a
250quadratic conjugate pair of newforms, the analytic rank of the
251Jacobian is twice the analytic rank of either newform (see \cite{GZ}).
252
253The second Birch and Swinnerton-Dyer conjecture has not been proven
254for the cases we consider.  In order to verify equation~\eqref{eqn1},
255we computed the five terms other than $\#\Sh(J,\Q)$ and solved for
256$\#\Sh(J,\Q)$. In each case, the value is an integer to within the
257accuracy of our calculations.  This number is a power of~2, which
258coincides with the independently computed size of the 2-torsion
259subgroup of~$\Sh(J,\Q)$. Hence, we have verified the second
260Birch and Swinnerton-Dyer conjecture for our curves at least
261numerically, if we assume that the Shafarevich-Tate group consists
262of 2-torsion only. (This is an ad hoc assumption based only
263on the fact that we do not know better.) See Section~\ref{Shah} for
264circumstances under which the verification is exact.
265
266The curves are listed in Table~\ref{table1},
267and the numerical results can be found in Table~\ref{table2}.
268
269
270\section{The Curves}
271\label{curves}
272
273Each of the genus~2 curves we consider is related to the Jacobian
274$J_0(N)$ of the modular curve $X_0(N)$ for some level $N$.  When only
275one of these genus~2 curves arises {}from a given level $N$, then we
276denote this curve by $C_N$; when there are two curves coming {}from level
277$N$ we use the notation $C_{N,A}$, $C_{N,B}$.  The relationship
278of, say, $C_N$ to $J_0(N)$ depends on the source.  Briefly, {}from
279Hasegawa \cite{Hs} we obtain quotients of $X_0(N)$ and {}from Wang
280\cite{Wan} we obtain curves whose Jacobians are quotients of $J_0(N)$.
281In both cases the Jacobian $J_N$ of $C_N$ is isogenous to a
2822-dimensional factor of $J_0(N)$. (When not referring to a specific
283curve, we will typically drop the subscript $N$ {}from $J$.)
284In this way we can also associate
285$C_N$ with a 2-dimensional subspace of $S_2(N)$, the space of cusp
286forms of weight~2 for $\Gamma_0(N)$.
287
288We now discuss the precise source of the genus~2 curves we will
289consider.  Hasegawa \cite{Hs} has provided exact equations for all
290genus~2 curves which are quotients of $X_0(N)$ by a subgroup of the
291Atkin-Lehner involutions.  There are 142 such curves.  We are
292particularly interested in those where the Jacobian corresponds to a
293subspace of $S_2(N)$ spanned by a quadratic conjugate pair of
294newforms. There are 21 of these with level $N \leq 200$.  For these
295curves we will provide evidence for the second conjecture.  There are
296seven more such curves with $N > 200$.  We can classify the other
2972-dimensional subspaces into four types.  There are
2982-dimensional subspaces of oldforms that are irreducible under the
299action of the Hecke algebra.  There are also 2-dimensional subspaces
300that are reducible under the action of the Hecke algebra and are
301spanned by two oldforms, two newforms or one of each. The Jacobians
302corresponding to the latter three kinds are always isogenous, over
303$\Q$, to the product of two elliptic curves. Given the small levels,
304these are elliptic curves for which Cremona \cite{Cr} has already
305provided evidence for the Birch and Swinnerton-Dyer conjectures.  In
306Table~\ref{Hasegawa}, we describe the kind of cusp forms spanning the
3072-dimensional subspace and the signs of their functional equations
308{}from the level at which they are newforms.  The analytic and
309Mordell-Weil ranks were always the smallest possible given those signs.
310
311The second set of curves was created by Wang \cite{Wan} and is further
312discussed in \cite{FM}.  This set consists of 28 curves that were
313constructed by considering the spaces $S_2(N)$ with $N \leq 200$.
314Whenever a subspace spanned by a pair of quadratic conjugate newforms
315was found, these newforms were integrated to produce a quotient
316abelian variety~$A$ of $J_0(N)$.  These quotients are {\em optimal} in the
317sense of \cite{Ma}, in that the kernel of the quotient map is
318connected.
319
320The period matrix for~$A$ was created using certain intersection
321numbers.  When all of the intersection numbers have the same value,
322then the polarization on~$A$ induced {}from the canonical polarization
323of~$J_0(N)$ is equivalent to a principal polarization. (Two
324polarizations are {\em equivalent} if they differ by an integer multiple.)
325Conversely, every 2-dimensional optimal quotient of $J_0(N)$ in which
326the induced polarization is equivalent to a principal polarization is
327found in this way.
328
329Using theta functions, numerical approximations were found for the
330Igusa invariants of the abelian surfaces. These numbers coincide with
331rational numbers of fairly small height within the limits of the
332precision used for the computations. Wang then constructed curves
333defined over~$\Q$ whose Igusa invariants are the rational numbers
334found. (There is one abelian surface at level $N = 177$ for which Wang
335was not able to find a curve.) If we assume that these rational
336numbers are the true Igusa invariants of the abelian surfaces, then it
337follows that Wang's curves have Jacobians isomorphic, over~$\Qbar$, to
338the principally polarized abelian surfaces in his list. Since the
339classification given by these invariants is only up to isomorphism
340over~$\Qbar$, the Jacobians of Wang's curves are not necessarily
341isomorphic to, but can be twists of, the optimal quotients
342of~$J_0(N)$ over~$\Q$ (see below).
343
344There are four curves in Hasegawa's list which do not show up in
345Wang's list (they are listed in Table~\ref{table1} with an $H$ in the
346last column).  Their Jacobians are quotients of~$J_0(N)$, but are not
347optimal quotients.  It is likely that there are modular genus~2 curves
348which neither are Atkin-Lehner quotients of~$X_0(N)$ (in Hasegawa's
349sense) nor have Jacobians that are optimal quotients. These curves
350could be found by looking at the optimal quotient abelian surfaces and
351checking whether they are isogenous to a principally polarized abelian
352surface over $\Q$.
353
354For 17 of the curves in Wang's list, the 2-dimensional subspace
355spanned by the newforms is the same as that giving one of Hasegawa's
356curves.  In all of those cases, the curve given by Wang's equation is
357isomorphic, over $\Q$, to that given by Hasegawa. This verifies Wang's
358equations for these 17 curves.  They are listed in Table~\ref{table1}
359with $HW$ in the last column.
360
361The remaining eleven curves (listed in Table~\ref{table1} with a
362$W$ in the last column) derive from the other eleven optimal
363quotients in Wang's list.  These are described in more detail in
365
366With the exception of curves $C_{63}$, $C_{117,A}$ and $C_{189}$, the
367Jacobians of all of our curves are absolutely simple, and the
368canonically polarized Jacobians have automorphism groups of size two.
369We showed that these Jacobians are absolutely simple using an argument
370like those in \cite{Le,Sto1}.  The automorphism group of the
371canonically polarized Jacobian of a hyperelliptic curve is isomorphic
372to the automorphism group of the curve (see \cite[Thm.\
37312.1]{Mi2}). Each automorphism of a hyperelliptic curve is induced by
374a linear fractional transformation on $x$-coordinates (see \cite[p.\
3751]{CF}). Each automorphism also permutes the six Weierstrass
376points. Once we believed we had found all of the automorphisms, we
377were able to show that there are no more by considering all linear
378fractional transformations sending three fixed Weierstrass points to
379any three Weierstrass points. In each case, we worked with sufficient
380accuracy to show that other linear fractional transformations did not
381permute the Weierstrass points.
382
383Let $\zeta_{3}$ denote a primitive third root of unity.  The
384Jacobians of curves $C_{63}$, $C_{117,A}$ and $C_{189}$ are each
385isogenous to the product of two elliptic curves over $\Q(\zeta_3)$,
386though not over $\Q$, where they are simple.  These genus~2 curves
387have automorphism groups of size 12.  In the following table we list
388the curve at the left.  On the right we give one of the elliptic
389curves which is a factor of its Jacobian. The second factor is the
390conjugate.
391$392\begin{array}{ll} 393C_{63}: & y^2 = x(x^2 + (9 - 12\zeta_{3})x - 48\zeta_{3}) \\ 394C_{117,A}: & y^2 = x(x^2 - (12 + 27\zeta_{3})x - (48 + 48\zeta_{3})) \\ 395C_{189}: & y^2 = x^3 + (66 - 3\zeta_{3})x^2 + (342 + 81\zeta_{3})x 396 + 105 + 21\zeta_{3} 397\end{array} 398$
399Note that these three Jacobians are examples of abelian varieties
400with extra twist' as discussed in~\cite{Cr2}, where they can be
401found in the tables on page~409.
402
403\subsection{Models for the Wang-only curves}
405
406As we have already noted, a modular genus~2 curve may be found by
407either, both, or neither of Wang's and \linebreak
408Hasegawa's techniques.
409Hasegawa's method allows for the exact determination, over $\Q$, of
410the equation of any modular genus~2 curve it has found.  On the other
411hand, if Wang's technique detects a modular genus~2 curve $C_N$, his
412method produces real approximations to a curve $C'_N$ which is defined
413over $\Q$ and is isomorphic to $C_N$ over $\Qbar$.  We will call
414$C'_N$ a {\em twisted modular genus~2 curve}.
415
416In this section we attempt to determine equations for the eleven
417modular genus~2 curves detected by Wang but not by Hasegawa.  If we
418assume that Wang's equations for the twisted modular genus~2 curves
419are correct, we find that we are able to determine the twists.  In
420turn, this gives us strong evidence that Wang's equations for the
421twisted curves were correct.  Undoing the twist, we determine probable
422equations for the modular genus~2 curves. We end by providing further
423evidence for the correctness of these equations.
424
425In what follows, we will use the notation of~\cite{Cr} and recommend
426it as a reference on the general results that we assume here and in
427Section~\ref{modular} and the appendix.
428Fix a level~$N$ and let
429$f(z) \in S_2(N)$.  Then $f$ has a Fourier expansion
430$f(z) = \sum\limits_{n=1}^{\infty} a_{n} e^{2 \pi i n z}\,.$
431For a newform~$f$, we have $a_1 \neq 0$; so we can normalize it by
432setting $a_1 = 1$. In our cases, the $a_n$'s are integers in a real
433quadratic field. For each prime~$p$ not dividing~$N$, the
434corresponding Euler factor of the $L$-series $L(f,s)$ is
435$1 - a_p p^{-s} + p^{1-2s}$.  Let $N(a_p)$ and $Tr(a_p)$ denote the
436norm and trace of~$a_p$.  The product of this Euler factor and its
437conjugate is
438$1 - Tr(a_p)\,p^{-s} + (N(a_p) + 2p)\,p^{-2s} 439 - p\,Tr(a_p)\,p^{-3s} + p^2\,p^{-4s}$.
440Therefore, the characteristic
441polynomial of the $p$-Frobenius on the corresponding abelian variety
442over $\F_{p}$ is
443$x^4 - Tr(a_p)\,x^3 + (N(a_p) + 2p)\,x^2 - p\,Tr(a_p)\,x + p^2$.
444Let $C$ be a curve, over $\Q$, whose Jacobian, over $\Q$, comes {}from
445the space spanned by $f$ and its conjugate.  Then we know that
446$p+1 - \#C(\F_{p}) = Tr(a_p)$ and
447$\frac{1}{2}(\#C(\F_{p})^{2} + \#C(\F_{p^2})) - (p+1)\# C(\F_{p}) - p = 448N(a_p)$ (see \cite[Lemma 3]{MS}).
449For the odd primes less than 200, not dividing $N$, we computed
450$\# C(\F_{p})$ and $\# C(\F_{p^2})$ for each curve given by one of
451Wang's equations. {}From these we could compute the characteristic
452polynomials of Frobenius and see if they agreed with those predicted
453by the $a_p$'s of the newforms.
454
455Of the eleven curves, the characteristic polynomials agreed for only
456four. In each of the remaining seven cases we found a twist of Wang's
457curve whose characteristic polynomials agreed with those predicted by
458the newform for all odd primes less than 200 not dividing $N$.  Four
459of these twists were quadratic and three were of higher degree.  It
460is these twists that appear in Table~\ref{table1}.
461
462We can provide further evidence that these equations are correct.
463For each curve given in Table~\ref{table1}, it is easy to determine
464the primes of singular reduction.  In Section~\ref{Tamagawa} we will
465provide techniques for determining which of those primes divides the
466conductor of its Jacobian.  In each case, the primes dividing the
467conductor of the Jacobian of the curve are exactly the primes
468dividing the level $N$; this is necessary.  With the exception of
469curve $C_{188}$, all the curves come {}from odd levels.  We used Liu's
470{\tt genus2reduction} program
471({\tt ftp://megrez.math.u-bordeaux.fr/pub/liu}) to compute the
472conductor of the curve. In each case (other than curve $C_{188}$),
473the conductor is the square of the level; this is also necessary. For
474curve $C_{188}$, the odd part of the conductor of the curve is the
475square of the odd part of the level.
476
477In addition, since the Jacobians of the Wang curves are optimal
478quotients, we can compute~$k\cdot\Omega$ (where $k$ is the Manin constant,
479conjectured to be 1)
480using the newforms.
481In each case, these agree (to within the accuracy of our computations)
482with the $\Omega$'s computed using the equations for the curves.
483We can also compute the value of~$c_p$ for optimal quotients from
484the newforms, when $p$ exactly divides~$N$ and the eigenvalue of the
485$p$th Atkin-Lehner involution is $-1$. When $p$ exactly divides~$N$
486and the eigenvalue of the $p$th Atkin-Lehner involution is~$+1$, the
487component group is either $0$, $\Z/2\Z$, or~$(\Z/2\Z)^2$. These results
488are always in agreement with the values computed using the equations
489for the curves. The algorithms based on the newforms are
490described in Section~\ref{modular}, those based on the
491equations of the curves are described in Section~\ref{algms}.
492
493Lastly, we were able to compute the Mordell-Weil ranks of the Jacobians
494of the curves given by ten of these eleven equations. In
495each case it agrees with the analytic rank of the Jacobian,
496as deduced {}from the newforms.
497
498It should be noted that curve~$C_{125,B}$ is the $\sqrt{5}$-twist of
499curve~$C_{125,A}$; the corresponding statement holds for the associated
5002-dimensional subspaces of~$S_2(125)$. Since curve~$C_{125,A}$ is
501a Hasegawa curve, this proves that the equation given in Table~\ref{table1}
502for curve~$C_{125,B}$ is correct.
503
504The $a_p$'s and other information concerning Wang's curves are
505currently kept in a database at the Institut f\"{u}r experimentelle
506Mathematik in Essen, Germany.  Most recently, this database was under
507the care of Michael M\"{u}ller.  William Stein also keeps a database
508of~$a_p$'s for newforms.
509
510\begin{rem}
511For the remainder of this paper we will assume that the equations for
512the curves given in Table~\ref{table1} are correct; that is, that
513they are equations for the curves whose Jacobians are isogenous
514to a factor of~$J_0(N)$ in the way described above.
515Some of the quantities can be computed either {}from the newform
516or {}from the equation for the curve.  We performed both computations
517whenever possible, and view this duplicate effort as an attempt to
518verify our implementation of the algorithms rather than an attempt
519to verify the equations in Table~\ref{table1}.  For most quantities,
520one method or the other is not guaranteed to produce a value; in this
521case, we simply quote the value {}from whichever method did succeed.
522The reader who is disturbed by this philosophy should
523ignore the Wang-only curves, since the equations for the Hasegawa
524curves can be proven to be correct.
525\end{rem}
526
527
528\section{Algorithms for genus~2 curves}
529\label{algms}
530
531In this section, we describe the algorithms that are based on the
532given models for the curves. We give algorithms that compute all
533terms on the right hand side of equation~\eqref{eqn1}, with the
534exception of the size of the Shafarevich-Tate group. We describe,
535however, how to find the size of its 2-torsion subgroup.
536
537\subsection{Torsion Subgroup}
538\label{torsion}
539
540The computation of the torsion subgroup of~$J(\Q)$ is straightforward.
541We used the technique described in~\cite[pp.~78--82]{CF}.
542This technique is not always effective, however. For an algorithm working
543in all cases see~\cite{Sto3}.
544
545\subsection{Mordell-Weil rank and $\Sh(J,\Q)[2]$}
546\label{MW}
547
548The group $J(\Q)$ is a finitely generated abelian group and so is
549isomorphic to $\Z^{r} \oplus J(\Q)\tors$ for some $r$ called the
550Mordell-Weil rank.
551As noted above (see Section~\ref{intro}), we justifiably use
552$r$ to denote both the analytic and Mordell-Weil ranks since they
553agree for all curves in Table~\ref{table1}.
554
555We used the algorithm described in \cite{FPS} to compute ${\rm 556Sel}^{2}_{\rm fake}(J,\Q)$ (notation {}from \cite{PSc}), which is a
557quotient of the 2-Selmer group ${\rm Sel}^{2}(J,\Q)$. More details
558on this algorithm can be found in \cite{Sto2}.  Theorem 13.2 of
559\cite{PSc} explains how to get ${\rm Sel}^{2}(J,\Q)$ {}from ${\rm 560Sel}^{2}_{\rm fake}(J,\Q)$.  Let $M[2]$ denote the 2-torsion of an
561abelian group $M$ and let dim$V$ denote the dimension of an $\F_{2}$
562vector space $V$.  We have
563$\dim {\rm Sel}^{2}(J,\Q) = r + \dim J(\Q)[2] + \dim \Sh(J,\Q)[2]$.
564In other words,
565$\dim\, \Sh (J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2].$
566
567It is interesting to note that in all 30 cases where
568$\dim \Sh(J,\Q)[2] \le 1$, we were able to compute the Mordell-Weil rank
569independently from the analytic rank.
570The
571cases where $\dim \Sh(J,\Q)[2] = 1$ are discussed in more
572detail in Section~\ref{Shah}.
573For both of the remaining cases we have $\dim \Sh(J,\Q)[2]=2$.
574One of these cases is
575$C_{125,B}$. For this curve we computed
576${\rm Sel}^{\sqrt{5}}(J_{125,B},\Q)$
577using the technique described in
578\cite{Sc}. {}From this, we were able to determine that the Mordell-Weil
579rank is 0 independently from the analytic rank.
580For the other case,
581$C_{133,A}$,
582we could show that $r$ had to be either~0
583or~2 {}from the equation, but we needed the analytic computation to
584show that $r=0$.
585
586\subsection{Regulator}
587\label{reg}
588
589When the Mordell-Weil rank is~0, then the regulator is~1. When the
590Mordell-Weil rank is positive, then to compute the regulator, we
591first need to find generators for $J(\Q)/J(\Q)\tors$. The regulator
592is the determinant of the canonical height pairing matrix on this set
593of generators. An algorithm for computing the generators and
594canonical heights is given in~\cite{FS}; it was used to find
595generators for $J(\Q)/J(\Q)\tors$ and to compute the regulators.  In
596that article, the algorithm for computing height constants at the
597infinite prime is not clearly explained and there are some errors in
598the examples. A clear algorithm for computing infinite height
599constants is given in~\cite{Sto3}. In~\cite{Sto4}, some improvements of
600the results and algorithms in~\cite{FS} and~\cite{Sto3} are discussed.
601The regulators in Table~\ref{table2} have been double-checked using
602these improved algorithms.
603
604\subsection{Tamagawa Numbers}
605\label{Tamagawa}
606
607Let $\OO$ be the integer ring in~$K$ which will be $\Q_{p}$ or
608$\Q_{p}\unr$ (the maximal unramified extension of $\Q_{p})$.
609Let $\JJ$ be the N\'{e}ron model of~$J$ over~$\OO$.
610Define $\JJ^{0}$ to be the open subgroup scheme of~$\JJ$ whose
611generic fiber is isomorphic to~$J$ over~$K$ and whose special fiber
612is the identity component of the closed fiber of~$\JJ$.
613The group $\JJ^{0}(\OO)$ is isomorphic to a subgroup of~$J(K)$ which
614we denote $J^{0}(K)$. The group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is
615the component group of~$\JJ$ over~$\OO_{\Q_{p}\unr}$.  We are
616interested in computing $c_p = \#J(\Q_{p})/J^{0}(\Q_{p})$, which is
617sometimes called the Tamagawa number.
618Since N\'{e}ron models are stable under unramified base extension,
619the $\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of
620$J^{0}(\Q_{p}\unr)$ is~$J^{0}(\Q_{p})$.
621Since $H^1(\Gal(\Q_{p}\unr/\Q_{p}), J^{0}(\Q_{p}\unr))$
622is trivial (see~\cite[p.\ 58]{Mi1}) we see that the
623$\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of
624$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is $J(\Q_{p})/J^{0}(\Q_{p})$.
625
626There exist several discussions in the literature on constructing the
627group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ starting with an integral
628model of the underlying curve.  For our purposes, we especially
629recommend Silverman's book~\cite{Si}, Chapter~IV, Sections 4 and~7.
630For a more detailed treatment, see~\cite[chap.\ 9]{BLR}.  In these
631books, one can find justifications for what we will do. While
632constructing such groups, we ran into a number of difficulties that
633we did not find described anywhere. For that reason, we will present
634examples of such difficulties that arose, as well as our methods of
635resolution.  We do not claim that we will describe all situations
636that could arise.
637
638When computing $c_p$ we need a proper, regular model~$\CC$ for~$C$
639over~$\Z_p$.  Let $\Z_p\unr$ denote the ring of integers of~$\Q_p\unr$
640and note that $\Z_p\unr$ is a pro-\'etale Galois extension
641of~$\Z_p$ with Galois group
642$\Gal(\Z_p\unr/\Z_p) = \Gal(\Q_p\unr/\Q_p)$.
643It follows that giving a model for~$C$ over~$\Z_p$ is equivalent to
644giving a model for~$C$ over~$\Z_p\unr$ that
645is equipped with a Galois action.  We have found it convenient to
646always work with the latter description.  Thus for us, giving a model
647over~$\Z_p$ will always mean giving a model over~$\Z_p\unr$ together
648with a Galois action.
649
650In order to find a proper, regular model for~$C$ over~$\Z_p$,
652consider the curves to be the two affine pieces $y^2+g(x)y=f(x)$ and
653$v^2 + u^3 g(1/u)v = u^6 f(1/u)$, glued together by $ux=1$, $v=u^3y$.
654We blow them up at all points that are not regular until we have a
655regular model.  (A point is {\em regular} if the cotangent space there has
656two generators.)  These curves are all proper, and this is not
657affected by blowing up.
658
659Let $\CC_p$ denote the special fiber of~$\CC$ over~$\Z_p\unr$.  The
660group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is isomorphic to a quotient
661of the degree~0 part of the free group on the irreducible components
662of~$\CC_{p}$. Let the irreducible components be denoted $\DD_{i}$ for
663$1\leq i\leq n$, and let the multiplicity of~$\DD_{i}$ in~$\CC_p$ be
664$d_{i}$.  Then the degree~0 part of the free group has the form
665$L = \{ \sum\limits_{i=1}^{n} \alpha_{i}\DD_{i} \mid 666 \sum\limits_{i=1}^{n} d_{i}\alpha_{i} = 0 \}\,.$
667
668In order to describe the group that we quotient out by, we must
669discuss the intersection pairing.  For components $\DD_{i}$ and~$\DD_{j}$
670of the special fiber, let $\DD_{i} \cdot \DD_{j}$ denote
671their intersection pairing. In all of the special fibers that arise
672in our examples, distinct components intersect transversally.  Thus,
673if $i \neq j$, then $\DD_{i} \cdot \DD_{j}$ equals the number of points
674at which $\DD_{i}$ and $\DD_{j}$ intersect.  The case of
675self-intersection ($i=j$) is discussed below.
676
677The kernel of the map {}from~$L$ to
678$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is generated by
679divisors of the form
680$[\DD_j] = \sum\limits_{i=1}^{n} (\DD_{j} \cdot \DD_{i}) \DD_{i}$
681for each component~$\DD_j$.  We can deduce $\DD_{j} \cdot \DD_{j}$ by
682noting that $[\DD_j]$ must be contained in the group~$L$. This follows
683{}from the fact that the intersection pairing of
684$\CC_{p} = \sum d_i\DD_{i}$ with any irreducible component is 0.
685
686\vspace{1mm}
687\noindent
688{\bf Example 1.} Curve $C_{65,B}$ over $\Z_{2}$.
689
690An equation for curve~$C_{65,B}$ is
691$y^2 = f(x) = -x^6 + 10x^5 - 32x^4 + 20x^3 + 40x^2 + 6x - 1$.  The Jacobian
692of this curve
693is a quotient of the Jacobian of~$X_0(65)$.  Though 65 is odd, this
694curve has singular reduction at~2.  Since the equation for this curve
695is conjectural (it is a Wang-only curve), it will be nice to verify
696that 2 does not divide the conductor of its Jacobian, i.e.\ that the
697Jacobian has good reduction at~2.  In addition, we will need a
698proper, regular model for this curve in order to find~$\Omega$.
699
700Note that $f(x)$ has a factor of
701$x^2 - 3x - 1$.  The special fiber of the arithmetic surface
702$y^2 = f(x)$ over~$\Z_{2}\unr$
703is given by
704$(y + x^3 + 1)^2 = 0 \pmod 2$; this is a genus~0 curve of multiplicity~2
705that we denote~$A$. This model is not regular at the two points
706$(x-\alpha, y, 2)$, where $\alpha$ is a root of $x^2 - 3x - 1$.  It is
707regular at infinity so we will blow up only the given affine cover.
708The current special fiber is in Figure~\ref{special2} and is labelled
709{\it Fiber~1}.
710
711We fix $\alpha$ and move $(x - \alpha, y, 2)$ to the origin using the
712substitution $x_0 = x-\alpha$. We get
713$y^2 = -x_0^6 + (-6\alpha + 10)x_0^5 + (5\alpha - 47)x_0^4 714 + (-28\alpha + 60)x_0^3 + (-11\alpha - 2)x_0^2 715 + (-24\alpha - 16)x_0 716$
717which we rewrite as the pair of equations
718\begin{align*}
719    g_{1}(x_{0},y,p)
720      &= -x_0^6 + (-3\alpha + 5) p x_0^5 + (5\alpha - 47) x_0^4
721           + (-7\alpha + 15) p^2 x_0^3 \\
722      & \qquad {} + (-11\alpha - 2) x_0^2 + (-3\alpha - 2) p^3 x_0 - y^2
723         \\
724      &= 0,\\
725    p &= 2.
726\end{align*}
727To blow up at $(x_0,y,p)$, we introduce projective coordinates
728$(x_1,y_1,p_1)$ with $x_{0} y_1 = x_{1} y$, $x_{0} p_{1} = x_{1} p$, and
729$y p_1 = y_{1} p$. We look in all three affine covers and check for regularity.
730
731\begin{description}
732\item[$x_{1} = 1$] We have $y = x_{0} y_{1}$, $p = x_{0} p_{1}$. We get
733  $g_2(x_{0},y_{1},p_{1}) = 0$, $x_{0} p_{1} = 2$, where
734  \begin{align*}
735     g_2(x_{0},y_{1},p_{1}) &= x_{0}^{-2}g_{1}(x_{0},x_{0}y_{1},x_{0}p_{1}) \\
736         &= -x_0^4 + (-3\alpha + 5) p_1 x_0^4 + (5\alpha - 47) x_0^2
737             + (-7\alpha + 15) p_1^2 x_0^3 \\
738	 & \qquad{} + (-11\alpha - 2) + (-3\alpha - 2) p_1^3 x_0^2 - y_1^2 \,.
739  \end{align*}
740  In the reduction we have either $x_{0} = 0$ or $p_1 = 0$.
741  \begin{description}
742    \item[$x_{0} = 0$] $(y_{1} + \alpha + 1)^2 = 0$.
743      This is a new component which we denote $B$. It has genus~0 and
744      multiplicity~2. We check regularity along~$B$ at
745      $(x_{0}, y_{1} + \alpha + 1, p_{1}-t, 2)$, with $t$ in $\Z_2\unr$, and
746      find that $B$ is nowhere regular.
747    \item[$p_{1} = 0$]
748      $(y_{1} + x_{0}^2 + \alpha x_{0} + (\alpha + 1))^2 = 0$.
749      Using the gluing maps, we see that this is~$A$.
750  \end{description}
751
752\item[$y_{1} = 1$] We get no new information {}from this affine cover.
753
754\item[$p_{1} = 1$] We have $x_{0} = x_{1} p$, $y = y_{1} p$. We get
755  $g_{3}(x_{1},y_{1},p) = p^{-2} g_{1}(x_{1}p,y_{1}p,p) = 0$, $p = 2$.
756  In the reduction we have
757  \begin{description}
758    \item[$p=0$] $(y_1 + (\alpha+1)x_1)^2 = 0$. Using the gluing maps, we
759      see that this is~$B$. It is nowhere regular.
760  \end{description}
761\end{description}
762
763The current special fiber is in
764Figure~\ref{special2} and is labelled {\it Fiber~2}. It is not regular
765along~$B$ and at the other point on~$A$ which we have not yet blown up.
766The component $B$ does not lie entirely in any one affine cover
767so we will blow up the affine covers $x_1 = 1$ and $p_1 = 1$ along~$B$.
768
769To blow up $x_1 = 1$ along~$B$ we make the substitution
770$y_2 = y_1 + \alpha + 1$ and replace each factor of~2 in a coefficient
771by~$x_0 p_1$. We have $g_{4}(x_0,y_2,p_1) = 0$ and $x_0 p_1 = 2$, and we
772want to blow up along the line $(x_0, y_2, 2)$.  Blowing up along a line
773is similar to blowing up at a point: since we are blowing up at
774$(x_0, y_2, 2) = (x_0, y_2)$, we introduce projective
775coordinates $x_3, y_3$ together with the relation $x_0 y_3 = x_3 y_2$.  We
776have two affine covers.
777
778\begin{description}
779  \item[$x_3 = 1$] We have $y_2 = y_{3} x_{0}$. We get
780    $g_{5}(x_{0},y_{3},p_{1}) = x_{0}^{-2} g_{4}(x_{0},y_{3}x_{0},p_1) = 0$
781    and $x_{0} p_{1} = 2$. In the reduction we have
782    \begin{description}
783      \item[$x_{0} = 0$]
784        $y_{3}^2 + (\alpha + 1) y_{3} p_{1} + \alpha p_{1}^3 + p_{1}^2 785 + \alpha + 1 = 0$.
786	This is~$B$. It is now a non-singular genus~1 curve.
787      \item[$p_{1} = 0$] $(x_0 + y_3 + \alpha)^2 = 0$. This is~$A$. The point
788        where $B$ meets~$A$ transversally is regular.
789    \end{description}
790
791  \item[$y_3 = 1$] We get no new information {}from this affine cover.
792\end{description}
793
794When we blow up $p_1 = 1$ along~$B$ we get essentially the same thing and
795all points are again regular.
796
797The other non-regular point on~$A$ is the conjugate of the one we
798blew up. Therefore, after performing the conjugate blow ups, it too
799will be a genus~1 component crossing~$A$ transversally. We denote
800this component $D$; it is conjugate to~$B$.
801
802
803\begin{figure}
804\caption{Special fibers of curve $C_{65,B}$ over $\Z_{2}$;
805         points not regular are thick}
806\label{special2}
807\begin{picture}(400,130)
808  \put(20,5){\begin{picture}(100,125)
809	       \thinlines
810	       \put(20,55){\line(1,0){60}}
811	       \put(85,55){\makebox(0,0){A}}
812	       \put(75,62){\makebox(0,0){2}}
813	       \put(40,55){\circle*{5}}
814	       \put(60,55){\circle*{5}}
815	       \put(50,5){\makebox(0,0){Fiber 1}}
816	     \end{picture}}
817  \put(145,5){\begin{picture}(100,125)
818		\thinlines
819		\put(50,5){\makebox(0,0){Fiber 2}}
820		\put(20,55){\line(1,0){60}}
821		\put(85,55){\makebox(0,0){A}}
822		\put(75,62){\makebox(0,0){2}}
823		\put(60,55){\circle*{5}}
824		\put(40,15){\line(0,1){80}}
825		\put(40.5,15){\line(0,1){80}}
826		\put(39.5,15){\line(0,1){80}}
827		\put(39,15){\line(0,1){80}}
828		\put(41,15){\line(0,1){80}}
829		\put(40,105){\makebox(0,0){B}}
830		\put(34,90){\makebox(0,0){2}}
831	      \end{picture}}
832  \put(270,5){\begin{picture}(100,125)
833		\thinlines
834		\put(20,55){\line(1,0){60}}
835		\put(85,55){\makebox(0,0){A}}
836		\put(75,62){\makebox(0,0){2}}
837		\put(40,15){\line(0,1){80}}
838		\put(40,105){\makebox(0,0){B}}
839		\put(60,15){\line(0,1){80}}
840		\put(60,105){\makebox(0,0){D}}
841		\put(50,5){\makebox(0,0){Fiber 3}}
842              \end{picture}}
843\end{picture}
844\end{figure}
845
846We now have a proper, regular model~$\CC$ of~$C$ over~$\Z_2$.
847Let $\CC_2$ be the special fiber of this model; a
848diagram of~$\CC_2$ is in Figure~\ref{special2} and is labelled
849{\it Fiber~3}.  We can use $\CC$ to show that the
850N\'eron model $\JJ$ of the Jacobian $J = J_{65,B}$ has good
851reduction at~2.
852
853We know that the reduction of~$\JJ^0$ is the extension of an abelian
854variety by a connected linear group.  Since $\CC$ is regular and
855proper, the abelian variety part of the reduction is the product of
856the Jacobians of the normalizations of the components of~$\CC_2$ (see
857\cite[9.3/11 and 9.5/4]{BLR}).  Thus, the abelian variety part is the
858product of the Jacobians of~$B$ and~$D$.  Since this is
8592-dimensional, the reduction of~$\JJ^0$ is an abelian variety.  In
860other words, since the sum of the genera of the components of the
861special fiber is equal to the dimension of~$J$, the reduction is an
862abelian variety.  It follows that $\JJ$ has good reduction at~2, that
863the conductor of~$J$ is odd, and that $c_2 = 1$.  As noted above, this
864gives further evidence that the equation given in Table~\ref{table1}
865is correct.
866
867
868\vspace{1mm}
869\noindent
870{\bf Example 2.} Curve $C_{63}$ over $\Z_{3}$.
871
872The Tamagawa number is often found using the intersection matrix and
873sub-determinants. This is not entirely satisfactory for cases where
874the special fiber has several components and a non-trivial Galois
875action. Here is an example of how to resolve this (see also~\cite{BL}).
876
877When we blow up curve~$C_{63}$ over~$\Z_{3}\unr$, we get
878the special fiber shown in Figure~\ref{special1}.
879Elements of $\Gal(\Q_{3}\unr/\Q_{3})$
880that do not fix the quadratic unramified extension of~$\Q_{3}$
881switch $H$ and~$I$. The other components are defined over~$\Q_{3}$.
882All components have genus~0. The group $J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$
883is isomorphic to a quotient of
884\begin{align*}
885  L = \{ \alpha A + \beta B + \delta D + \epsilon E + \phi F + \gamma G
886          &+ \eta H + \iota I \\
887          &\mid \alpha + \beta + 2\delta + 2\epsilon + 4\phi + 2\gamma
888	         + 2\eta +  2\iota = 0 \} \,.
889\end{align*}
890
891The kernel is generated by the following divisors.
892\begin{center}
893  \begin{tabular}{*{2}{@{[}c@{]$\;=\;$}r@{\hspace{2cm}}}}
894    $A$ & $-2A + E$ &            $B$ & $-2B + E$ \\
895    $D$ & $-D + E$ &             $E$ & $A + B + D - 4E + F$  \\
896    $F$ & $E - 2F + G + H + I$ & $G$ & $F - 2G$ \\
897    $H$ & $F - 2H$ &             $I$ & $F - 2I$
898  \end{tabular}
899\end{center}
900
901\begin{figure}
902\caption{Special fiber of curve $C_{63}$ over $\Z_{3}$}
903\label{special1}
904\begin{picture}(400,130)
905  \put(100,5){\begin{picture}(200,125)
906		\thinlines
907		\put(20,50){\line(1,0){160}}
908		\put(40,20){\line(0,1){60}}
909		\put(60,20){\line(0,1){60}}
910		\put(80,20){\line(0,1){60}}
911		\put(150,10){\line(0,1){100}}
912		\put(120,70){\line(1,0){60}}
913		\put(120,90){\line(1,0){60}}
914		\put(120,30){\line(1,0){60}}
915		\put(40,88){\makebox(0,0){G}}
916		\put(60,88){\makebox(0,0){H}}
917		\put(80,88){\makebox(0,0){I}}
918		\put(150,118){\makebox(0,0){E}}
919		\put(185,50){\makebox(0,0){F}}
920		\put(185,90){\makebox(0,0){A}}
921		\put(185,70){\makebox(0,0){B}}
922		\put(185,30){\makebox(0,0){D}}
923		\put(35,70){\makebox(0,0){2}}
924		\put(55,70){\makebox(0,0){2}}
925		\put(75,70){\makebox(0,0){2}}
926		\put(165,55){\makebox(0,0){4}}
927		\put(165,35){\makebox(0,0){2}}
928		\put(145,104){\makebox(0,0){2}}
929              \end{picture}}
930\end{picture}
931\end{figure}
932
933When we project away {}from~$A$, we find that
934$J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$ is isomorphic to
935\begin{align*}
936  \langle B, D, E, F, G, H, I
937            &\mid E = 0, E = 2B, D = E, 4E = B + D + F, \\
938            &\quad 2F = E + G +  H + I, F = 2G = 2H = 2I \rangle.
939\end{align*}
940At this point, it is straightforward to simplify the representation by
941elimination. Note that we projected away {}from~$A$, which is
942Galois-invariant. It is best to continue eliminating Galois-invariant
943elements first. We find that this group is isomorphic to
944$\langle H, I \mid 2H = 2I = 0 \rangle$ and elements of
945$\Gal(\Q_{3}\unr/\Q_{3})$ that do not fix the quadratic unramified
946extension of~$\Q_{3}$ switch $H$ and~$I$. Therefore
947$J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr) \cong \Z/2\Z \oplus \Z/2\Z$ and
948$c_3 = \#J(\Q_{3})/J^{0}(\Q_{3}) = 2$.
949
950\subsection{Computing $\Omega$}
951\label{Omega}
952
953By an {\em integral differential} (or {\em integral form}) on $J$ we mean the
954pullback to $J$ of a global relative differential form on the N\'eron
955model of $J$ over $\Z$.  The set of integral $n$-forms on $J$ is a
956full-rank lattice in the vector space of global holomorphic $n$-forms
957on $J$.  Since $J$ is an abelian variety of dimension 2, the integral
9581-forms are a free $\Z$-module of rank 2 and the integral 2-forms are
959a free $\Z$-module of rank 1. Moreover, the wedge of a basis for the
960integral 1-forms is a generator for the integral 2-forms.  The
961quantity $\Omega$ is the integral, over the real points of $J$, of a
962generator for the integral 2-forms.  (We choose the generator that
964
965We now translate this into a computation on the curve $C$.  Let
966$\{\omega_1, \omega_2\}$ be a $\Q$-basis for the holomorphic
967differentials on $C$ and let $\{\gamma_1, \gamma_2, \gamma_3, 968\gamma_4\}$ be a $\Z$-basis for the homology of $C(\C)$.  Create a
969$2\times 4$ complex matrix $M_{\C} = [ \int_{\gamma_j}\omega_i]$ by
970integrating the differentials over the homology and let $M_{\R} = 971\Tr_{\C/\R}(M_{\C})$ be the $2\times 4$ real matrix whose entries are
972traces {}from the complex matrix.  The columns of $M_{\R}$ generate a
973lattice $\Lambda$ in $\R^2$.  If we make the standard identification
974between the holomorphic 1-forms on $J$ and the holomorphic
975differentials on $C$ (see \cite{Mi2}), then the notation
976$\int_{J(\R)} |\omega_1 \wedge \omega_2|$ makes sense and its value
977can be computed as the area of a fundamental domain for $\Lambda$.
978
979If $\{\omega_1, \omega_2\}$ is a basis for the integral 1-forms on
980$J$, then $\int_{J(\R)} |\omega_1 \wedge \omega_2| = \Omega$.  On the
981other hand, the computation of $M_{\C}$ is simplest if we choose
982$\omega_1 = dX/Y$, and $\omega_2=X\,dX/Y$ with respect to a model for
983$C$ of the form $Y^2=F(X)$; in this case we obtain $\Omega$ by a
984simple change-of-basis calculation.  This assumes, of course, that we
985know how to express a basis for the integral 1-forms in terms of the
986basis $\{\omega_1, \omega_2\}$; this is addressed in more detail
987below.
988
989It is worth mentioning an alternate strategy.  Instead of finding a
990$\Z$-basis for the homology of $C(\C)$ one could find a $\Z$-basis
991$\{\gamma'_1, \gamma'_2\}$ for the subgroup of the homology that is
992fixed by complex conjugation (call this the real homology).
993Integrating would give us a $2\times 2$ real matrix $M'_{\R}$ and the
994determinant of $M'_{\R}$ would equal the integral of $\omega_1 995\wedge \omega_2$ over the connected component of $J(\R)$.
996In other words, the number of real connected components of $J$ is
997equal to the index of the $\C/\R$-traces in the real homology.
998
999We now come to the question of determining the differentials on $C$
1000which correspond to the integral 1-forms on $J$.  Call these the
1001integral differentials on $C$.  This computation can be done one
1002prime at a time.  At each prime $p$ this is equivalent to determining
1003a $\Z_p\unr$-basis for the global relative differentials on any
1004proper, regular model for $C$ over $\Z_p\unr$.  In fact a more
1005general class of models can be used; see the discussion of models
1006with rational singularities in \cite[\S 6.7]{BLR} and \cite[\S
10074.1]{Li}.
1008
1009We start with the model $y^2 + g(x)y=f(x)$ given in
1010Table~\ref{table1}.  Note that the substitution $X=x$ and $Y=2y+g(x)$
1011gives us a model of the form $Y^2=F(X)$.  For integration purposes,
1012our preferred differentials are $dX/Y=dx/(2y+g(x))$ and
1013$X\,dX/Y=x\,dx/(2y+g(x))$.  It is not hard to show that at primes of
1014non-singular reduction for the $y^2 + g(x)y=f(x)$ model, these
1015differentials will generate the integral 1-forms.  For each prime $p$
1016of singular reduction we give the following algorithm.  All steps
1017take place over $\Z_p\unr$.
1018
1019\begin{description}
1020  \item[Step 1]
1021    Compute explicit equations for a proper, regular model $\CC$.
1022
1023  \item[Step 2]
1024    Diagram the configuration of the special fiber of $\CC$.
1025
1026  \item[Step 3] (Optional)
1027    Identify exceptional components and blow them down in the
1028    configuration diagram. Repeat step 3 as necessary.
1029
1030  \item[Step 4] (Optional)
1031    Remove components with genus 0 and self-intersection $-2$.
1032    Since $C$ has genus greater than 1,
1033    there will be a component that is not of this kind.
1034    (This
1035    step corresponds to contracting the given components to create a
1036    non-proper model with rational singularities.  We will not need a
1037    diagram of the resulting configuration.)
1038
1039  \item[Step 5]
1040    Determine a $\Z_p\unr$-basis for the integral differentials.  It
1041    suffices to check this on a dense open subset of each surviving
1042    component.  Note that we have explicit equations for a dense open
1043    subset of each of these components {}from the model $\CC$ in step 1.  A
1044    pair of differentials $\{\eta_1, \eta_2\}$ will be a basis for the
1045    integral differentials (at $p$) if the following three statements are
1046    true.
1047    \begin{description}
1048      \item[a]
1049        The pair $\{\eta_1, \eta_2\}$ is a basis for the holomorphic
1050        differentials on $C$.
1051      \item[b]
1052        The reductions of $\eta_1$ and $\eta_2$ produce well-defined
1053        differentials mod $p$ on an open subset of each surviving component.
1054      \item[c]
1055        If $a_1\eta_1+a_2\eta_2 = 0 \pmod{p}$ on all surviving components,
1056	then $p|a_1$ and $p|a_2$.
1057    \end{description}
1058\end{description}
1059
1060Techniques for explicitly computing a proper, regular model are
1061discussed in Section~\ref{Tamagawa}.  A configuration diagram should
1062include the genus, multiplicity and self-intersection number of
1063each component and the number and type of intersections between
1064components.  Note that when an exceptional component is blown down,
1065all of the self-intersection numbers of the components intersecting
1066it will go up (towards 0).  In particular, components which were not
1067exceptional before may become exceptional in the new configuration.
1068
1069Steps 3 and 4 are intended to make this algorithm more efficient for
1070a human.  They are entirely optional.  For a computer implementation
1071it may be easier to simply check every component than to worry about
1072manipulating configurations.
1073
1074The curves in Table~\ref{table1} are given as $y^2 + g(x)y=f(x)$.  We
1075assumed, at first, that $dx/(2y+g(x))$ and $x\,dx/(2y+g(x))$ generate
1076the integral differentials.  We integrated these differentials around
1077each of the four paths generating the complex homology and found a
1078provisional $\Omega$. Then we checked the proper, regular models to
1079determine if these differentials really do generate the integral
1080differentials and adjusted $\Omega$ when necessary.  There were
1081three curves where we needed to adjust $\Omega$.  We describe the
1082adjustment for curve $C_{65,B}$ in the following example.  For curve
1083$C_{63}$, we used the differentials $3\,dx/(2y+g(x))$ and
1084$x\,dx/(2y+g(x))$.  For curve $C_{65,A}$, we used the differentials
1085$3\,dx/(2y+g(x))$ and $3x\,dx/(2y+g(x))$.
1086
1087\vspace{2mm}
1088\noindent
1089{\bf Example 3.} Curve $C_{65,B}$.
1090
1091The primes of singular reduction for curve $C_{65,B}$ are 2, 5 and
109213.  In Example 1 of Section~\ref{Tamagawa}, we found a proper,
1093regular model $\CC$ for $C$ over $\Z_2\unr$.  The configuration for
1094the special fiber of $\CC$ is sketched in Figure~\ref{special2} under
1095the label {\it Fiber 3}.  Component $A$ is exceptional and can be
1096blown down to produce a model in which $B$ and $D$ cross
1097transversally.  Since $B$ and $D$ both have genus 1, we cannot
1098eliminate either of these components.  Furthermore, it suffices to
1099check $B$, since $D$ is its Galois conjugate.
1100
1101To get {}from the equation of the curve listed in Table~\ref{table1}
1102to an affine containing an open subset of $B$ we need to make the
1103substitutions $x=x_0 - \alpha$ and $y=x_0 (y_{3}x_0 - \alpha - 1)$.
1104We also have $x_{0}p_{1}=2$.  Using the substitutions and the
1105relation $dx_{0}/x_0 = -dp_{1}/p_1$, we get
1106$\frac{dx}{2y} = \frac{-dp_1}{2p_1(y_3 x_0 - \alpha - 1)} 1107 \text{\quad and\quad} 1108 \frac{x\,dx}{2y} 1109 = \frac{-(x_0 + \alpha)\,dp_1}{2p_1(y_3 x_0 - \alpha - 1)} \,. 1110$
1111Note that $p_1 - t$ is a uniformizer at $p_1 = t$ almost everywhere
1112on~$B$.  When we multiply each differential by~2, then the
1113denominator of each is almost everywhere non-zero; thus, $dx/y$ and
1114$x\,dx/y$ are integral at~$2$.  Moreover, although the linear
1115combination $(x-\alpha)\,dx/y$ is identically zero on~$B$, it is not
1116identically zero on~$D$ (its Galois conjugate is not identically zero
1117on~$B$).  Thus, our new basis is correct at~2.  We multiply the
1118provisional $\Omega$ by~4 to get a new provisional $\Omega$ which is
1119correct at~$2$.
1120
1121Similar (but somewhat simpler) computations at the primes $5$ and~$13$
1122show that no adjustment is needed at these primes.  Thus, $dx/y$
1123and $x\,dx/y$ form a basis for the integral differentials of curve
1124$C_{65,B}$, and the correct value of $\Omega$ is 4 times our original
1125guess.
1126
1127\section{Modular algorithms}
1128\label{modular}
1129
1130In this section, we describe the algorithms that were used to compute
1131some of the data from the newforms. This includes the analytic rank
1132and leading coefficient of the $L$-series. For optimal quotients,
1133the value of~$k\cdot\Omega$ can also be found ($k$ is the Manin constant),
1134as well as partial information
1135on the Tamagawa numbers~$c_p$ and the size of the torsion subgroup.
1136
1137\subsection{Analytic rank of $L(J,s)$ and leading coefficient at $s=1$}
1138\label{l}
1139
1140Fix a Jacobian~$J$ corresponding to the 2-dimensional subspace of
1141$S_2(N)$ spanned by quadratic conjugate, normalized newforms~$f$
1142and~$\overline{f}$.  Let $W_N$ be the Fricke involution. The newforms~$f$
1143and~$\overline{f}$ have the same eigenvalue~$\epsilon_N$ with respect
1144to~$W_N$, namely $+1$ or~$-1$. In the notation of Section~\ref{curves}, let
1145$L(f,s) = \sum\limits_{n=1}^{\infty} \frac{a_n}{n^s}$
1146be the $L$-series of~$f$; then $L(\overline{f},s)$ is the Dirichlet
1147series whose coefficients are the conjugates of the
1148coefficients of~$L(f,s)$. (Recall that the~$a_n$ are integers in some
1149real quadratic field.)  The order of~$L(f,s)$ at~$s = 1$ is even
1150when $\epsilon_N = -1$ and odd when $\epsilon_N = +1$.  We have
1151$L(J,s) = L(f,s) L(\overline{f},s)$. Thus the analytic rank of $J$ is~0
1152modulo~4 when $\epsilon_N = -1$ and 2 modulo~4 when $\epsilon_N = +1$.
1153We found that the ranks were all 0 or~2. To prove that the analytic
1154rank of~$J$ is~0, we need to show $L(f,1) \neq 0$ and
1155$L(\overline{f},1) \neq 0$. In the case that $\epsilon_N = +1$, to
1156prove that the analytic rank is~2, we need to show that $L'(f,1) \neq 0$
1157and $L'(\overline{f},1) \neq 0$.  When $\epsilon_N = -1$, we can
1158evaluate $L(f,1)$ as in~\cite[\S~2.11]{Cr}.  When $\epsilon_N = +1$, we
1159can evaluate $L'(f,1)$ as in~\cite[\S~2.13]{Cr}.  Each appropriate
1160$L(f,1)$ or~$L'(f,1)$ was at least~$0.1$ and the errors in our
1161approximations were all less than~$10^{-67}$. In this way we
1162determined the analytic ranks, which we denote~$r$.  As noted in the
1163introduction, the analytic rank equals the Mordell-Weil rank if $r = 0$
1164or~$r = 2$.  Thus, we can simply call $r$ the rank, without fear of
1165ambiguity.
1166
1167To compute the leading coefficient of~$L(J,s)$ at~$s = 1$, we note that
1168$\lim_{s \to 1} L(J,s)/(s-1)^r = L^{(r)}(J,1)/r!$.
1169In the $r=0$ case, we simply have $L(J,1) = L(f,1)L(\overline{f},1)$.
1170In the $r=2$ case, we have
1171$L''(J,s) 1172 = L''(f,s)L(\overline{f},s) + 2L'(f,s)L'(\overline{f},s) 1173 + L(f,s)L''(\overline{f},s)$.
1174Evaluating both sides
1175at $s=1$ we get $\frac{1}{2}L''(J,1) = L'(f,1)L'(\overline{f},1)$.
1176
1177\subsection{Computing $k\cdot\Omega$}\label{modomega}
1178Let $J$, $f$ and $\overline{f}$ be as in Section~\ref{l} and
1179denote by $V$ the 2-dimensional space spanned by $f$ and
1180$\overline{f}$.
1181In computing $\Omega$ from an equation for the curve,
1182we use a basis of integral
1183differentials (see Section~\ref{Omega}) for $J$.
1185use a basis $\{\omega_1,\omega_2\}$
1186for the subgroup of $V$ consisting of forms whose
1187$q$-expansion coefficients lie in $\Z$, and we will
1188obtain the quantity $k\cdot\Omega$. It can be shown
1189that $k$ is a rational number.  This rational number
1190is called the {\em Manin constant}, and
1191it is conjectured to equal~$1$.
1192
1193We can compute $k\cdot\Omega$ using a generalization
1194to dimension 2~of the algorithm for computing periods
1195described in  \cite[\S2.10]{Cr}.  This is because
1196$k\cdot\Omega$ is the volume of the real points
1197of the quotient of $\C\times\C$ by the
1198lattice of period integrals
1199$(\int_\gamma \omega_1, \int_\gamma\omega_2)$
1200with $\gamma$ in the integral homology
1201$H_1(X_0(N),\Z)$.
1202When $L(J,1)\neq 0$ the
1203method of \cite[\S2.11]{Cr} coupled with
1204Sections~\ref{l} and~\ref{bsdratio} can also be used
1205to compute $k\cdot\Omega$.
1206
1207\subsection{Computing $L(J,1)/(k\cdot\Omega)$}\label{bsdratio}
1208We compute the rational number $L(J,1)/(k\cdot\Omega)$, for optimal
1209quotients,
1210using the algorithm in \cite{AS}.
1211This algorithm generalizes the algorithm described in
1212\cite[\S2.8]{Cr} to dimension greater than 1.
1213
1214\subsection{Tamagawa numbers}
1215In this section we assume that $p$ is a prime which
1216exactly divides the conductor $N$ of $J$.
1217Under these conditions, Grothendieck \cite{Gr} gave a
1218description of the component group of $J$ in
1219terms of a monodromy pairing on certain character groups.
1220(For more details, see Ribet \cite[\S2]{Ri}.)
1221If, in addition,  $J$ is a new optimal quotient of $J_0(N)$, one
1222deduces the following. When
1223the eigenvalue for $f$ of the Atkin-Lehner involution $w_p$ is
1224$+1$, then the rational component group of $J$ is a subgroup of
1225$(\Z/2\Z)^2$. Furthermore, when the eigenvalue of $w_p$ is $-1$,
1226the algorithm described in \cite{Ste} can be used to compute
1227the value of~$c_p$.
1228
1229\subsection{Torsion subgroup}
1230\label{modtors}
1231
1232To compute an integer divisible by the order of the
1233torsion subgroup of $J$ we make use of the following two observations.
1234First, it is a consequence of the Eichler-Shimura relation
1235\cite[\S7.9]{Sh} that if $p$ is a prime not dividing the
1236conductor $N$ of $J$ and $f(T)$ is the characteristic polynomial
1237of the endomorphism $T_p$
1238of $J$, then $\#J(\F_p) = f(p+1)$ (see \cite[\S2.4]{Cr}
1239for an algorithm to compute $f(T)$).
1240Second, if $p$ is an odd prime at which $J$ has good reduction,
1241then the natural map $J(\Q)\tors\rightarrow J(\F_p)$ is injective
1242(see \cite[p.\ 70]{CF}). This does not depend on whether $J$ is an
1243optimal quotient.
1244To obtain a lower bound on the torsion subgroup for optimal quotients,
1245we use modular symbols and the Abel-Jacobi theorem \cite[IV.2]{La}
1246to compute the order of the image of the rational point
1247$(0)-(\infty)\in J_0(N)$.
1248
1249\section{Tables}
1250\label{tables}
1251
1252In Table~\ref{table1}, we list the 32 curves described in
1253Section~\ref{curves}. We give the level $N$ {}from which each curve
1254arose, an integral model for the curve, and list the source(s) {}from
1255which it came ($H$ for Hasegawa \cite{Ha}, $W$ for Wang \cite{Wan}).
1256Throughout the paper, the curves are denoted $C_N$ (or $C_{N,A}$, $C_{N,B}$).
1257
1258\begin{table}
1259\begin{center}
1260\begin{tabular}{|l|rcl|c|}
1261\hline
1262\multicolumn{1}{|c|}{$N$}
1263      & \multicolumn{3}{|c|}{Equation} & Source\\ \hline\hline
126423    & $y^2 + (x^3 + x + 1)y$   & $=$ &
1265        $-2 x^5 - 3 x^2 + 2 x - 2$                           & HW \\
126629    & $y^2 + (x^3 + 1)y$       & $=$ &
1267        $-x^5 - 3 x^4 + 2 x^2 + 2 x - 2$                     & HW \\
126831    & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1269        $-x^5 - 5 x^4 - 5 x^3 + 3 x^2 + 2 x - 3$             & HW \\
127035    & $y^2 + (x^3 + x)y$       & $=$ &
1271        $-x^5 - 8 x^3 - 7 x^2 - 16 x - 19$                   & H  \\ \hline
127239    & $y^2 + (x^3 + 1)y$       & $=$ &
1273        $-5 x^4 - 2 x^3 + 16 x^2 - 12 x + 2$                 & H  \\
127463    & $y^2 + (x^3 - 1)y$       & $=$ &
1275        $14 x^3 - 7$                                         & W  \\
127665,A  & $y^2 + (x^3 + 1)y$       & $=$ &
1277        $-4 x^6 + 9 x^4 + 7 x^3 + 18 x^2 - 10$               & W  \\
127865,B  & $y^2$                    & $=$ &
1279        $-x^6 + 10 x^5 - 32 x^4 + 20 x^3 + 40 x^2 + 6 x - 1$ & W  \\ \hline
128067    & $y^2 + (x^3 + x + 1)y$   & $=$ &
1281        $x^5 - x$                                            & HW \\
128273    & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1283        $-x^5 - 2 x^3 + x$                                   & HW \\
128485    & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1285        $x^4 + x^3 + 3 x^2 - 2 x + 1$                        & H  \\
128687    & $y^2 + (x^3 + x + 1)y$   & $=$ &
1287        $-x^4 + x^3 - 3 x^2 + x - 1$                         & HW \\ \hline
128893    & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1289        $-2 x^5 + x^4 + x^3$                                 & HW \\
1290103   & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1291        $x^5 + x^4$                                          & HW \\
1292107   & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1293        $x^4 - x^2 - x - 1$                                  & HW \\
1294115   & $y^2 + (x^3 + x + 1)y$   & $=$ &
1295        $2 x^3 + x^2 + x$                                    & HW \\ \hline
1296117,A & $y^2 + (x^3 - 1)y$       & $=$ &
1297        $3 x^3 - 7$                                          & W  \\
1298117,B & $y^2 + (x^3 + 1)y$       & $=$ &
1299        $-x^6 - 3 x^4 - 5 x^3 - 12 x^2 - 9 x - 7$            & W  \\
1300125,A & $y^2 + (x^3 + x + 1)y$   & $=$ &
1301        $x^5 + 2 x^4 + 2 x^3 + x^2 - x - 1$                  & HW \\
1302125,B & $y^2 + (x^3 + x + 1)y$   & $=$ &
1303        $x^6 + 5 x^5 + 12 x^4 + 12 x^3 + 6 x^2 - 3 x - 4$    & W  \\ \hline
1304133,A & $y^2 + (x^3 + x + 1)y$   & $=$ &
1305        $-2 x^6 + 7 x^5 - 2 x^4 - 19 x^3 + 2 x^2 + 18 x + 7$ & W  \\
1306133,B & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1307        $-x^5 + x^4 - 2 x^3 + 2 x^2 - 2 x$                   & HW \\
1308135   & $y^2 + (x^3 + x + 1)y$   & $=$ &
1309        $x^4 - 3 x^3 + 2 x^2 - 8 x - 3$                      & W  \\
1310147   & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1311        $x^5 + 2 x^4 + x^3 + x^2 + 1$                        & HW \\ \hline
1312161   & $y^2 + (x^3 + x + 1)y$   & $=$ &
1313        $x^3 + 4 x^2 + 4 x + 1$                              & HW \\
1314165   & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1315        $x^5 + 2 x^4 + 3 x^3 + x^2 - 3 x$                    & H  \\
1316167   & $y^2 + (x^3 + x + 1)y$   & $=$ &
1317        $-x^5 - x^3 - x^2 - 1$                               & HW \\
1318175   & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1319        $-x^5 - x^4 - 2 x^3 - 4 x^2 - 2 x - 1$               & W  \\ \hline
1320177   & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1321        $x^5 + x^4 + x^3$                                    & HW \\
1322188   & $y^2$                    & $=$ &
1323        $x^5 - x^4 + x^3 + x^2 - 2 x + 1$                    & W  \\
1324189   & $y^2 + (x^3 - 1)y$       & $=$ &
1325        $x^3 - 7$                                            & W  \\
1326191   & $y^2 + (x^3 + x + 1)y$   & $=$ &
1327        $-x^3 + x^2 + x$                                     & HW \\ \hline
1328\end{tabular}
1329\end{center}
1330\caption{Levels, integral models and sources for curves}
1331\label{table1}
1332\end{table}
1333
1334In Table~\ref{table2}, we list the curve~$C_N$ simply by~$N$, the
1335level {}from which it arose.  Let $r$ denote the rank.  We
1336list ${\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)$ where $L(J,s)$ is the
1337$L$-series for the Jacobian $J$ of~$C_N$ and round off the results to
1338five digits.  The symbol $\Omega$ was defined in Section~\ref{Omega}
1339and is also rounded to five digits.  Let Reg denote the regulator,
1340also rounded to five digits.  We list the $c_{p}$'s by primes of
1341increasing order dividing the level~$N$.  We denote $J(\Q)\tors = \Phi$
1342and list its size.  We use $\Sh ?$ to denote the size of
1343$({\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)) \cdot 1344 (\#J(\Q)\tors)^2/(\Omega\cdot {\rm Reg} \cdot \prod c_{p})$,
1345rounded to the nearest integer.  We will refer to this as the {\em conjectured
1346size of} $\Sh(J,\Q)$. The last column gives a bound on the accuracy of the
1347computations; all values of $\Sh ?$ were at least this close to the
1348nearest integer before rounding.
1349
1350\newcommand{\mcc}[1]{\multicolumn{1}{|c|}{#1}}
1351\newcommand{\mcd}[1]{\multicolumn{2}{|c|}{#1}}
1352
1353\begin{table}
1354\begin{center}
1355\begin{tabular}{|l|c|r@{.}l|r@{.}l|l|l|c|c|l|}
1356\hline
1357\mcc{$N$} & $r$
1358& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
1359& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & $\Phi$ & $\Sh$? & \mcc{error}
1360\\ \hline\hline
136123    & 0 & 0&24843 &  2&7328 & 1         & 11    & 11 & 1 & $< 10^{-120}$ \\
136229    & 0 & 0&29152 &  2&0407 & 1         & 7     &  7 & 1 & $< 10^{-50}$ \\
136331    & 0 & 0&44929 &  2&2464 & 1         & 5     &  5 & 1 & $< 10^{-49}$ \\
136435    & 0 & 0&37275 &  2&9820 & 1         & 16,2  & 16 & 1 & $< 10^{-25}$ \\
1365\hline
136639    & 0 & 0&38204 & 10&697  & 1         & 28,1  & 28 & 1 & $< 10^{-25}$ \\
136763    & 0 & 0&75328 &  4&5197 & 1         & 2,3   &  6 & 1 & $< 10^{-49}$ \\
136865,A  & 0 & 0&45207 &  6&3289 & 1         & 7,1   & 14 & 2 & $< 10^{-48}$ \\
136965,B  & 0 & 0&91225 &  5&4735 & 1         & 1,3   &  6 & 2 & $< 10^{-50}$ \\
1370\hline
137167    & 2 & 0&23410 & 20&465  & 0.011439  & 1     &  1 & 1 & $< 10^{-50}$ \\
137273    & 2 & 0&25812 & 24&093  & 0.010713  & 1     &  1 & 1 & $< 10^{-49}$ \\
137385    & 2 & 0&34334 &  9&1728 & 0.018715  & 4,2   &  2 & 1 & $< 10^{-26}$ \\
137487    & 0 & 1&4323  &  7&1617 & 1         & 5,1   &  5 & 1 & $< 10^{-49}$ \\
1375\hline
137693    & 2 & 0&33996 & 18&142  & 0.0046847 & 4,1   &  1 & 1 & $< 10^{-49}$ \\
1377103   & 2 & 0&37585 & 16&855  & 0.022299  & 1     &  1 & 1 & $< 10^{-49}$ \\
1378107   & 2 & 0&53438 & 11&883  & 0.044970  & 1     &  1 & 1 & $< 10^{-49}$ \\
1379115   & 2 & 0&41693 & 10&678  & 0.0097618 & 4,1   &  1 & 1 & $< 10^{-50}$ \\
1380\hline
1381117,A & 0 & 1&0985  &  3&2954 & 1         & 4,3   &  6 & 1 & $< 10^{-49}$ \\
1382117,B & 0 & 1&9510  &  1&9510 & 1         & 4,1   &  2 & 1 & $< 10^{-49}$ \\
1383125,A & 2 & 0&62996 & 13&026  & 0.048361  & 1     &  1 & 1 & $< 10^{-50}$ \\
1384125,B & 0 & 2&0842  &  2&6052 & 1         & 5     &  5 & 4 & $< 10^{-49}$ \\
1385\hline
1386133,A & 0 & 2&2265  &  2&7832 & 1         & 5,1   &  5 & 4 & $< 10^{-49}$ \\
1387133,B & 2 & 0&43884 & 15&318  & 0.028648  & 1,1   &  1 & 1 & $< 10^{-49}$ \\
1388135   & 0 & 1&5110  &  4&5331 & 1         & 3,1   &  3 & 1 & $< 10^{-49}$ \\
1389147   & 2 & 0&61816 & 13&616  & 0.045400  & 2,2   &  2 & 1 & $< 10^{-50}$ \\
1390\hline
1391161   & 2 & 0&82364 & 11&871  & 0.017345  & 4,1   &  1 & 1 & $< 10^{-47}$ \\
1392165   & 2 & 0&68650 &  9&5431 & 0.071936  & 4,2,2 &  4 & 1 & $< 10^{-26}$ \\
1393167   & 2 & 0&91530 &  7&3327 & 0.12482   & 1     &  1 & 1 & $< 10^{-47}$ \\
1394175   & 0 & 0&97209 &  4&8605 & 1         & 1,5   &  5 & 1 & $< 10^{-44}$ \\
1395\hline
1396177   & 2 & 0&90451 & 13&742  & 0.065821  & 1,1   &  1 & 1 & $< 10^{-45}$ \\
1397188   & 2 & 1&1708  & 11&519  & 0.011293  & 9,1   &  1 & 1 & $< 10^{-44}$ \\
1398189   & 0 & 1&2982  &  3&8946 & 1         & 1,3   &  3 & 1 & $< 10^{-43}$ \\
1399191   & 2 & 0&95958 & 17&357  & 0.055286  & 1     &  1 & 1 & $< 10^{-44}$ \\
1400\hline
1401\end{tabular}
1402\end{center}
1403\caption{Conjectured sizes of $\Sh (J,\Q)$}
1404\label{table2}
1405\end{table}
1406
1407In Table~\ref{table3} are generators of $J(\Q)/J(\Q)\tors$ for the
1408curves whose Jacobians have Mordell-Weil rank~2. The generators are
1409given as divisor classes. Whenever possible, we have chosen
1410generators of the form $[P - Q]$ where $P$ and~$Q$ are rational
1411points on the curve. Curve~167 is the only example where this is not
1412the case, since the degree zero divisors supported on the (known)
1413rational points on~$C_{167}$ generate a subgroup of index two in the
1414full Mordell-Weil group.
1415Affine points are given by their $x$ and $y$ coordinates in the model
1416given in Table~\ref{table1}.  There are two points at infinity in the
1417normalization of the curves described by our equations, with the
1418exception of curve~$C_{188}$. These are denoted by $\infty_a$, where
1419$a$ is the value of the function $y/x^3$ on the point in question.
1420The (only) point at infinity on curve~$C_{188}$ is simply
1421denoted~$\infty$.
1422
1423\begin{table}
1424\begin{center}
1425\begin{tabular}{|l|l|l|}
1426\hline
1427 \mcc{$N$} & \mcd{Generators of $J(\Q)/J(\Q)\tors$} \\ \hline\hline
1428 67   & $[(0, 0) - \infty_{-1}]$ &
1429        $[(0, 0) - (0, -1)]$ \\
1430 73   & $[(0, -1) - \infty_{-1}]$ &
1431        $[(0, 0) - \infty_{-1}]$ \\
1432 85   & $[(1, 1) - \infty_{-1}]$ &
1433        $[(-1, 3) - \infty_{0}]$ \\
1434 93   & $[(-1, 1) - \infty_{0}]$ &
1435        $[(1, -3) - (-1, -2)]$ \\ \hline
1436103   & $[(0, 0) - \infty_{-1}]$ &
1437        $[(0, -1) - (0,0)]$ \\
1438107   & $[\infty_{-1} - \infty_{0}]$ &
1439        $[(-1, -1) - \infty_{-1}]$ \\
1440115   & $[(1, -4) - \infty_{0}]$ &
1441        $[(1, 1) - (-2, 2)]$ \\
1442125,A & $[\infty_{-1} - \infty_{0}]$ &
1443        $[(-1, 0) - \infty_{-1}]$ \\ \hline
1444133,B & $[\infty_{-1} - \infty_{0}]$ &
1445        $[(0, -1) - \infty_{-1}]$ \\
1446147   & $[\infty_{-1} - \infty_{0}]$ &
1447        $[(-1, -1) - \infty_{0}]$ \\
1448161   & $[(1, 2) - (-1, 1)]$ &
1449        $[(\frac{1}{2}, -3) - (1, 2)]$ \\
1450165   & $[(1, 1) - \infty_{-1}]$ &
1451        $[(0, 0) - \infty_{0} ]$ \\ \hline
1452167   & $[(-1 ,1) - \infty_{0}]$ &
1453        $[(i, 0) + (-i, 0) - \infty_{0} - \infty_{-1}]$ \\
1454177   & $[(0, -1) - \infty_{0}]$ &
1455        $[(0, 0) - (0, -1)]$ \\
1456188   & $[(0, -1) - \infty]$ &
1457        $[(0, 1) - (1, -2)]$ \\
1458191   & $[\infty_{-1} - \infty_{0}]$ &
1459        $[(0, -1) - \infty_{0}]$ \\
1460\hline
1461\end{tabular}
1462\end{center}
1463\caption{Generators of $J(\Q)/J(\Q)\tors$ in rank 2 cases}
1464\label{table3}
1465\end{table}
1466
1467In Table~\ref{table4} are the reduction types, {}from the
1468classification of~\cite{NU}, of the special fibers of the minimal,
1469proper, regular models of the curves for each of the primes of
1470singular reduction for the curve. They are the same as the primes
1471dividing the level except that curve~$C_{65,A}$ has singular
1472reduction at the prime~3 and curve~$C_{65,B}$ has singular reduction
1473at the prime~2.
1474
1475\begin{table}
1476\begin{center}
1477\begin{tabular}{|l|l|l|l|l||l|l|l|l|l|}
1478\hline
1479\mcc{$N$} & Prime & Type & Prime & Type &
1480\mcc{$N$} & Prime & Type & Prime & Type
1481\\ \hline\hline
148223   & 23 & ${\rm I}_{3-2-1}$ & & &
1483  117,A &  3 & ${\rm III}-{\rm III}^{\ast}-0$
1484        & 13 & ${\rm I}_{1-1-1}$ \\
148529   & 29 & ${\rm I}_{3-1-1}$ & & &
1486  117,B &  3 & ${\rm I}_{3-1-1}^{\ast}$
1487        & 13 & ${\rm I}_{1-1-0}$ \\
148831   & 31 & ${\rm I}_{2-1-1}$ & & &
1489  125,A &  5 & ${\rm VIII}-1$ & & \\
149035   &  5 & ${\rm I}_{3-2-2}$
1491     &  7 & ${\rm I}_{2-1-0}$ &
1492  125,B &  5 &  ${\rm IX}-3$ & & \\ \hline
149339   &  3 & ${\rm I}_{6-2-2}$
1494     & 13 & ${\rm I}_{1-1-0}$ &
1495  133,A &  7 & ${\rm I}_{2-1-1}$
1496        & 19 & ${\rm I}_{1-1-0}$ \\
149763   &  3 & $2{\rm I}_{0}^{\ast}-0$
1498     &  7 & ${\rm I}_{1-1-1}$ &
1499  133,B &  7 & ${\rm I}_{1-1-0}$
1500        & 19 & ${\rm I}_{1-1-0}$ \\
150165,A &  3 & ${\rm I}_{0}-{\rm I}_{0}-1$
1502     &  5 & ${\rm I}_{3-1-1}$ &
1503  135   &  3 & III
1504        &  5 & ${\rm I}_{3-1-0}$ \\
150565,A & 13 & ${\rm I}_{1-1-0}$ & & &
1506  147   &  3 & ${\rm I}_{2-1-0}$
1507        &  7 & VII \\ \hline
150865,B &  2 & ${\rm I}_{0}-{\rm I}_{0}-1$
1509     &  5 & ${\rm I}_{3-1-0}$ &
1510  161   &  7 & ${\rm I}_{2-2-0}$
1511        & 23 & ${\rm I}_{1-1-0}$ \\
151265,B & 13 & ${\rm I}_{1-1-1}$ & & &
1513  165   &  3 & ${\rm I}_{2-2-0}$
1514        &  5 & ${\rm I}_{2-1-0}$  \\
151567   & 67 & ${\rm I}_{1-1-0}$ & & &
1516  165   & 11 & ${\rm I}_{2-1-0}$ & & \\
151773   & 73 & ${\rm I}_{1-1-0}$ & & &
1518  167   & 167 & ${\rm I}_{1-1-0}$ & & \\ \hline
151985   &  5 & ${\rm I}_{2-2-0}$
1520     & 17 & ${\rm I}_{2-1-0}$ &
1521  175   &  5 & ${\rm II}-{\rm II}-0$
1522        &  7 & ${\rm I}_{2-1-1}$ \\
152387   &  3 & ${\rm I}_{2-1-1}$
1524     & 29 & ${\rm I}_{1-1-0}$ &
1525  177   &  3 & ${\rm I}_{1-1-0}$
1526        & 59 & ${\rm I}_{1-1-0}$ \\
152793   &  3 & ${\rm I}_{2-2-0}$
1528     & 31 & ${\rm I}_{1-1-0}$ &
1529  188   &  2 & ${\rm IV}-{\rm IV}-0$
1530        & 47 & ${\rm I}_{1-1-0}$ \\
1531103  & 103 & ${\rm I}_{1-1-0}$ & & &
1532  189   &  3 & ${\rm II}-{\rm IV}^{\ast}-0$
1533        &  7 & ${\rm I}_{1-1-1}$ \\ \hline
1534107  & 107 & ${\rm I}_{1-1-0}$ & & &
1535  191   & 191 & ${\rm I}_{1-1-0}$ & & \\
1536115  &  5 & ${\rm I}_{2-2-0}$
1537     & 23 & ${\rm I}_{1-1-0}$ & & & & & \\ \hline
1538\end{tabular}
1539\end{center}
1540\caption{Namikawa and Ueno classification of special fibers}
1541\label{table4}
1542\end{table}
1543
1544
1545\section{Discussion of Shafarevich-Tate groups and evidence for the
1546second conjecture}
1547\label{Shah}
1548
1549{}From Section~\ref{MW} we have
1550$\dim \Sh(J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2]$.
1551With the exception of curves $C_{65,A}$, $C_{65,B}$, $C_{125,B}$, and
1552$C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 0$. Thus we expect
1553$\#\Sh(J,\Q)$ to be an odd square. In each case, the conjectured
1554size of $\Sh(J,\Q)$ is~1.  For curves $C_{65,A}$, $C_{65,B}$,
1555$C_{125,B}$ and $C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 1, 1, 2$
1556and~2 and the conjectured size of $\Sh(J,\Q) = 2, 2, 4$ and~4,
1557respectively.  We see that in each case, the (conjectured) size of
1558the odd part of $\Sh(J,\Q)$ is~1 and the 2-part is accounted for by
1559its 2-torsion.
1560
1561For the optimal quotients, we computed the value of
1562the rational number
1563$L(J,1)/(k\cdot\Omega)$. Thus we can verify exactly that
1564equation~\eqref{eqn1} holds if all of the following
1565three conditions are met:
1566a) the rank is 0, b) $\Sh(J,\Q) = \Sh(J,\Q)[2]$, and c) the
1567Manin constant $k$ is 1 or bounded away from 1. The Manin constants
1568are 1 to within the accuracy of our calculations
1569(they are defined in Section~\ref{modomega}). Thus, if
1570these can be proven to be 1 or bounded away from 1 by some
1571amount greater than our degree of accuracy ($10^{-14}$), then
1572we have a proof that they are exactly 1.
1573
1574It is also interesting to consider deficient primes.  A prime $p$ is
1575deficient with respect to a curve $C$ of genus~2, if $C$ has no
1576degree 1 rational divisor over~$\Q_{p}$.  {}From~\cite{PSt}, the
1577number of deficient primes has the same parity as $\dim \Sh(J,\Q)[2]$.
1578Curve $C_{65,A}$ has one deficient prime~$3$. Curve
1579$C_{65,B}$ has one deficient prime~$2$. Curve $C_{117,B}$ has two
1580deficient primes $3$ and~$\infty$.  The rest of the curves have no
1581deficient primes.
1582
1583Since we have found $r$ (analytic rank) independent points on each
1584Jacobian, we have a direct proof that the Mordell-Weil rank must
1585equal the analytic rank if $\dim \Sh(J,\Q)[2] = 0$.  For
1586curves $C_{65,A}$ and $C_{65,B}$, the presence of an odd number of
1587deficient primes gives us a
1588similar result.  For $C_{125,B}$ we used a $\sqrt{5}$-Selmer group
1589to get a similar result.
1590Thus, we have an independent proof of equality
1591between analytic and Mordell-Weil ranks for all curves except
1592$C_{133,A}$.
1593
1594The 2-Selmer groups have the same dimensions for the pairs
1595$C_{125,A}$, $C_{125,B}$ and $C_{133,A}$, $C_{133,B}$.  For each
1596pair, the Mordell-Weil rank is~2 for one curve and the 2-torsion of
1597the Shafarevich-Tate group has dimension~2 for the other. In
1598addition, the two Jacobians, when canonically embedded into~$J_0(N)$,
1599intersect in their 2-torsion subgroups, and one can check that their
16002-Selmer groups become equal under the identification of
1601$H^1(\Q, J_{N,A}[2])$ with $H^1(\Q, J_{N,B}[2])$ induced by the identification
1602of the 2-torsion subgroups.  Thus these are examples of the principle
1603of a visible part of a Shafarevich-Tate group' as discussed
1604in~\cite{CM}.
1605
1606\vspace{5mm}
1607\begin{center}
1608{\sc Appendix: Other Hasegawa curves}
1609\end{center}
1610
1611In Table~\ref{Hasegawa} is data concerning all 142 of Hasegawa's
1612curves in the order presented in his paper. Let us explain the
1613entries.  The first column in each set of three columns gives the
1614level, $N$. The second column gives a classification of the cusp
1615forms spanning the 2-dimensional subspace of $S_2(N)$ corresponding
1616to the Jacobian.  When that subspace is irreducible with respect to
1617the action of the Hecke algebra and is spanned by two newforms or two
1618oldforms, we write $2n$ or $2o$, respectively.  When that subspace is
1619reducible and is spanned by two oldforms, two newforms or one of
1620each, we write $oo$, $nn$ and $on$, respectively. The third column
1621contains the sign of the functional equation at the level $M$ at
1622which the cusp form is a newform. This is the negative of
1623$\epsilon_M$ (described in Section~\ref{l}).  The order of the two
1624signs in the third column agrees with that of the forms listed in the
1625second column.  We include this information for those who would like
1626to further study these curves.  The curves with $N<200$ classified as
1627$2n$ appeared already in Table~\ref{table1}.
1628
1629The smallest possible Mordell-Weil ranks corresponding to $++$, $+-$,
1630$-+$ and $--$, predicted by the first Birch and Swinnerton-Dyer
1631conjecture, are $0$, $1$, $1$ and $2$ respectively. In all cases,
1632those were, in fact, the Mordell-Weil ranks. This was determined by
1633computing 2-Selmer groups with a computer program based on
1634\cite{Sto2}.  Of course, these are cases where the first Birch and
1635Swinnerton-Dyer conjecture is already known to hold.  In the cases
1636where the Mordell-Weil rank is positive, the Mordell-Weil group has a
1637subgroup of finite index generated by degree zero divisors supported
1638on rational points with $x$-coordinates with numerators bounded by 7
1639(in absolute value) and denominators by 12 with one exception.  On
1640the second curve with $N=138$, the divisor class
1641$[(3+2\sqrt{2},80+56\sqrt{2}) + (3-2\sqrt{2},80-56\sqrt{2})-2\infty]$
1642generates a subgroup of finite index in the Mordell-Weil group.
1643
1644\vfill
1645
1646\begin{table}
1647\begin{center}
1648\begin{tabular}{|c|c|c||c|c|c||c|c|c||c|c|c||c|c|c|}
1649\hline
165022 & $oo$ & $++$ & 58 & $nn$ & $+-$ & 87 & $2o$ & $++$ & 129 & $on$ & $--$ &
1651198 & $2o$ & $+-$  \\
165223 & $2n$ & $++$ & 60 & $oo$ & $++$ & 88 & $on$ & $+-$ & 130 & $on$ & $-+$ &
1653204 & $2o$ & $+-$  \\
165426 & $nn$ & $++$ & 60 & $2o$ & $++$ & 90 & $on$ & $++$ & 132 & $oo$ & $++$ &
1655205 & $2n$ & $--$  \\
165628 & $oo$ & $++$ & 60 & $2o$ & $++$ & 90 & $oo$ & $++$ & 133 & $2n$ & $--$ &
1657206 & $2o$ & $--$  \\
165829 & $2n$ & $++$ & 62 & $2o$ & $++$ & 90 & $oo$ & $++$ & 134 & $2o$ & $--$ &
1659209 & $2n$ & $--$  \\
166030 & $on$ & $++$ & 66 & $nn$ & $++$ & 90 & $oo$ & $++$ & 135 & $on$ & $+-$ &
1661210 & $on$ & $+-$  \\
166230 & $oo$ & $++$ & 66 & $2o$ & $++$ & 91 & $nn$ & $--$ & 138 & $nn$ & $+-$ &
1663213 & $2n$ & $--$  \\
166430 & $on$ & $++$ & 66 & $2o$ & $++$ & 93 & $2n$ & $--$ & 138 & $on$ & $+-$ &
1665215 & $on$ & $--$  \\
166631 & $2n$ & $++$ & 66 & $on$ & $++$ & 98 & $oo$ & $++$ & 140 & $oo$ & $++$ &
1667221 & $2n$ & $--$  \\
166833 & $on$ & $++$ & 67 & $2n$ & $--$ & 100 & $oo$ & $++$ & 142 & $nn$ & $+-$
1669& 230 & $2o$ & $--$  \\ \hline
167035 & $2n$ & $++$ & 68 & $oo$ & $++$ & 102 & $on$ & $+-$ & 143 & $on$ & $+-$
1671& 255 & $2o$ & $--$ \\
167237 & $nn$ & $+-$ & 69 & $2o$ & $++$ & 102 & $on$ & $+-$ & 146 & $2o$ & $--$
1673& 266 & $2o$ & $--$ \\
167438 & $on$ & $++$ & 70 & $on$ & $++$ & 103 & $2n$ & $--$ & 147 & $2n$ & $--$
1675& 276 & $2o$ & $+-$ \\
167639 & $2n$ & $++$ & 70 & $2o$ & $++$ & 104 & $2o$ & $++$ & 150 & $on$ & $++$
1677& 284 & $2o$ & $+-$ \\
167840 & $on$ & $++$ & 70 & $2o$ & $++$ & 106 & $on$ & $--$ & 153 & $on$ & $+-$
1679& 285 & $on$ & $--$ \\
168040 & $oo$ & $++$ & 70 & $2o$ & $++$ & 107 & $2n$ & $--$ & 154 & $on$ & $--$
1681& 286 & $on$ & $--$ \\
168242 & $on$ & $++$ & 72 & $on$ & $++$ & 110 & $on$ & $++$ & 156 & $oo$ & $++$
1683& 287 & $2n$ & $--$ \\
168442 & $oo$ & $++$ & 72 & $oo$ & $++$ & 111 & $oo$ & $+-$ & 158 & $on$ & $--$
1685& 299 & $2n$ & $--$ \\
168642 & $on$ & $++$ & 73 & $2n$ & $--$ & 112 & $on$ & $+-$ & 161 & $2n$ & $--$
1687& 330 & $2o$ & $--$ \\
168842 & $oo$ & $++$ & 74 & $oo$ & $+-$ & 114 & $oo$ & $+-$ & 165 & $2n$ & $--$
1689& 357 & $2n$ & $--$ \\ \hline
169044 & $2o$ & $++$ & 77 & $on$ & $+-$ & 115 & $2n$ & $--$ & 166 & $on$ & $--$
1691& 380 & $2o$ & $+-$ \\
169246 & $2o$ & $++$ & 78 & $oo$ & $++$ & 116 & $2o$ & $+-$ & 167 & $2n$ & $--$
1693& 390 & $on$ & $--$ \\
169448 & $on$ & $++$ & 78 & $2o$ & $++$ & 117 & $2o$ & $++$ & 168 & $2o$ & $++$
1695& & & \\
169648 & $oo$ & $++$ & 80 & $oo$ & $++$ & 120 & $oo$ & $++$ & 170 & $2o$ & $--$
1697& & & \\
169850 & $nn$ & $++$ & 84 & $oo$ & $++$ & 120 & $on$ & $++$ & 177 & $2n$ & $--$
1699& & & \\
170052 & $oo$ & $++$ & 84 & $oo$ & $++$ & 121 & $on$ & $+-$ & 180 & $2o$ & $++$
1701& & & \\
170252 & $oo$ & $++$ & 84 & $oo$ & $++$ & 122 & $on$ & $--$ & 184 & $on$ & $+-$
1703& & & \\
170454 & $on$ & $++$ & 84 & $oo$ & $++$ & 125 & $2n$ & $--$ & 186 & $2o$ & $--$
1705& & & \\
170657 & $on$ & $+-$ & 85 & $2n$ & $--$ & 126 & $oo$ & $++$ & 190 & $on$ & $+-$
1707& & & \\
170857 & $on$ & $+-$ & 87 & $2n$ & $++$ & 126 & $on$ & $++$ & 191 & $2n$ & $--$
1709& & & \\
1710 \hline
1711\end{tabular}
1712\end{center}
1713\caption{Spaces of cusp forms associated to Hasegawa's curves}
1714\label{Hasegawa}
1715\end{table}
1716
1717\pagebreak
1718\begin{thebibliography}{99}
1719
1720\bibitem[AS]{AS}
1721  A.\ Agash\'{e} and W.A.\ Stein: Some abelian varieties with visible
1722  Shafarevich-Tate groups. Preprint, 1999.
1723\bibitem[BSD]{BSD}
1724  B.\ Birch and H.P.F.\ Swinnerton-Dyer: Notes on elliptic curves.
1725  II. {\it J. reine angew. Math.}, Vol.~{\bf  218}, 79--108, 1965. MR 31
1726  \#3419
1727\bibitem[BL]{BL}
1728  S.\ Bosch and Q.\ Liu:  Rational points of the group of components
1729  of a N\'{e}ron model. To appear in {\it Manuscripta Math}.
1730\bibitem[BLR]{BLR}
1731  S.\ Bosch, W.\ L\"{u}tkebohmert and M.\ Raynaud: N\'{e}ron models.
1732  Springer-Verlag, Berlin, 1990. MR {\bf 91i}:14034
1733\bibitem[Ca]{Ca}
1734  J.W.S.\ Cassels: Arithmetic on curves of genus 1. VIII. On conjectures
1735  of Birch and Swinnerton-Dyer.
1736  {\it J. reine angew. Math.}, Vol.~{\bf  217}, 180--199, 1965.
1737  MR 31 \#3420
1738\bibitem[CF]{CF}
1739  J.W.S.\ Cassels and E.V.\ Flynn: Prolegomena to a middlebrow
1740  arithmetic of curves of genus~2. (London Math. Soc., Lecture Note Series
1741  230),
1742  Cambridge Univ. Press, Cambridge, 1996. MR {\bf 97i}:11071
1743\bibitem[Cr1]{Cr2}
1744  J.E.\ Cremona: Abelian varieties with extra twist, cusp forms, and
1745  elliptic curves over imaginary quadratic fields,
1746  {\it J. London Math.\ Soc.\ (2)}, Vol.~{\bf 45}, 404--416, 1992.
1747  MR {\bf 93h}:11056
1748\bibitem[Cr2]{Cr}
1749  J.E.\ Cremona: Algorithms for modular elliptic curves. 2nd edition.
1750  Cambridge Univ. Press, Cambridge, 1997. MR {\bf 93m}:11053
1751\bibitem[CM]{CM}
1752  J.E.\ Cremona and B.\ Mazur:
1753  Visualizing elements in the Shafarevich-Tate group.
1754  To appear in {\it Experiment.\ Math.}
1755\bibitem[FPS]{FPS}
1756  E.V.\ Flynn, B.\ Poonen and E.F.\ Schaefer: Cycles of quadratic
1757  polynomials and rational points on a genus-two curve. {\it Duke Math.\ J.},
1758  Vol.~{\bf 90}, 435--463, 1997. MR {\bf 98j}:11048
1759\bibitem[FS]{FS}
1760  E.V.\ Flynn and N.P.\ Smart: Canonical heights on the Jacobians
1761  of curves of genus~2 and the infinite descent. {\it Acta Arith.},
1762  Vol.~{\bf 79}, 333--352, 1997. MR {\bf 98f}:11066
1763\bibitem[FM]{FM}
1764  G.\ Frey and M.\ M\"{u}ller: Arithmetic of modular curves and applications.
1765  In {\it Algorithmic algebra and number theory}, Ed.\ Matzat et al.,
1766  Springer-Verlag, Berlin, 11--48, 1999.
1767\bibitem[GZ]{GZ}
1768B.H.\ Gross and D.B.\ Zagier:
1769  Heegner points and derivatives of $L$-series.
1770  {\it Invent. Math.},
1771  Vol.~{\bf 84}, 225--320, 1986. MR {\bf 87j}:11057
1772\bibitem[Gr]{Gr}
1773  A.\ Grothendieck: Groupes de monodromie en g\'eom\'etrie alg\'ebrique.
1774    SGA 7 I, Expos\'{e} IX. (Lecture Notes in Math. Vol.
1775    {\bf 288}, 313--523.)  Berlin--Heidelberg--New York: Springer 1972.
1776    MR 50 \#7134
1777\bibitem[Ha]{Ha}
1778  R.\ Hartshorne: Algebraic geometry. (Grad.\ Texts in Math.\ 52),
1779  Springer-Verlag, New York, 1977.
1780  MR 57 \#3116
1781\bibitem[Hs]{Hs}
1782  Y.\ Hasegawa: Table of quotient curves of modular curves $X_0(N)$
1783  with genus~2. {\it Proc.\ Japan.\ Acad.}, Vol.~{\bf 71}, 235--239, 1995.
1784  MR {\bf 97e}:11071
1785\bibitem[Ko]{Ko}
1786  V.A.\ Kolyvagin: Finiteness of $E(\Q)$ and $\Sh (E,\Q)$ for a subclass
1787  of Weil curves. {\it Izv.\ Akad.\ Nauk SSSR Ser.\ Mat.}, Vol.~{\bf 52},
1788  522--540, 1988. MR {\bf 89m}:11056
1789\bibitem[KL]{KL}
1790  V.A.\ Kolyvagin and D.Y.\ Logachev: Finiteness of the Shafarevich-Tate
1791  group and the group of rational points for some modular abelian varieties.
1792  {\it Leningrad Math J.}, Vol.~{\bf 1}, 1229--1253, 1990. MR {\bf 91c}:11032
1793\bibitem[La]{La}
1794  S.\ Lang: Introduction to modular forms. Springer-Verlag, Berlin, 1976.
1795 MR 55 \#2751
1796\bibitem[Le]{Le}
1797  F.\ Lepr\'{e}vost: Jacobiennes de certaines courbes de genre 2:
1798  torsion et simplicit\'e. {\it J. Th\'eor. Nombres Bordeaux}, Vol.~{\bf 7},
1799  283--306, 1995. MR {\bf 98a}:11078
1800\bibitem[Li]{Li}
1801  Q.\ Liu: Conducteur et discriminant minimal de courbes de genre 2.
1802  {\it Compos.\ Math.}, Vol.~{\bf 94}, 51--79, 1994. MR {\bf 96b}:14038
1803\bibitem[Ma]{Ma}
1804  B.\ Mazur: Rational isogenies of prime degree (with an appendix by D.
1805  Goldfeld). {\it Invent.\ Math.}, Vol.~{\bf 44}, 129--162, 1978.
1806  MR {\bf 80h}:14022
1807\bibitem[MS]{MS}
1808  J.R.\ Merriman and N.P.\ Smart: Curves of genus~2 with good reduction
1809  away {}from 2 with a rational Weierstrass point. {\it Math.\ Proc.\ Cambridge
1810  Philos.\ Soc.}, Vol.~{\bf 114}, 203--214, 1993. MR {\bf 94h}:14031
1811\bibitem[Mi1]{Mi1}
1812  J.S.\ Milne: Arithmetic duality theorems. Academic Press, Boston, 1986.
1813  MR {\bf 88e}:14028
1814\bibitem[Mi2]{Mi2}
1815  J.S.\ Milne: Jacobian varieties. In: {\it Arithmetic geometry},
1816  Ed.\ G.\ Cornell, G. and J.H.\ Silverman,  Springer-Verlag, New York,
1817  167--212, 1986. MR {\bf 89b}:14029
1818\bibitem[NU]{NU}
1819  Y.\ Namikawa and K.\ Ueno: The complete classification of fibres in
1820  pencils of curves of genus two. {\it Manuscripta Math.}, Vol.~{\bf 9},
1821  143--186, 1973. MR 51 \#5595
1822\bibitem[PSc]{PSc}
1823  B.\ Poonen and E.F.\ Schaefer: Explicit descent for Jacobians of
1824  cyclic covers of the projective line. {\it J. reine angew. Math.}, Vol.
1825  {\bf 488}, 141--188, 1997. MR {\bf 98k}:11087
1826\bibitem[PSt]{PSt}
1827  B.\ Poonen and M.\ Stoll: The Cassels-Tate pairing on polarized
1828  abelian varieties. To appear in {\it Ann.\ Math.}
1829\bibitem[Ri]{Ri}
1830  K.\ Ribet: On modular representations of $\Gal(\Qbar/\Q)$
1831arising from modular forms. {\it Invent. math.}
1832  Vol.~{\bf 100}, 431--476, 1990. MR {\bf 91g}:11066
1833\bibitem[Sc]{Sc}
1834  E.F.\ Schaefer: Computing a Selmer group of a Jacobian using functions
1835  on the curve. {\it Math.\ Ann.}, Vol. {\bf 310}, 447-471, 1998.
1836  MR {\bf 99h}:11063
1837\bibitem[Sh]{Sh}
1838  G.\ Shimura: Introduction to the arithmetic theory of
1839automorphic functions. Princeton University Press, 1994.
1840MR {\bf 95e}:11048
1841\bibitem[Si]{Si}
1842  J.H.\ Silverman: Advanced topics in the arithmetic of elliptic curves.
1843  (Grad.\ Texts in Math.\ 151), Springer-Verlag, New York,
1844  1994. MR {\bf 96b}:11074
1845\bibitem[Ste]{Ste}
1846  W.A.\ Stein: Component groups of optimal quotients of Jacobians.
1847  Preprint, 1999.
1848\bibitem[Sto1]{Sto1}
1849  M.\  Stoll:  Two simple 2-dimensional abelian varieties
1850      defined over~$\Q$ with Mordell-Weil rank at least~$19$.
1851  {\it C. R. Acad. Sci. Paris, S\'erie I}, Vol.~{\bf 321}, 1341--1344, 1995.
1852  MR {\bf 96j}:11084
1853\bibitem[Sto2]{Sto2}
1854  M.\ Stoll: Implementing 2-descent in genus~2. Preprint.
1855\bibitem[Sto3]{Sto3}
1856  M.\ Stoll: On the height constant for curves of genus two. To appear
1857  in {\it Acta Arith}.
1858\bibitem[Sto4]{Sto4}
1859  M.\ Stoll: On the height constant for curves of genus two, II.
1860  In preparation.
1861\bibitem[Ta]{Ta}
1862  J.\ Tate: On the conjectures of Birch and Swinneron-Dyer and a geometric
1863  analog. {\it S\'{e}minaire Bourbaki}, Vol.~{\bf 306}, 1965/1966.
1864  MR 1 610977
1865\bibitem[Wal1]{Wal1}
1866J.-L.\ Waldspurger: Correspondances de Shimura,
1867      Proceedings of the International Congress of Mathematicians,
1868             Vol.~{\bf 1, 2} (Warsaw, 1983),
1869      525--531, 1984. MR {\bf 86m}:11036
1870\bibitem[Wal2]{Wal2}
1871J.-L.\ Waldspurger: Sur les coefficients de Fourier des formes
1872      modulaires de poids demi-entier.
1873      {\it J. Math. Pures Appl. (9)}, Vol.~{\bf 60},
1874375--484, 1981. MR {\bf 83h}:10061
1875\bibitem[Wan]{Wan}
1876  X.\ Wang: 2-dimensional simple factors of $J_0(N)$. {\it Manuscripta
1877  Math.}, Vol.~{\bf 87}, 179--197, 1995. MR {\bf 96h}:11059
1878\end{thebibliography}
1879
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