CoCalc Public Fileswww / papers / evidence / evidence2.tex
Author: William A. Stein
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60\begin{document}
61
62\title[Modular Jacobians]{Empirical evidence for the Birch and
63Swinnerton-Dyer conjectures for
64modular Jacobians of genus~2 curves}
65
66\author{E.\ Victor Flynn}
67\address{Department of Mathematical Sciences, University of
68Liverpool, P.O.Box 147,
69Liverpool L69 3BX, England}
70\email{evflynn@liverpool.ac.uk}
71
72\author{Franck Lepr\'{e}vost}
73\address{Universit\'{e} Grenoble I, Institut Fourier, BP 74, F-38402 Saint
74Martin d'H\{e}res Cedex, France}
75\email{leprevot@math.jussieu.fr}
76
77\author{Edward F.\ Schaefer}
78\address{Department of Mathematics and Computer Science,
79Santa Clara University, Santa Clara, CA 95053, USA}
80\email{eschaefe@math.scu.edu}
81
82\author{William A.\ Stein}
83\address{Department of Mathematics, Harvard University, One Oxford Street,
84Cambridge, MA  02138, USA}
85\email{was@math.berkeley.edu}
86
87\author{Michael Stoll}
89D\"{u}sseldorf, Germany}
90\email{stoll@math.uni-duesseldorf.de}
91
92\author{Joseph L.\ Wetherell}
93\address{Department of Mathematics, University of Southern California,
941042 W.\ 36th Place, Los Angeles, CA  90089-1113, USA}
95\email{jlwether@alum.mit.edu}
96
97\subjclass{Primary 11G40; Secondary 11G10, 11G30, 14H25, 14H40,14H45}
98\keywords{Birch and Swinnerton-Dyer conjecture, genus~2, Jacobian, modular
99abelian variety}
100
101\thanks{The first author thanks the Nuffield Foundation
102(Grant SCI/180/96/71/G) for financial support.
103The second author did some of the research at
104the Max-Planck Institut f\"ur Mathematik and
105the Technische Universit\"at Berlin.
106The third author thanks the National Security Agency (Grant
107MDA904-99-1-0013).
108The fourth author was supported by a Sarah M. Hallam fellowship.
109The fifth author did some of the research at
110the Max-Planck-Institut f\"ur Mathematik.
111The sixth author thanks the National Science Foundation
112(Grant DMS-9705959).
113The authors had useful conversations with John Cremona, Qing Liu,
114Karl Rubin and
115Peter Swinnerton-Dyer and are grateful to Xiangdong Wang and Michael
116M\"{u}ller for making data available to them.}
117
118\date{June 5, 2000}
119
120\begin{abstract}
121This paper provides empirical evidence for the Birch and
122Swinnerton-Dyer conjectures for modular Jacobians of genus~2 curves.
123The second of these conjectures relates six quantities associated to
124a Jacobian over the rational numbers.  One of these
125six quantities is
126the size of the Shafarevich-Tate group.
127Unable to compute that, we
128computed the five other quantities and solved for the last one.  In
129all 32~cases, the result is very close to an integer that is a power
130of~2.  In addition, this power of~2 agrees with the size of the
1312-torsion of the Shafarevich-Tate group, which we could compute.
132\end{abstract}
133
134\maketitle
135\markboth{FLYNN, LEPR\'{E}VOST, SCHAEFER, STEIN, STOLL, AND WETHERELL}%
136         {GENUS~2 BIRCH AND SWINNERTON-DYER CONJECTURE}
137
138\pagebreak
139
140
141\section{Introduction}
142\label{intro}
143
144\normalsize
145\baselineskip=18pt
146
147The conjectures of Birch and Swinnerton-Dyer, originally stated
148for elliptic curves over~$\Q$, have been a constant source of
149motivation for the study of elliptic curves, with the ultimate
150goal being to find a proof.
151This has resulted not only in a better
152theoretical understanding, but also in the development of better
153algorithms for computing the analytic and arithmetic
154invariants that are so intriguingly related by them. We now know
155that the first and, up to a non-zero rational factor, the
156second conjecture hold for modular elliptic curves over~$\Q$
157\footnote{It has recently been announced by
158Brueil, Conrad, Diamond and Taylor that they have extended Wiles'
159results and shown
160that all elliptic curves over~$\Q$ are modular.}
161in the
162analytic rank~0 and~1 cases (see \cite{GZ,Ko,Wal1,Wal2}).
163Furthermore,
164a number of people have provided numerical evidence for the
165conjectures for a large number of elliptic curves; see
166for example~\cite{BGZ,BSD,Ca,Cr}.
167
168By now, our theoretical and algorithmic knowledge of curves of
169genus~2 and their Jacobians has reached a state that makes it
170possible to conduct similar investigations. The Birch and
171Swinnerton-Dyer conjectures have been generalized to arbitrary
172abelian varieties over number fields by Tate~\cite{Ta}. If
173$J$ is the Jacobian of a genus~2 curve over $\Q$,
174then the first conjecture
175states that the order of vanishing of the $L$-series of the Jacobian at
176$s=1$ (the {\em analytic rank}) is equal to the Mordell-Weil rank of the
177Jacobian. The second conjecture is that
178\begin{equation} \label{eqn1}
179  \lim\limits_{s \to 1} (s-1)^{-r} L(J,s) =
180     \Omega \cdot {\rm Reg} \cdot \prod\limits_{p} c_{p}
181        \cdot \#\Sh(J,\Q ) \cdot (\#J(\Q)\tors)^{-2} \,.
182\end{equation}
183In this equation, $L(J,s)$ is the $L$-series of the Jacobian
184$J$, and $r$ is its analytic rank.  We use $\Omega$ to denote the
185integral over $J(\R)$ of a particular differential 2-form; the
186precise choice of this differential is described in
187Section~\ref{Omega}.  ${\rm Reg}$ is the regulator of $J(\Q)$.  For
188primes $p$, we use $c_{p}$ to denote the size of $J(\Q_p)/J^0(\Q_p)$,
189where $J^0(\Q_p)$ is defined in Section~\ref{Tamagawa}.  We let
190$\Sh(J,\Q)$ be the Shafarevich-Tate group of $J$ over $\Q$, and we let
191$J(\Q)\tors$ denote the torsion subgroup of $J(\Q)$.
192
193As in the case of elliptic curves, the first conjecture assumes
194that the $L$-series can be analytically continued to $s = 1$,
195and the second conjecture additionally assumes that the
196Shafarevich-Tate group is finite. Neither of these assumptions is
197known to hold for arbitrary genus~2 curves. The analytic
198continuation of the $L$-series, however, is known to exist for
199modular abelian varieties over~$\Q$, where an abelian
200variety is called {\em modular} if it is a quotient of the Jacobian~$J_0(N)$
201of the modular curve~$X_0(N)$ for some level~$N$. For simplicity,
202we will also call a genus~2 curve {\em modular} when its Jacobian is
203modular in this sense. So it is certainly a good idea to look
204at modular genus~2 curves over~$\Q$, since we then at least know that the
205statement of the first conjecture makes sense. Moreover, for many modular
206abelian varieties it is also known that the Shafarevich-Tate group
207is finite, therefore the statement of the second conjecture also
208makes sense. As it turns out, all of our examples belong to this
209class.
210An additional benefit of choosing modular genus~2 curves is
211that one can find lists of such curves in the literature.
212In fact, we have no way of verifying the Birch and Swinnerton-Dyer
213conjectures for Jacobians of genus 2 curves that are not modular, since
214there is no known way of computing the analytic rank or
215the leading coefficient of the $L$-series at $s=1$.
216
217
219Swinnerton-Dyer conjectures for such modular genus~2 curves. Since there
220is no known effective way of computing the size of the Shafarevich-Tate
221group, we computed the other five terms in equation~\eqref{eqn1}
222(in two different ways, if possible). This required several different
223algorithms, some of which were developed or improved while we were
224working on this paper. If one of these algorithms
225is already well described in the literature, then we simply cite it.
226Otherwise, we describe it here in some detail (in particular,
227algorithms for computing $\Omega$ and
228$c_p$).
229
230For modular abelian varieties associated to newforms whose
231$L$-series have analytic rank~0 or~1, the first Birch and Swinnerton-Dyer
232conjecture has been proven. In such cases, the
233Shafarevich-Tate group is also known to be finite and the second conjecture
234has been proven, up to a non-zero rational factor. This all
235follows {}from results in
236\cite{GZ,KL,Wal1,Wal2}.
237In our examples, all of the analytic
238ranks are either~0 or~1.  Thus we already know that the first
239conjecture holds.  Since the Jacobians we consider are associated to a
240quadratic conjugate pair of newforms, the analytic rank of the
241Jacobian is twice the analytic rank of either newform (see \cite{GZ}).
242
243The second Birch and Swinnerton-Dyer conjecture has not been proven
244for the cases we consider.  In order to verify equation~\eqref{eqn1},
245we computed the five terms other than $\#\Sh(J,\Q)$ and solved for
246$\#\Sh(J,\Q)$. In each case, the value is an integer to within the
247accuracy of our calculations.  This number is a power of~2, which
248coincides with the independently computed size of the 2-torsion
249subgroup of~$\Sh(J,\Q)$. Hence, we have verified the second
250Birch and Swinnerton-Dyer conjecture for our curves at least
251numerically, if we assume that the Shafarevich-Tate group consists
252of 2-torsion only. (This is an ad hoc assumption based only
253on the fact that we do not know better.) See Section~\ref{Shah} for
254circumstances under which the verification is exact.
255
256The curves are listed in Table~\ref{table1},
257and the numerical results can be found in Table~\ref{table2}.
258
259
260\section{The Curves}
261\label{curves}
262
263Each of the genus~2 curves we consider is related to the Jacobian
264$J_0(N)$ of the modular curve $X_0(N)$ for some level $N$.  When only
265one of these genus~2 curves arises {}from a given level $N$, then we
266denote this curve by $C_N$; when there are two curves coming {}from level
267$N$ we use the notation $C_{N,A}$, $C_{N,B}$.  The relationship
268of, say, $C_N$ to $J_0(N)$ depends on the source.  Briefly, {}from
269Hasegawa \cite{Hs} we obtain quotients of $X_0(N)$ and {}from Wang
270\cite{Wan} we obtain curves whose Jacobians are quotients of $J_0(N)$.
271In both cases the Jacobian $J_N$ of $C_N$ is isogenous to a
2722-dimensional factor of $J_0(N)$. (When not referring to a specific
273curve, we will typically drop the subscript $N$ {}from $J$.)
274In this way we can also associate
275$C_N$ with a 2-dimensional subspace of $S_2(N)$, the space of cusp
276forms of weight~2 for $\Gamma_0(N)$.
277
278We now discuss the precise source of the genus~2 curves we will
279consider.  Hasegawa \cite{Hs} has provided exact equations for all
280genus~2 curves which are quotients of $X_0(N)$ by a subgroup of the
281Atkin-Lehner involutions.  There are 142 such curves.  We are
282particularly interested in those where the Jacobian corresponds to a
283subspace of $S_2(N)$ spanned by a quadratic conjugate pair of
284newforms. There are 21 of these with level $N \leq 200$.  For these
285curves we will provide evidence for the second conjecture.  There are
286seven more such curves with $N > 200$.  We can classify the other
2872-dimensional subspaces into four types.  There are
2882-dimensional subspaces of oldforms that are irreducible under the
289action of the Hecke algebra.  There are also 2-dimensional subspaces
290that are reducible under the action of the Hecke algebra and are
291spanned by two oldforms, two newforms or one of each. The Jacobians
292corresponding to the latter three kinds are always isogenous, over
293$\Q$, to the product of two elliptic curves. Given the small levels,
294these are elliptic curves for which Cremona \cite{Cr} has already
295provided evidence for the Birch and Swinnerton-Dyer conjectures.  In
296Table~\ref{Hasegawa}, we describe the kind of cusp forms spanning the
2972-dimensional subspace and the signs of their functional equations
298{}from the level at which they are newforms.  The analytic and
299Mordell-Weil ranks were always the smallest possible given those signs.
300
301The second set of curves was created by Wang \cite{Wan} and is further
302discussed in \cite{FM}.  This set consists of 28 curves that were
303constructed by considering the spaces $S_2(N)$ with $N \leq 200$.
304Whenever a subspace spanned by a pair of quadratic conjugate newforms
305was found, these newforms were integrated to produce a quotient
306abelian variety~$A$ of $J_0(N)$.  These quotients are {\em optimal} in the
307sense of \cite{Ma}, in that the kernel of the quotient map is
308connected.
309
310The period matrix for~$A$ was created using certain intersection
311numbers.  When all of the intersection numbers have the same value,
312then the polarization on~$A$ induced {}from the canonical polarization
313of~$J_0(N)$ is equivalent to a principal polarization. (Two
314polarizations are {\em equivalent} if they differ by an integer multiple.)
315Conversely, every 2-dimensional optimal quotient of $J_0(N)$ in which
316the induced polarization is equivalent to a principal polarization is
317found in this way.
318
319Using theta functions, numerical approximations were found for the
320Igusa invariants of the abelian surfaces. These numbers coincide with
321rational numbers of fairly small height within the limits of the
322precision used for the computations. Wang then constructed curves
323defined over~$\Q$ whose Igusa invariants are the rational numbers
324found. (There is one abelian surface at level $N = 177$ for which Wang
325was not able to find a curve.) If we assume that these rational
326numbers are the true Igusa invariants of the abelian surfaces, then it
327follows that Wang's curves have Jacobians isomorphic, over~$\Qbar$, to
328the principally polarized abelian surfaces in his list. Since the
329classification given by these invariants is only up to isomorphism
330over~$\Qbar$, the Jacobians of Wang's curves are not necessarily
331isomorphic to, but can be twists of, the optimal quotients
332of~$J_0(N)$ over~$\Q$ (see below).
333
334There are four curves in Hasegawa's list which do not show up in
335Wang's list (they are listed in Table~\ref{table1} with an $H$ in the
336last column).  Their Jacobians are quotients of~$J_0(N)$, but are not
337optimal quotients.  It is likely that there are modular genus~2 curves
338which neither are Atkin-Lehner quotients of~$X_0(N)$ (in Hasegawa's
339sense) nor have Jacobians that are optimal quotients. These curves
340could be found by looking at the optimal quotient abelian surfaces and
341checking whether they are isogenous to a principally polarized abelian
342surface over $\Q$.
343
344For 17 of the curves in Wang's list, the 2-dimensional subspace
345spanned by the newforms is the same as that giving one of Hasegawa's
346curves.  In all of those cases, the curve given by Wang's equation is
347isomorphic, over $\Q$, to that given by Hasegawa. This verifies Wang's
348equations for these 17 curves.  They are listed in Table~\ref{table1}
349with $HW$ in the last column.
350
351The remaining eleven curves (listed in Table~\ref{table1} with a
352$W$ in the last column) derive from the other eleven optimal
353quotients in Wang's list.  These are described in more detail in
355
356With the exception of curves $C_{63}$, $C_{117,A}$ and $C_{189}$, the
357Jacobians of all of our curves are absolutely simple, and the
358canonically polarized Jacobians have automorphism groups of size two.
359We showed that these Jacobians are absolutely simple using an argument
360like those in \cite{Le,Sto1}.  The automorphism group of the
361canonically polarized Jacobian of a hyperelliptic curve is isomorphic
362to the automorphism group of the curve (see \cite[Thm.\
36312.1]{Mi2}). Each automorphism of a hyperelliptic curve is induced by
364a linear fractional transformation on $x$-coordinates (see \cite[p.\
3651]{CF}). Each automorphism also permutes the six Weierstrass
366points. Once we believed we had found all of the automorphisms, we
367were able to show that there are no more by considering all linear
368fractional transformations sending three fixed Weierstrass points to
369any three Weierstrass points. In each case, we worked with sufficient
370accuracy to show that other linear fractional transformations did not
371permute the Weierstrass points.
372
373Let $\zeta_{3}$ denote a primitive third root of unity.  The
374Jacobians of curves $C_{63}$, $C_{117,A}$ and $C_{189}$ are each
375isogenous to the product of two elliptic curves over $\Q(\zeta_3)$,
376though not over $\Q$, where they are simple.  These genus~2 curves
377have automorphism groups of size 12.  In the following table we list
378the curve at the left.  On the right we give one of the elliptic
379curves which is a factor of its Jacobian. The second factor is the
380conjugate.
381$382\begin{array}{ll} 383C_{63}: & y^2 = x(x^2 + (9 - 12\zeta_{3})x - 48\zeta_{3}) \\ 384C_{117,A}: & y^2 = x(x^2 - (12 + 27\zeta_{3})x - (48 + 48\zeta_{3})) \\ 385C_{189}: & y^2 = x^3 + (66 - 3\zeta_{3})x^2 + (342 + 81\zeta_{3})x 386 + 105 + 21\zeta_{3} 387\end{array} 388$
389Note that these three Jacobians are examples of abelian varieties
390with extra twist' as discussed in~\cite{Cr2}, where they can be
391found in the tables on page~409.
392
393\subsection{Models for the Wang-only curves}
395
396As we have already noted, a modular genus~2 curve may be found by
397either, both, or neither of Wang's and
398Hasegawa's techniques.
399Hasegawa's method allows for the exact determination, over $\Q$, of
400the equation of any modular genus~2 curve it has found.  On the other
401hand, if Wang's technique detects a modular genus~2 curve $C_N$, his
402method produces real approximations to a curve $C'_N$ which is defined
403over $\Q$ and is isomorphic to $C_N$ over $\Qbar$.  We will call
404$C'_N$ a {\em twisted modular genus~2 curve}.
405
406In this section we attempt to determine equations for the eleven
407modular genus~2 curves detected by Wang but not by Hasegawa.  If we
408assume that Wang's equations for the twisted modular genus~2 curves
409are correct, we find that we are able to determine the twists.  In
410turn, this gives us strong evidence that Wang's equations for the
411twisted curves were correct.  Undoing the twist, we determine probable
412equations for the modular genus~2 curves. We end by providing further
413evidence for the correctness of these equations.
414
415In what follows, we will use the notation of~\cite{Cr} and recommend
416it as a reference on the general results that we assume here and in
417Section~\ref{modular} and the appendix.
418Fix a level~$N$ and let
419$f(z) \in S_2(N)$.  Then $f$ has a Fourier expansion
420$f(z) = \sum\limits_{n=1}^{\infty} a_{n} e^{2 \pi i n z}\,.$
421For a newform~$f$, we have $a_1 \neq 0$; so we can normalize it by
422setting $a_1 = 1$. In our cases, the $a_n$'s are integers in a real
423quadratic field. For each prime~$p$ not dividing~$N$, the
424corresponding Euler factor of the $L$-series $L(f,s)$ is
425$1 - a_p p^{-s} + p^{1-2s}$.  Let $N(a_p)$ and $Tr(a_p)$ denote the
426norm and trace of~$a_p$.  The product of this Euler factor and its
427conjugate is
428$1 - Tr(a_p)\,p^{-s} + (N(a_p) + 2p)\,p^{-2s} 429 - p\,Tr(a_p)\,p^{-3s} + p^2\,p^{-4s}$.
430Therefore, the characteristic
431polynomial of the $p$-Frobenius on the corresponding abelian variety
432over $\F_{p}$ is
433$x^4 - Tr(a_p)\,x^3 + (N(a_p) + 2p)\,x^2 - p\,Tr(a_p)\,x + p^2$.
434Let $C$ be a curve, over $\Q$, whose Jacobian, over $\Q$, comes {}from
435the space spanned by $f$ and its conjugate.  Then we know that
436$p+1 - \#C(\F_{p}) = Tr(a_p)$ and
437$\frac{1}{2}(\#C(\F_{p})^{2} + \#C(\F_{p^2})) - (p+1)\# C(\F_{p}) - p = 438N(a_p)$ (see \cite[Lemma 3]{MS}).
439For the odd primes less than 200, not dividing $N$, we computed
440$\# C(\F_{p})$ and $\# C(\F_{p^2})$ for each curve given by one of
441Wang's equations. {}From these we could compute the characteristic
442polynomials of Frobenius and see if they agreed with those predicted
443by the $a_p$'s of the newforms.
444
445Of the eleven curves, the characteristic polynomials agreed for only
446four. In each of the remaining seven cases we found a twist of Wang's
447curve whose characteristic polynomials agreed with those predicted by
448the newform for all odd primes less than 200 not dividing $N$.  Four
449of these twists were quadratic and three were of higher degree.  It
450is these twists that appear in Table~\ref{table1}.
451
452We can provide further evidence that these equations are correct.
453For each curve given in Table~\ref{table1}, it is easy to determine
454the primes of singular reduction.  In Section~\ref{Tamagawa} we will
455provide techniques for determining which of those primes divides the
456conductor of its Jacobian.  In each case, the primes dividing the
457conductor of the Jacobian of the curve are exactly the primes
458dividing the level $N$; this is necessary.  With the exception of
459curve $C_{188}$, all the curves come {}from odd levels.  We used Liu's
460{\tt genus2reduction} program
461({\tt ftp://megrez.math.u-bordeaux.fr/pub/liu}) to compute the
462conductor of the curve. In each case (other than curve $C_{188}$),
463the conductor is the square of the level; this is also necessary. For
464curve $C_{188}$, the odd part of the conductor of the curve is the
465square of the odd part of the level.
466
467In addition, since the Jacobians of the Wang curves are optimal
468quotients, we can compute~$k\cdot\Omega$ (where $k$ is the Manin constant,
469conjectured to be 1)
470using the newforms.
471In each case, these agree (to within the accuracy of our computations)
472with the $\Omega$'s computed using the equations for the curves.
473We can also compute the value of~$c_p$ for optimal quotients from
474the newforms, when $p$ exactly divides~$N$ and the eigenvalue of the
475$p$th Atkin-Lehner involution is $-1$. When $p$ exactly divides~$N$
476and the eigenvalue of the $p$th Atkin-Lehner involution is~$+1$, the
477component group is either $0$, $\Z/2\Z$, or~$(\Z/2\Z)^2$. These results
478are always in agreement with the values computed using the equations
479for the curves. The algorithms based on the newforms are
480described in Section~\ref{modular}, those based on the
481equations of the curves are described in Section~\ref{algms}.
482
483Lastly, we were able to compute the Mordell-Weil ranks of the Jacobians
484of the curves given by ten of these eleven equations. In
485each case it agrees with the analytic rank of the Jacobian,
486as deduced {}from the newforms.
487
488It should be noted that curve~$C_{125,B}$ is the $\sqrt{5}$-twist of
489curve~$C_{125,A}$; the corresponding statement holds for the associated
4902-dimensional subspaces of~$S_2(125)$. Since curve~$C_{125,A}$ is
491a Hasegawa curve, this proves that the equation given in Table~\ref{table1}
492for curve~$C_{125,B}$ is correct.
493
494The $a_p$'s and other information concerning Wang's curves are
495currently kept in a database at the Institut f\"{u}r experimentelle
496Mathematik in Essen, Germany.  Most recently, this database was under
497the care of Michael M\"{u}ller.  William Stein also keeps a database
498of~$a_p$'s for newforms.
499
500\begin{remark}
501For the remainder of this paper we will assume that the equations for
502the curves given in Table~\ref{table1} are correct; that is, that
503they are equations for the curves whose Jacobians are isogenous
504to a factor of~$J_0(N)$ in the way described above.
505Some of the quantities can be computed either {}from the newform
506or {}from the equation for the curve.  We performed both computations
507whenever possible, and view this duplicate effort as an attempt to
508verify our implementation of the algorithms rather than an attempt
509to verify the equations in Table~\ref{table1}.  For most quantities,
510one method or the other is not guaranteed to produce a value; in this
511case, we simply quote the value {}from whichever method did succeed.
512The reader who is disturbed by this philosophy should
513ignore the Wang-only curves, since the equations for the Hasegawa
514curves can be proven to be correct.
515\end{remark}
516
517
518\section{Algorithms for genus~2 curves}
519\label{algms}
520
521In this section, we describe the algorithms that are based on the
522given models for the curves. We give algorithms that compute all
523terms on the right hand side of equation~\eqref{eqn1}, with the
524exception of the size of the Shafarevich-Tate group. We describe,
525however, how to find the size of its 2-torsion subgroup. Note that these
526algorithms are for general genus 2 curves and do not depend on modularity.
527
528\subsection{Torsion Subgroup}
529\label{torsion}
530
531The computation of the torsion subgroup of~$J(\Q)$ is straightforward.
532We used the technique described in~\cite[pp.~78--82]{CF}.
533This technique is not always effective, however. For an algorithm working
534in all cases see~\cite{Sto3}.
535
536\subsection{Mordell-Weil rank and $\Sh(J,\Q)[2]$}
537\label{MW}
538
539The group $J(\Q)$ is a finitely generated abelian group and so is
540isomorphic to $\Z^{r} \oplus J(\Q)\tors$ for some $r$ called the
541Mordell-Weil rank.
542As noted above (see Section~\ref{intro}), we justifiably use
543$r$ to denote both the analytic and Mordell-Weil ranks since they
544agree for all curves in Table~\ref{table1}.
545
546We used the algorithm described in \cite{FPS} to compute ${\rm 547Sel}^{2}_{\rm fake}(J,\Q)$ (notation {}from \cite{PSc}), which is a
548quotient of the 2-Selmer group ${\rm Sel}^{2}(J,\Q)$. More details
549on this algorithm can be found in \cite{Sto2}.  Theorem 13.2 of
550\cite{PSc} explains how to get ${\rm Sel}^{2}(J,\Q)$ {}from ${\rm 551Sel}^{2}_{\rm fake}(J,\Q)$.  Let $M[2]$ denote the 2-torsion of an
552abelian group $M$ and let dim$V$ denote the dimension of an $\F_{2}$
553vector space $V$.  We have
554$\dim {\rm Sel}^{2}(J,\Q) = r + \dim J(\Q)[2] + \dim \Sh(J,\Q)[2]$.
555In other words,
556$\dim\, \Sh (J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2].$
557
558It is interesting to note that in all 30 cases where
559$\dim \Sh(J,\Q)[2] \le 1$, we were able to compute the Mordell-Weil rank
560independently from the analytic rank.
561The
562cases where $\dim \Sh(J,\Q)[2] = 1$ are discussed in more
563detail in Section~\ref{Shah}.
564For both of the remaining cases we have $\dim \Sh(J,\Q)[2]=2$.
565One of these cases is
566$C_{125,B}$. For this curve we computed
567${\rm Sel}^{\sqrt{5}}(J_{125,B},\Q)$
568using the technique described in
569\cite{Sc}. {}From this, we were able to determine that the Mordell-Weil
570rank is 0 independently from the analytic rank.
571For the other case,
572$C_{133,A}$,
573we could show that $r$ had to be either~0
574or~2 {}from the equation, but we needed the analytic computation to
575show that $r=0$.
576
577\subsection{Regulator}
578\label{reg}
579
580When the Mordell-Weil rank is~0, then the regulator is~1. When the
581Mordell-Weil rank is positive, then to compute the regulator, we
582first need to find generators for $J(\Q)/J(\Q)\tors$. The regulator
583is the determinant of the canonical height pairing matrix on this set
584of generators. An algorithm for computing the generators and
585canonical heights is given in~\cite{FS}; it was used to find
586generators for $J(\Q)/J(\Q)\tors$ and to compute the regulators.  In
587that article, the algorithm for computing height constants at the
588infinite prime is not clearly explained and there are some errors in
589the examples. A clear algorithm for computing infinite height
590constants is given in~\cite{Sto3}. In~\cite{Sto4}, some improvements of
591the results and algorithms in~\cite{FS} and~\cite{Sto3} are discussed.
592The regulators in Table~\ref{table2} have been double-checked using
593these improved algorithms.
594
595\subsection{Tamagawa Numbers}
596\label{Tamagawa}
597
598Let $\OO$ be the integer ring in~$K$ which will be $\Q_{p}$ or
599$\Q_{p}\unr$ (the maximal unramified extension of $\Q_{p})$.
600Let $\JJ$ be the N\'{e}ron model of~$J$ over~$\OO$.
601Define $\JJ^{0}$ to be the open subgroup scheme of~$\JJ$ whose
602generic fiber is isomorphic to~$J$ over~$K$ and whose special fiber
603is the identity component of the closed fiber of~$\JJ$.
604The group $\JJ^{0}(\OO)$ is isomorphic to a subgroup of~$J(K)$ which
605we denote $J^{0}(K)$. The group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is
606the component group of~$\JJ$ over~$\OO_{\Q_{p}\unr}$.  We are
607interested in computing $c_p = \#J(\Q_{p})/J^{0}(\Q_{p})$, which is
608sometimes called the Tamagawa number.
609Since N\'{e}ron models are stable under unramified base extension,
610the $\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of
611$J^{0}(\Q_{p}\unr)$ is~$J^{0}(\Q_{p})$.
612Since $H^1(\Gal(\Q_{p}\unr/\Q_{p}), J^{0}(\Q_{p}\unr))$
613is trivial (see~\cite[p.\ 58]{Mi1}) we see that the
614$\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of
615$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is $J(\Q_{p})/J^{0}(\Q_{p})$.
616
617There exist several discussions in the literature on constructing the
618group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ starting with an integral
619model of the underlying curve.  For our purposes, we especially
620recommend Silverman's book~\cite{Si}, Chapter~IV, Sections 4 and~7.
621For a more detailed treatment, see~\cite[chap.\ 9]{BLR} and~\cite[\S 2]{Ed2}.
622One can find justifications for what we will do in these sources. While
623constructing such groups, we ran into a number of difficulties that
624we did not find described anywhere. For that reason, we will present
625examples of such difficulties that arose, as well as our methods of
626resolution.  We do not claim that we will describe all situations
627that could arise.
628
629When computing $c_p$ we need a proper, regular model~$\CC$ for~$C$
630over~$\Z_p$.  Let $\Z_p\unr$ denote the ring of integers of~$\Q_p\unr$
631and note that $\Z_p\unr$ is a pro-\'etale Galois extension
632of~$\Z_p$ with Galois group
633$\Gal(\Z_p\unr/\Z_p) = \Gal(\Q_p\unr/\Q_p)$.
634It follows that giving a model for~$C$ over~$\Z_p$ is equivalent to
635giving a model for~$C$ over~$\Z_p\unr$ that
636is equipped with a Galois action.  We have found it convenient to
637always work with the latter description.  Thus for us, giving a model
638over~$\Z_p$ will always mean giving a model over~$\Z_p\unr$ together
639with a Galois action.
640
641In order to find a proper, regular model for~$C$ over~$\Z_p$,
643consider the curves to be the two affine pieces $y^2+g(x)y=f(x)$ and
644$v^2 + u^3 g(1/u)v = u^6 f(1/u)$, glued together by $ux=1$, $v=u^3y$.
645We blow them up at all points that are not regular until we have a
646regular model.  (A point is {\em regular} if the cotangent space there has
647two generators.)  These curves are all proper, and this is not
648affected by blowing up.
649
650Let $\CC_p$ denote the special fiber of~$\CC$ over~$\Z_p\unr$.  The
651group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is isomorphic to a quotient
652of the degree~0 part of the free group on the irreducible components
653of~$\CC_{p}$. Let the irreducible components be denoted $\DD_{i}$ for
654$1\leq i\leq n$, and let the multiplicity of~$\DD_{i}$ in~$\CC_p$ be
655$d_{i}$.  Then the degree~0 part of the free group has the form
656$L = \{ \sum\limits_{i=1}^{n} \alpha_{i}\DD_{i} \mid 657 \sum\limits_{i=1}^{n} d_{i}\alpha_{i} = 0 \}\,.$
658
659In order to describe the group that we quotient out by, we must
660discuss the intersection pairing.  For components $\DD_{i}$ and~$\DD_{j}$
661of the special fiber, let $\DD_{i} \cdot \DD_{j}$ denote
662their intersection pairing. In all of the special fibers that arise
663in our examples, distinct components intersect transversally.  Thus,
664if $i \neq j$, then $\DD_{i} \cdot \DD_{j}$ equals the number of points
665at which $\DD_{i}$ and $\DD_{j}$ intersect.  The case of
666self-intersection ($i=j$) is discussed below.
667
668The kernel of the map {}from~$L$ to
669$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is generated by
670divisors of the form
671$[\DD_j] = \sum\limits_{i=1}^{n} (\DD_{j} \cdot \DD_{i}) \DD_{i}$
672for each component~$\DD_j$.  We can deduce $\DD_{j} \cdot \DD_{j}$ by
673noting that $[\DD_j]$ must be contained in the group~$L$. This follows
674{}from the fact that the intersection pairing of
675$\CC_{p} = \sum d_i\DD_{i}$ with any irreducible component is 0.
676
677\vspace{1mm}
678\noindent
679{\bf Example 1.} Curve $C_{65,B}$ over $\Z_{2}$.
680
681The Jacobian
682of $C_{65,B}$
683is a quotient of the Jacobian of~$X_0(65)$.
684Since 65 is odd, $J_0(65)$ has good reducation at~2; however,
685$C_{65,B}$ has singular
686reduction at~2.  Since the equation for this curve
687is conjectural (it is a Wang-only curve), it will be nice to verify
688that 2 does not divide the conductor of its Jacobian, i.e.\ that the
689Jacobian has good reduction at~2.  In addition, we will need a
690proper, regular model for this curve in order to find~$\Omega$.
691
692We start with the arithmetic surface over~$\Z_{2}\unr$ given by the
693two pieces
694$y^2 = f(x) = -x^6 + 10x^5 - 32x^4 + 20x^3 + 40x^2 + 6x - 1$ and
695$v^2 = u^6 f(1/u)$. (Here and in the following we will not specify the
696gluing maps.) This arithmetic surface is regular at $u=0$ so we
697focus our attention on the first affine piece. The special fiber of
698$y^2 = f(x)$ over~$\Z_{2}\unr$
699is given by
700$(y + x^3 + 1)^2 = 0 \pmod 2$; this is a genus~0 curve of multiplicity~2
701that we denote~$A$. This model is not regular at the two points
702$(x-\alpha, y, 2)$, where $\alpha$ is a root of $x^2 - 3x - 1$.
703The current special fiber is in Figure~\ref{special2} and is labelled
704{\it Fiber~1}.
705
706We fix $\alpha$ and move $(x - \alpha, y, 2)$ to the origin using the
707substitution $x_0 = x-\alpha$. We get
708$y^2 = -x_0^6 + (-6\alpha + 10)x_0^5 + (5\alpha - 47)x_0^4 709 + (-28\alpha + 60)x_0^3 + (-11\alpha - 2)x_0^2 710 + (-24\alpha - 16)x_0 711$
712which we rewrite as the pair of equations
713\begin{align*}
714    g_{1}(x_{0},y,p)
715      &= -x_0^6 + (-3\alpha + 5) p x_0^5 + (5\alpha - 47) x_0^4
716           + (-7\alpha + 15) p^2 x_0^3 \\
717      & \qquad {} + (-11\alpha - 2) x_0^2 + (-3\alpha - 2) p^3 x_0 - y^2
718         \\
719      &= 0,\\
720    p &= 2.
721\end{align*}
722To blow up at $(x_0,y,p)$, we introduce projective coordinates
723$(x_1,y_1,p_1)$ with $x_{0} y_1 = x_{1} y$, $x_{0} p_{1} = x_{1} p$, and
724$y p_1 = y_{1} p$. We look in three affine pieces that cover the blow-up
725of $g_1(x_{0},y,p)=0,$ $p=2$
726and check for regularity.
727
728\begin{description}
729\item[$x_{1} = 1$] We have $y = x_{0} y_{1}$, $p = x_{0} p_{1}$. We get
730  $g_2(x_{0},y_{1},p_{1}) = 0$, $x_{0} p_{1} = 2$, where
731  \begin{align*}
732     g_2(x_{0},y_{1},p_{1}) &= x_{0}^{-2}g_{1}(x_{0},x_{0}y_{1},x_{0}p_{1}) \\
733         &= -x_0^4 + (-3\alpha + 5) p_1 x_0^4 + (5\alpha - 47) x_0^2
734             + (-7\alpha + 15) p_1^2 x_0^3 \\
735	 & \qquad{} + (-11\alpha - 2) + (-3\alpha - 2) p_1^3 x_0^2 - y_1^2 \,.
736  \end{align*}
737  In the reduction we have either $x_{0} = 0$ or $p_1 = 0$.
738  \begin{description}
739    \item[$x_{0} = 0$] $(y_{1} + \alpha + 1)^2 = 0$.
740      This is a new component which we denote $B$. It has genus~0 and
741      multiplicity~2. We check regularity along~$B$ at
742      $(x_{0}, y_{1} + \alpha + 1, p_{1}-t, 2)$, with $t$ in $\Z_2\unr$, and
743      find that $B$ is nowhere regular.
744    \item[$p_{1} = 0$]
745      $(y_{1} + x_{0}^2 + \alpha x_{0} + (\alpha + 1))^2 = 0$.
746      Using the gluing maps, we see that this is~$A$.
747  \end{description}
748
749\item[$y_{1} = 1$] We get no new information {}from this affine piece.
750
751\item[$p_{1} = 1$] We have $x_{0} = x_{1} p$, $y = y_{1} p$. We get
752  $g_{3}(x_{1},y_{1},p) = p^{-2} g_{1}(x_{1}p,y_{1}p,p) = 0$, $p = 2$.
753  In the reduction we have
754  \begin{description}
755    \item[$p=0$] $(y_1 + (\alpha+1)x_1)^2 = 0$. Using the gluing maps, we
756      see that this is~$B$. It is nowhere regular.
757  \end{description}
758\end{description}
759
760The current special fiber is in
761Figure~\ref{special2} and is labelled {\it Fiber~2}. It is not regular
762along~$B$ and at the other point on~$A$ which we have not yet blown up.
763The component $B$ does not lie entirely in any one affine piece
764so we will blow up the affine pieces $x_1 = 1$ and $p_1 = 1$ along~$B$.
765
766To blow up $x_1 = 1$ along~$B$ we make the substitution
767$y_2 = y_1 + \alpha + 1$ and replace each factor of~2 in a coefficient
768by~$x_0 p_1$. We have $g_{4}(x_0,y_2,p_1) = 0$ and $x_0 p_1 = 2$, and we
769want to blow up along the line $(x_0, y_2, 2)$.  Blowing up along a line
770is similar to blowing up at a point: since we are blowing up at
771$(x_0, y_2, 2) = (x_0, y_2)$, we introduce projective
772coordinates $x_3, y_3$ together with the relation $x_0 y_3 = x_3 y_2$.  We
773consider two affine pieces that cover the blow-up of $x_1 = 1$.
774
775\begin{description}
776  \item[$x_3 = 1$] We have $y_2 = y_{3} x_{0}$. We get
777    $g_{5}(x_{0},y_{3},p_{1}) = x_{0}^{-2} g_{4}(x_{0},y_{3}x_{0},p_1) = 0$
778    and $x_{0} p_{1} = 2$. In the reduction we have
779    \begin{description}
780      \item[$x_{0} = 0$]
781        $y_{3}^2 + (\alpha + 1) y_{3} p_{1} + \alpha p_{1}^3 + p_{1}^2 782 + \alpha + 1 = 0$.
783	This is~$B$. It is now a non-singular genus~1 curve.
784      \item[$p_{1} = 0$] $(x_0 + y_3 + \alpha)^2 = 0$. This is~$A$. The point
785        where $B$ meets~$A$ transversally is regular.
786    \end{description}
787
788  \item[$y_3 = 1$] We get no new information {}from this affine piece.
789\end{description}
790
791When we blow up $p_1 = 1$ along~$B$ we get essentially the same thing and
792all points are again regular.
793
794The other non-regular point on~$A$ is the conjugate of the one we
795blew up. Therefore, after performing the conjugate blow ups, it too
796will be a genus~1 component crossing~$A$ transversally. We denote
797this component $D$; it is conjugate to~$B$.
798
799
800\begin{figure}
801\caption{Special fibers of curve $C_{65,B}$ over $\Z_{2}$;
802         points not regular are thick}
803\label{special2}
804\begin{picture}(400,130)
805  \put(20,5){\begin{picture}(100,125)
806	       \thinlines
807	       \put(20,55){\line(1,0){60}}
808	       \put(85,55){\makebox(0,0){A}}
809	       \put(75,62){\makebox(0,0){2}}
810	       \put(40,55){\circle*{5}}
811	       \put(60,55){\circle*{5}}
812	       \put(50,5){\makebox(0,0){Fiber 1}}
813	     \end{picture}}
814  \put(145,5){\begin{picture}(100,125)
815		\thinlines
816		\put(50,5){\makebox(0,0){Fiber 2}}
817		\put(20,55){\line(1,0){60}}
818		\put(85,55){\makebox(0,0){A}}
819		\put(75,62){\makebox(0,0){2}}
820		\put(60,55){\circle*{5}}
821		\put(40,15){\line(0,1){80}}
822		\put(40.5,15){\line(0,1){80}}
823		\put(39.5,15){\line(0,1){80}}
824		\put(39,15){\line(0,1){80}}
825		\put(41,15){\line(0,1){80}}
826		\put(40,105){\makebox(0,0){B}}
827		\put(34,90){\makebox(0,0){2}}
828	      \end{picture}}
829  \put(270,5){\begin{picture}(100,125)
830		\thinlines
831		\put(20,55){\line(1,0){60}}
832		\put(85,55){\makebox(0,0){A}}
833		\put(75,62){\makebox(0,0){2}}
834		\put(40,15){\line(0,1){80}}
835		\put(40,105){\makebox(0,0){B}}
836		\put(60,15){\line(0,1){80}}
837		\put(60,105){\makebox(0,0){D}}
838		\put(50,5){\makebox(0,0){Fiber 3}}
839              \end{picture}}
840\end{picture}
841\end{figure}
842
843We now have a proper, regular model~$\CC$ of~$C$ over~$\Z_2$.
844Let $\CC_2$ be the special fiber of this model; a
845diagram of~$\CC_2$ is in Figure~\ref{special2} and is labelled
846{\it Fiber~3}.  We can use $\CC$ to show that the
847N\'eron model $\JJ$ of the Jacobian $J = J_{65,B}$ has good
848reduction at~2.
849
850We know that the reduction of~$\JJ^0$ is the extension of an abelian
851variety by a connected linear group.  Since $\CC$ is regular and
852proper, the abelian variety part of the reduction is the product of
853the Jacobians of the normalizations of the components of~$\CC_2$ (see
854\cite[9.3/11 and 9.5/4]{BLR}).  Thus, the abelian variety part is the
855product of the Jacobians of~$B$ and~$D$.  Since this is
8562-dimensional, the reduction of~$\JJ^0$ is an abelian variety.  In
857other words, since the sum of the genera of the components of the
858special fiber is equal to the dimension of~$J$, the reduction is an
859abelian variety.  It follows that $\JJ$ has good reduction at~2, that
860the conductor of~$J$ is odd, and that $c_2 = 1$.  As noted above, this
861gives further evidence that the equation given in Table~\ref{table1}
862is correct.
863
864
865\vspace{1mm}
866\noindent
867{\bf Example 2.} Curve $C_{63}$ over $\Z_{3}$.
868
869The Tamagawa number is often found using the intersection matrix and
870sub-determinants. This is not entirely satisfactory for cases where
871the special fiber has several components and a non-trivial Galois
872action. Here is an example of how to resolve this (see also~\cite{BL}).
873
874When we blow up curve~$C_{63}$ over~$\Z_{3}\unr$, we get
875the special fiber shown in Figure~\ref{special1}.
876Elements of $\Gal(\Q_{3}\unr/\Q_{3})$
877that do not fix the quadratic unramified extension of~$\Q_{3}$
878switch $H$ and~$I$. The other components are defined over~$\Q_{3}$.
879All components have genus~0. The group $J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$
880is isomorphic to a quotient of
881\begin{align*}
882  L = \{ \alpha A + \beta B + \delta D + \epsilon E + \phi F + \gamma G
883          &+ \eta H + \iota I \\
884          &\mid \alpha + \beta + 2\delta + 2\epsilon + 4\phi + 2\gamma
885	         + 2\eta +  2\iota = 0 \} \,.
886\end{align*}
887
888The kernel is generated by the following divisors.
889\begin{center}
890  \begin{tabular}{*{2}{@{[}c@{]$\;=\;$}r@{\hspace{2cm}}}}
891    $A$ & $-2A + E$ &            $B$ & $-2B + E$ \\
892    $D$ & $-D + E$ &             $E$ & $A + B + D - 4E + F$  \\
893    $F$ & $E - 2F + G + H + I$ & $G$ & $F - 2G$ \\
894    $H$ & $F - 2H$ &             $I$ & $F - 2I$
895  \end{tabular}
896\end{center}
897
898\begin{figure}
899\caption{Special fiber of curve $C_{63}$ over $\Z_{3}$}
900\label{special1}
901\begin{picture}(400,130)
902  \put(100,5){\begin{picture}(200,125)
903		\thinlines
904		\put(20,50){\line(1,0){160}}
905		\put(40,20){\line(0,1){60}}
906		\put(60,20){\line(0,1){60}}
907		\put(80,20){\line(0,1){60}}
908		\put(150,10){\line(0,1){100}}
909		\put(120,70){\line(1,0){60}}
910		\put(120,90){\line(1,0){60}}
911		\put(120,30){\line(1,0){60}}
912		\put(40,88){\makebox(0,0){G}}
913		\put(60,88){\makebox(0,0){H}}
914		\put(80,88){\makebox(0,0){I}}
915		\put(150,118){\makebox(0,0){E}}
916		\put(185,50){\makebox(0,0){F}}
917		\put(185,90){\makebox(0,0){A}}
918		\put(185,70){\makebox(0,0){B}}
919		\put(185,30){\makebox(0,0){D}}
920		\put(35,70){\makebox(0,0){2}}
921		\put(55,70){\makebox(0,0){2}}
922		\put(75,70){\makebox(0,0){2}}
923		\put(165,55){\makebox(0,0){4}}
924		\put(165,35){\makebox(0,0){2}}
925		\put(145,104){\makebox(0,0){2}}
926              \end{picture}}
927\end{picture}
928\end{figure}
929
930When we project away {}from~$A$, we find that
931$J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$ is isomorphic to
932\begin{align*}
933  \langle B, D, E, F, G, H, I
934            &\mid E = 0, E = 2B, D = E, 4E = B + D + F, \\
935            &\quad 2F = E + G +  H + I, F = 2G = 2H = 2I \rangle.
936\end{align*}
937At this point, it is straightforward to simplify the representation by
938elimination. Note that we projected away {}from~$A$, which is
939Galois-invariant. It is best to continue eliminating Galois-invariant
940elements first. We find that this group is isomorphic to
941$\langle H, I \mid 2H = 2I = 0 \rangle$ and elements of
942$\Gal(\Q_{3}\unr/\Q_{3})$ that do not fix the quadratic unramified
943extension of~$\Q_{3}$ switch $H$ and~$I$. Therefore
944$J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr) \cong \Z/2\Z \oplus \Z/2\Z$ and
945$c_3 = \#J(\Q_{3})/J^{0}(\Q_{3}) = 2$.
946
947\subsection{Computing $\Omega$}
948\label{Omega}
949
950By an {\em integral differential} (or {\em integral form}) on $J$ we mean the
951pullback to $J$ of a global relative differential form on the N\'eron
952model of $J$ over $\Z$.  The set of integral $n$-forms on $J$ is a
953full-rank lattice in the vector space of global holomorphic $n$-forms
954on $J$.  Since $J$ is an abelian variety of dimension 2, the integral
9551-forms are a free $\Z$-module of rank 2 and the integral 2-forms are
956a free $\Z$-module of rank 1. Moreover, the wedge of a basis for the
957integral 1-forms is a generator for the integral 2-forms.  The
958quantity $\Omega$ is the integral, over the real points of $J$, of a
959generator for the integral 2-forms.  (We choose the generator that
961
962We now translate this into a computation on the curve $C$.  Let
963$\{\omega_1, \omega_2\}$ be a $\Q$-basis for the holomorphic
964differentials on $C$ and let $\{\gamma_1, \gamma_2, \gamma_3, 965\gamma_4\}$ be a $\Z$-basis for the homology of $C(\C)$.  Create a
966$2\times 4$ complex matrix $M_{\C} = [ \int_{\gamma_j}\omega_i]$ by
967integrating the differentials over the homology and let $M_{\R} = 968\Tr_{\C/\R}(M_{\C})$ be the $2\times 4$ real matrix whose entries are
969traces {}from the complex matrix.  The columns of $M_{\R}$ generate a
970lattice $\Lambda$ in $\R^2$.  If we make the standard identification
971between the holomorphic 1-forms on $J$ and the holomorphic
972differentials on $C$ (see \cite{Mi2}), then the notation
973$\int_{J(\R)} |\omega_1 \wedge \omega_2|$ makes sense and its value
974can be computed as the area of a fundamental domain for $\Lambda$.
975
976If $\{\omega_1, \omega_2\}$ is a basis for the integral 1-forms on
977$J$, then $\int_{J(\R)} |\omega_1 \wedge \omega_2| = \Omega$.  On the
978other hand, the computation of $M_{\C}$ is simplest if we choose
979$\omega_1 = dX/Y$, and $\omega_2=X\,dX/Y$ with respect to a model for
980$C$ of the form $Y^2=F(X)$; in this case we obtain $\Omega$ by a
981simple change-of-basis calculation.  This assumes, of course, that we
982know how to express a basis for the integral 1-forms in terms of the
983basis $\{\omega_1, \omega_2\}$; this is addressed in more detail
984below.
985
986It is worth mentioning an alternate strategy.  Instead of finding a
987$\Z$-basis for the homology of $C(\C)$ one could find a $\Z$-basis
988$\{\gamma'_1, \gamma'_2\}$ for the subgroup of the homology that is
989fixed by complex conjugation (call this the real homology).
990Integrating would give us a $2\times 2$ real matrix $M'_{\R}$ and the
991determinant of $M'_{\R}$ would equal the integral of $\omega_1 992\wedge \omega_2$ over the connected component of $J(\R)$.
993In other words, the number of real connected components of $J$ is
994equal to the index of the $\C/\R$-traces in the real homology.
995
996We now come to the question of determining the differentials on $C$
997which correspond to the integral 1-forms on $J$.  Call these the
998integral differentials on $C$.  This computation can be done one
999prime at a time.  At each prime $p$ this is equivalent to determining
1000a $\Z_p\unr$-basis for the global relative differentials on any
1001proper, regular model for $C$ over $\Z_p\unr$.  In fact a more
1002general class of models can be used; see the discussion of models
1003with rational singularities in \cite[\S 6.7]{BLR} and \cite[\S
10044.1]{Li}.
1005
1006We start with the model $y^2 + g(x)y=f(x)$ given in
1007Table~\ref{table1}.  Note that the substitution $X=x$ and $Y=2y+g(x)$
1008gives us a model of the form $Y^2=F(X)$.  For integration purposes,
1009our preferred differentials are $dX/Y=dx/(2y+g(x))$ and
1010$X\,dX/Y=x\,dx/(2y+g(x))$.  It is not hard to show that at primes of
1011non-singular reduction for the $y^2 + g(x)y=f(x)$ model, these
1012differentials will generate the integral 1-forms.  For each prime $p$
1013of singular reduction we give the following algorithm.  All steps
1014take place over $\Z_p\unr$.
1015
1016\begin{description}
1017  \item[Step 1]
1018    Compute explicit equations for a proper, regular model $\CC$.
1019
1020  \item[Step 2]
1021    Diagram the configuration of the special fiber of $\CC$.
1022
1023  \item[Step 3] (Optional)
1024    Identify exceptional components and blow them down in the
1025    configuration diagram. Repeat step 3 as necessary.
1026
1027  \item[Step 4] (Optional)
1028    Remove components with genus 0 and self-intersection $-2$.
1029    Since $C$ has genus greater than 1,
1030    there will be a component that is not of this kind.
1031    (This
1032    step corresponds to contracting the given components to create a
1033    non-proper model with rational singularities.  We will not need a
1034    diagram of the resulting configuration.)
1035
1036  \item[Step 5]
1037    Determine a $\Z_p\unr$-basis for the integral differentials.  It
1038    suffices to check this on a dense open subset of each surviving
1039    component.  Note that we have explicit equations for a dense open
1040    subset of each of these components {}from the model $\CC$ in step 1.  A
1041    pair of differentials $\{\eta_1, \eta_2\}$ will be a basis for the
1042    integral differentials (at $p$) if the following three statements are
1043    true.
1044    \begin{description}
1045      \item[a]
1046        The pair $\{\eta_1, \eta_2\}$ is a basis for the holomorphic
1047        differentials on $C$.
1048      \item[b]
1049        The reductions of $\eta_1$ and $\eta_2$ produce well-defined
1050        differentials mod $p$ on an open subset of each surviving component.
1051      \item[c]
1052        If $a_1\eta_1+a_2\eta_2 = 0 \pmod{p}$ on all surviving components,
1053	then $p|a_1$ and $p|a_2$.
1054    \end{description}
1055\end{description}
1056
1057Techniques for explicitly computing a proper, regular model are
1058discussed in Section~\ref{Tamagawa}.  A configuration diagram should
1059include the genus, multiplicity and self-intersection number of
1060each component and the number and type of intersections between
1061components.  Note that when an exceptional component is blown down,
1062all of the self-intersection numbers of the components intersecting
1063it will go up (towards 0).  In particular, components which were not
1064exceptional before may become exceptional in the new configuration.
1065
1066Steps 3 and 4 are intended to make this algorithm more efficient for
1067a human.  They are entirely optional.  For a computer implementation
1068it may be easier to simply check every component than to worry about
1069manipulating configurations.
1070
1071The curves in Table~\ref{table1} are given as $y^2 + g(x)y=f(x)$.  We
1072assumed, at first, that $dx/(2y+g(x))$ and $x\,dx/(2y+g(x))$ generate
1073the integral differentials.  We integrated these differentials around
1074each of the four paths generating the complex homology and found a
1075provisional $\Omega$. Then we checked the proper, regular models to
1076determine if these differentials really do generate the integral
1077differentials and adjusted $\Omega$ when necessary.  There were
1078three curves where we needed to adjust $\Omega$.  We describe the
1079adjustment for curve $C_{65,B}$ in the following example.  For curve
1080$C_{63}$, we used the differentials $3\,dx/(2y+g(x))$ and
1081$x\,dx/(2y+g(x))$.  For curve $C_{65,A}$, we used the differentials
1082$3\,dx/(2y+g(x))$ and $3x\,dx/(2y+g(x))$.
1083
1084\vspace{2mm}
1085\noindent
1086{\bf Example 3.} Curve $C_{65,B}$.
1087
1088The primes of singular reduction for curve $C_{65,B}$ are 2, 5 and
108913.  In Example 1 of Section~\ref{Tamagawa}, we found a proper,
1090regular model $\CC$ for $C$ over $\Z_2\unr$.  The configuration for
1091the special fiber of $\CC$ is sketched in Figure~\ref{special2} under
1092the label {\it Fiber 3}.  Component $A$ is exceptional and can be
1093blown down to produce a model in which $B$ and $D$ cross
1094transversally.  Since $B$ and $D$ both have genus 1, we cannot
1095eliminate either of these components.  Furthermore, it suffices to
1096check $B$, since $D$ is its Galois conjugate.
1097
1098To get {}from the equation of the curve listed in Table~\ref{table1}
1099to an affine containing an open subset of $B$ we need to make the
1100substitutions $x=x_0 - \alpha$ and $y=x_0 (y_{3}x_0 - \alpha - 1)$.
1101We also have $x_{0}p_{1}=2$.  Using the substitutions and the
1102relation $dx_{0}/x_0 = -dp_{1}/p_1$, we get
1103$\frac{dx}{2y} = \frac{-dp_1}{2p_1(y_3 x_0 - \alpha - 1)} 1104 \text{\quad and\quad} 1105 \frac{x\,dx}{2y} 1106 = \frac{-(x_0 + \alpha)\,dp_1}{2p_1(y_3 x_0 - \alpha - 1)} \,. 1107$
1108Note that $p_1 - t$ is a uniformizer at $p_1 = t$ almost everywhere
1109on~$B$.  When we multiply each differential by~2, then the
1110denominator of each is almost everywhere non-zero; thus, $dx/y$ and
1111$x\,dx/y$ are integral at~$2$.  Moreover, although the linear
1112combination $(x-\alpha)\,dx/y$ is identically zero on~$B$, it is not
1113identically zero on~$D$ (its Galois conjugate is not identically zero
1114on~$B$).  Thus, our new basis is correct at~2.  We multiply the
1115provisional $\Omega$ by~4 to get a new provisional $\Omega$ which is
1116correct at~$2$.
1117
1118Similar (but somewhat simpler) computations at the primes $5$ and~$13$
1119show that no adjustment is needed at these primes.  Thus, $dx/y$
1120and $x\,dx/y$ form a basis for the integral differentials of curve
1121$C_{65,B}$, and the correct value of $\Omega$ is 4 times our original
1122guess.
1123
1124\section{Modular algorithms}
1125\label{modular}
1126
1127In this section, we describe the algorithms that were used to compute
1128some of the data from the newforms. This includes the analytic rank
1129and leading coefficient of the $L$-series. For optimal quotients,
1130the value of~$k\cdot\Omega$ can also be found ($k$ is the Manin constant),
1131as well as partial information
1132on the Tamagawa numbers~$c_p$ and the size of the torsion subgroup.
1133
1134\subsection{Analytic rank of $L(J,s)$ and leading coefficient at $s=1$}
1135\label{l}
1136
1137Fix a Jacobian~$J$ corresponding to the 2-dimensional subspace of
1138$S_2(N)$ spanned by quadratic conjugate, normalized newforms~$f$
1139and~$\overline{f}$.  Let $W_N$ be the Fricke involution. The newforms~$f$
1140and~$\overline{f}$ have the same eigenvalue~$\epsilon_N$ with respect
1141to~$W_N$, namely $+1$ or~$-1$. In the notation of Section~\ref{curves}, let
1142$L(f,s) = \sum\limits_{n=1}^{\infty} \frac{a_n}{n^s}$
1143be the $L$-series of~$f$; then $L(\overline{f},s)$ is the Dirichlet
1144series whose coefficients are the conjugates of the
1145coefficients of~$L(f,s)$. (Recall that the~$a_n$ are integers in some
1146real quadratic field.)  The order of~$L(f,s)$ at~$s = 1$ is even
1147when $\epsilon_N = -1$ and odd when $\epsilon_N = +1$.  We have
1148$L(J,s) = L(f,s) L(\overline{f},s)$. Thus the analytic rank of $J$ is~0
1149modulo~4 when $\epsilon_N = -1$ and 2 modulo~4 when $\epsilon_N = +1$.
1150We found that the ranks were all 0 or~2. To prove that the analytic
1151rank of~$J$ is~0, we need to show $L(f,1) \neq 0$ and
1152$L(\overline{f},1) \neq 0$. In the case that $\epsilon_N = +1$, to
1153prove that the analytic rank is~2, we need to show that $L'(f,1) \neq 0$
1154and $L'(\overline{f},1) \neq 0$.  When $\epsilon_N = -1$, we can
1155evaluate $L(f,1)$ as in~\cite[\S 2.11]{Cr}.  When $\epsilon_N = +1$, we
1156can evaluate $L'(f,1)$ as in~\cite[\S 2.13]{Cr}.  Each appropriate
1157$L(f,1)$ or~$L'(f,1)$ was at least~$0.1$ and the errors in our
1158approximations were all less than~$10^{-67}$. In this way we
1159determined the analytic ranks, which we denote~$r$.  As noted in the
1160introduction, the analytic rank equals the Mordell-Weil rank if $r = 0$
1161or~$r = 2$.  Thus, we can simply call $r$ the rank, without fear of
1162ambiguity.
1163
1164To compute the leading coefficient of~$L(J,s)$ at~$s = 1$, we note that
1165$\lim_{s \to 1} L(J,s)/(s-1)^r = L^{(r)}(J,1)/r!$.
1166In the $r=0$ case, we simply have $L(J,1) = L(f,1)L(\overline{f},1)$.
1167In the $r=2$ case, we have
1168$L''(J,s) 1169 = L''(f,s)L(\overline{f},s) + 2L'(f,s)L'(\overline{f},s) 1170 + L(f,s)L''(\overline{f},s)$.
1171Evaluating both sides
1172at $s=1$ we get $\frac{1}{2}L''(J,1) = L'(f,1)L'(\overline{f},1)$.
1173
1174\subsection{Computing $k\cdot\Omega$}\label{modomega}
1175Let $J$, $f$ and $\overline{f}$ be as in Section~\ref{l} and assume
1176$J$ is an optimal quotient.  Let $V$ be the 2-dimensional space
1177spanned by $f$ and $\overline{f}$. Choose a basis
1178$\{\omega_1,\omega_2\}$ for the subgroup of $V$ consisting of forms
1179whose $q$-expansion coefficients lie in $\Z$.  Let $k\cdot\Omega$ be
1180the volume of the real points of the quotient of $\C\times\C$ by the
1181lattice of period integrals $(\int_\gamma \omega_1, 1182\int_\gamma\omega_2)$ with $\gamma$ in the integral homology
1183$H_1(X_0(N),\Z)$.
1184The rational number $k$
1185is called
1186the {\em Manin constant}.  In practice we compute $k\cdot\Omega$
1187using modular symbols and a generalization to dimension 2~of the
1188algorithm for computing periods described in \cite[\S2.10]{Cr}.  When
1189$L(J,1)\neq 0$ the method of \cite[\S2.11]{Cr} coupled with
1190Sections~\ref{l} and~\ref{bsdratio} can also be used to compute
1191$k\cdot\Omega$.
1192
1193A slight generalization of the argument of
1194Proposition 2 of \cite{Ed1} proves that $k$ is, in fact, an integer.
1195This generalization can be found in \cite{AS2}, where
1196one also finds a conjecture that~$k$ must equal~$1$ for all optimal quotients of
1197Jacobians of modular curves, which generalizes the longstanding conjecture of Manin
1198that~$k$ equals~$1$ for all optimal elliptic curves.  In unpublished work, Edixhoven
1199has partially proven Manin's conjecture.
1200
1201The computations of the present paper verify that $k$ equals~$1$ for the
1202optimal quotients that we are considering. Specifically, we computed
1203$k\cdot\Omega$ as above and $\Omega$ as described in Section~\ref{Omega}.
1204The quotient of the two values was always well within $0.5$ of $1$.
1205
1206\subsection{Computing $L(J,1)/(k\cdot\Omega)$}\label{bsdratio}
1207We compute the rational number $L(J,1)/(k\cdot\Omega)$, for optimal
1208quotients,
1209using the algorithm in \cite{AS1}.
1210This algorithm generalizes the algorithm described in
1211\cite[\S2.8]{Cr} to dimension greater than 1.
1212
1213\subsection{Tamagawa numbers}
1214In this section we assume that $p$ is a prime which
1215exactly divides the conductor $N$ of $J$.
1216Under these conditions, Grothendieck \cite{Gr} gave a
1217description of the component group of $J$ in
1218terms 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 $J_0(N)$, one
1221deduces the following. When
1222the eigenvalue for $f$ of the Atkin-Lehner involution $w_p$ is
1223$+1$, then the rational component group of $J$ is a subgroup of
1224$(\Z/2\Z)^2$. Furthermore, when the eigenvalue of $w_p$ is $-1$,
1225the algorithm described in \cite{Ste} can be used to compute
1226the value of~$c_p$.
1227
1228\subsection{Torsion subgroup}
1229\label{modtors}
1230
1231To compute an integer divisible by the order of the
1232torsion subgroup of $J$ we make use of the following two observations.
1233First, it is a consequence of the Eichler-Shimura relation
1234\cite[\S7.9]{Sh} that if $p$ is a prime not dividing the
1235conductor $N$ of $J$ and $f(T)$ is the characteristic polynomial
1236of 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 injective
1241(see \cite[p.\ 70]{CF}). This does not depend on whether $J$ is an
1242optimal 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 point
1246$(0)-(\infty)\in J_0(N)$.
1247
1248\section{Tables}
1249\label{tables}
1250
1251In Table~\ref{table1}, we list the 32 curves described in
1252Section~\ref{curves}. We give the level $N$ {}from which each curve
1253arose, an integral model for the curve, and list the source(s) {}from
1254which it came ($H$ for Hasegawa \cite{Ha}, $W$ for Wang \cite{Wan}).
1255Throughout the paper, the curves are denoted $C_N$ (or $C_{N,A}$, $C_{N,B}$).
1256
1257\begin{table}
1258\begin{center}
1259\begin{tabular}{|l|rcl|c|}
1260\hline
1261\multicolumn{1}{|c|}{$N$}
1262      & \multicolumn{3}{|c|}{Equation} & Source\\ \hline\hline
126323    & $y^2 + (x^3 + x + 1)y$   & $=$ &
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  \\ \hline
127139    & $y^2 + (x^3 + 1)y$       & $=$ &
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  \\ \hline
127967    & $y^2 + (x^3 + x + 1)y$   & $=$ &
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 \\ \hline
128793    & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
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 \\ \hline
1295117,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  \\ \hline
1303133,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 \\ \hline
1311161   & $y^2 + (x^3 + x + 1)y$   & $=$ &
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  \\ \hline
1319177   & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
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 \\ \hline
1327\end{tabular}
1328\end{center}
1329\caption{Levels, integral models and sources for curves}
1330\label{table1}
1331\end{table}
1332
1333In Table~\ref{table2}, we list the curve~$C_N$ simply by~$N$, the
1334level {}from which it arose.  Let $r$ denote the rank.  We
1335list ${\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)$ where $L(J,s)$ is the
1336$L$-series for the Jacobian $J$ of~$C_N$ and round off the results to
1337five 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 of
1340increasing order dividing the level~$N$.  We denote $J(\Q)\tors = \Phi$
1341and list its size.  We use $\Sh ?$ to denote the size of
1342$({\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)) \cdot 1343 (\#J(\Q)\tors)^2/(\Omega\cdot {\rm Reg} \cdot \prod c_{p})$,
1344rounded to the nearest integer.  We will refer to this as the {\em conjectured
1345size of} $\Sh(J,\Q)$.
1346For rank~$0$ optimal quotients this integer equals the (a priori)
1347rational number $(L(J,1)/(k\cdot\Omega))\cdot((\#J(\Q)\tors)^2/\prod c_{p})$;
1348of course there is no rounding error in this computation. For all other cases
1349the last column gives a bound on the accuracy of the
1350computations; all values of $\Sh ?$ were at least this close to the
1351nearest integer before rounding.
1352
1353\newcommand{\mcc}[1]{\multicolumn{1}{|c|}{#1}}
1354\newcommand{\mcd}[1]{\multicolumn{2}{|c|}{#1}}
1355
1356\begin{table}
1357\begin{center}
1358\begin{tabular}{|l|c|r@{.}l|r@{.}l|l|l|c|c|l|}
1359\hline
1360\mcc{$N$} & $r$
1361& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
1362& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & $\Phi$ & $\Sh$? & \mcc{error}
1363\\ \hline\hline
136423    & 0 & 0&24843 &  2&7328 & 1         & 11    & 11 & 1 &  \\
136529    & 0 & 0&29152 &  2&0407 & 1         & 7     &  7 & 1 &  \\
136631    & 0 & 0&44929 &  2&2464 & 1         & 5     &  5 & 1 &  \\
136735    & 0 & 0&37275 &  2&9820 & 1         & 16,2  & 16 & 1 & $< 10^{-25}$ \\
1368\hline
136939    & 0 & 0&38204 & 10&697  & 1         & 28,1  & 28 & 1 & $< 10^{-25}$ \\
137063    & 0 & 0&75328 &  4&5197 & 1         & 2,3   &  6 & 1 &  \\
137165,A  & 0 & 0&45207 &  6&3289 & 1         & 7,1   & 14 & 2 &  \\
137265,B  & 0 & 0&91225 &  5&4735 & 1         & 1,3   &  6 & 2 &  \\
1373\hline
137467    & 2 & 0&23410 & 20&465  & 0.011439  & 1     &  1 & 1 & $< 10^{-50}$ \\
137573    & 2 & 0&25812 & 24&093  & 0.010713  & 1     &  1 & 1 & $< 10^{-49}$ \\
137685    & 2 & 0&34334 &  9&1728 & 0.018715  & 4,2   &  2 & 1 & $< 10^{-26}$ \\
137787    & 0 & 1&4323  &  7&1617 & 1         & 5,1   &  5 & 1 &  \\
1378\hline
137993    & 2 & 0&33996 & 18&142  & 0.0046847 & 4,1   &  1 & 1 & $< 10^{-49}$ \\
1380103   & 2 & 0&37585 & 16&855  & 0.022299  & 1     &  1 & 1 & $< 10^{-49}$ \\
1381107   & 2 & 0&53438 & 11&883  & 0.044970  & 1     &  1 & 1 & $< 10^{-49}$ \\
1382115   & 2 & 0&41693 & 10&678  & 0.0097618 & 4,1   &  1 & 1 & $< 10^{-50}$ \\
1383\hline
1384117,A & 0 & 1&0985  &  3&2954 & 1         & 4,3   &  6 & 1 &  \\
1385117,B & 0 & 1&9510  &  1&9510 & 1         & 4,1   &  2 & 1 &  \\
1386125,A & 2 & 0&62996 & 13&026  & 0.048361  & 1     &  1 & 1 & $< 10^{-50}$ \\
1387125,B & 0 & 2&0842  &  2&6052 & 1         & 5     &  5 & 4 &  \\
1388\hline
1389133,A & 0 & 2&2265  &  2&7832 & 1         & 5,1   &  5 & 4 &  \\
1390133,B & 2 & 0&43884 & 15&318  & 0.028648  & 1,1   &  1 & 1 & $< 10^{-49}$ \\
1391135   & 0 & 1&5110  &  4&5331 & 1         & 3,1   &  3 & 1 &  \\
1392147   & 2 & 0&61816 & 13&616  & 0.045400  & 2,2   &  2 & 1 & $< 10^{-50}$ \\
1393\hline
1394161   & 2 & 0&82364 & 11&871  & 0.017345  & 4,1   &  1 & 1 & $< 10^{-47}$ \\
1395165   & 2 & 0&68650 &  9&5431 & 0.071936  & 4,2,2 &  4 & 1 & $< 10^{-26}$ \\
1396167   & 2 & 0&91530 &  7&3327 & 0.12482   & 1     &  1 & 1 & $< 10^{-47}$ \\
1397175   & 0 & 0&97209 &  4&8605 & 1         & 1,5   &  5 & 1 &  \\
1398\hline
1399177   & 2 & 0&90451 & 13&742  & 0.065821  & 1,1   &  1 & 1 & $< 10^{-45}$ \\
1400188   & 2 & 1&1708  & 11&519  & 0.011293  & 9,1   &  1 & 1 & $< 10^{-44}$ \\
1401189   & 0 & 1&2982  &  3&8946 & 1         & 1,3   &  3 & 1 &  \\
1402191   & 2 & 0&95958 & 17&357  & 0.055286  & 1     &  1 & 1 & $< 10^{-44}$ \\
1403\hline
1404\end{tabular}
1405\end{center}
1406\caption{Conjectured sizes of $\Sh (J,\Q)$}
1407\label{table2}
1408\end{table}
1409
1410In Table~\ref{table3} are generators of $J(\Q)/J(\Q)\tors$ for the
1411curves whose Jacobians have Mordell-Weil rank~2. The generators are
1412given as divisor classes. Whenever possible, we have chosen
1413generators of the form $[P - Q]$ where $P$ and~$Q$ are rational
1414points on the curve. Curve~167 is the only example where this is not
1415the case, since the degree zero divisors supported on the (known)
1416rational points on~$C_{167}$ generate a subgroup of index two in the
1417full Mordell-Weil group.
1418Affine points are given by their $x$ and $y$ coordinates in the model
1419given in Table~\ref{table1}.  There are two points at infinity in the
1420normalization of the curves described by our equations, with the
1421exception of curve~$C_{188}$. These are denoted by $\infty_a$, where
1422$a$ is the value of the function $y/x^3$ on the point in question.
1423The (only) point at infinity on curve~$C_{188}$ is simply
1424denoted~$\infty$.
1425
1426\begin{table}
1427\begin{center}
1428\begin{tabular}{|l|l|l|}
1429\hline
1430 \mcc{$N$} & \mcd{Generators of $J(\Q)/J(\Q)\tors$} \\ \hline\hline
1431 67   & $[(0, 0) - \infty_{-1}]$ &
1432        $[(0, 0) - (0, -1)]$ \\
1433 73   & $[(0, -1) - \infty_{-1}]$ &
1434        $[(0, 0) - \infty_{-1}]$ \\
1435 85   & $[(1, 1) - \infty_{-1}]$ &
1436        $[(-1, 3) - \infty_{0}]$ \\
1437 93   & $[(-1, 1) - \infty_{0}]$ &
1438        $[(1, -3) - (-1, -2)]$ \\ \hline
1439103   & $[(0, 0) - \infty_{-1}]$ &
1440        $[(0, -1) - (0,0)]$ \\
1441107   & $[\infty_{-1} - \infty_{0}]$ &
1442        $[(-1, -1) - \infty_{-1}]$ \\
1443115   & $[(1, -4) - \infty_{0}]$ &
1444        $[(1, 1) - (-2, 2)]$ \\
1445125,A & $[\infty_{-1} - \infty_{0}]$ &
1446        $[(-1, 0) - \infty_{-1}]$ \\ \hline
1447133,B & $[\infty_{-1} - \infty_{0}]$ &
1448        $[(0, -1) - \infty_{-1}]$ \\
1449147   & $[\infty_{-1} - \infty_{0}]$ &
1450        $[(-1, -1) - \infty_{0}]$ \\
1451161   & $[(1, 2) - (-1, 1)]$ &
1452        $[(\frac{1}{2}, -3) - (1, 2)]$ \\
1453165   & $[(1, 1) - \infty_{-1}]$ &
1454        $[(0, 0) - \infty_{0} ]$ \\ \hline
1455167   & $[(-1 ,1) - \infty_{0}]$ &
1456        $[(i, 0) + (-i, 0) - \infty_{0} - \infty_{-1}]$ \\
1457177   & $[(0, -1) - \infty_{0}]$ &
1458        $[(0, 0) - (0, -1)]$ \\
1459188   & $[(0, -1) - \infty]$ &
1460        $[(0, 1) - (1, -2)]$ \\
1461191   & $[\infty_{-1} - \infty_{0}]$ &
1462        $[(0, -1) - \infty_{0}]$ \\
1463\hline
1464\end{tabular}
1465\end{center}
1466\caption{Generators of $J(\Q)/J(\Q)\tors$ in rank 2 cases}
1467\label{table3}
1468\end{table}
1469
1470In Table~\ref{table4} are the reduction types, {}from the
1471classification of~\cite{NU}, of the special fibers of the minimal,
1472proper, regular models of the curves for each of the primes of
1473singular reduction for the curve. They are the same as the primes
1474dividing the level except that curve~$C_{65,A}$ has singular
1475reduction at the prime~3 and curve~$C_{65,B}$ has singular reduction
1476at the prime~2.
1477
1478\begin{table}
1479\begin{center}
1480\begin{tabular}{|l|l|l|l|l||l|l|l|l|l|}
1481\hline
1482\mcc{$N$} & Prime & Type & Prime & Type &
1483\mcc{$N$} & Prime & Type & Prime & Type
1484\\ \hline\hline
148523   & 23 & ${\rm I}_{3-2-1}$ & & &
1486  117,A &  3 & ${\rm III}-{\rm III}^{\ast}-0$
1487        & 13 & ${\rm I}_{1-1-1}$ \\
148829   & 29 & ${\rm I}_{3-1-1}$ & & &
1489  117,B &  3 & ${\rm I}_{3-1-1}^{\ast}$
1490        & 13 & ${\rm I}_{1-1-0}$ \\
149131   & 31 & ${\rm I}_{2-1-1}$ & & &
1492  125,A &  5 & ${\rm VIII}-1$ & & \\
149335   &  5 & ${\rm I}_{3-2-2}$
1494     &  7 & ${\rm I}_{2-1-0}$ &
1495  125,B &  5 &  ${\rm IX}-3$ & & \\ \hline
149639   &  3 & ${\rm I}_{6-2-2}$
1497     & 13 & ${\rm I}_{1-1-0}$ &
1498  133,A &  7 & ${\rm I}_{2-1-1}$
1499        & 19 & ${\rm I}_{1-1-0}$ \\
150063   &  3 & $2{\rm I}_{0}^{\ast}-0$
1501     &  7 & ${\rm I}_{1-1-1}$ &
1502  133,B &  7 & ${\rm I}_{1-1-0}$
1503        & 19 & ${\rm I}_{1-1-0}$ \\
150465,A &  3 & ${\rm I}_{0}-{\rm I}_{0}-1$
1505     &  5 & ${\rm I}_{3-1-1}$ &
1506  135   &  3 & III
1507        &  5 & ${\rm I}_{3-1-0}$ \\
150865,A & 13 & ${\rm I}_{1-1-0}$ & & &
1509  147   &  3 & ${\rm I}_{2-1-0}$
1510        &  7 & VII \\ \hline
151165,B &  2 & ${\rm I}_{0}-{\rm I}_{0}-1$
1512     &  5 & ${\rm I}_{3-1-0}$ &
1513  161   &  7 & ${\rm I}_{2-2-0}$
1514        & 23 & ${\rm I}_{1-1-0}$ \\
151565,B & 13 & ${\rm I}_{1-1-1}$ & & &
1516  165   &  3 & ${\rm I}_{2-2-0}$
1517        &  5 & ${\rm I}_{2-1-0}$  \\
151867   & 67 & ${\rm I}_{1-1-0}$ & & &
1519  165   & 11 & ${\rm I}_{2-1-0}$ & & \\
152073   & 73 & ${\rm I}_{1-1-0}$ & & &
1521  167   & 167 & ${\rm I}_{1-1-0}$ & & \\ \hline
152285   &  5 & ${\rm I}_{2-2-0}$
1523     & 17 & ${\rm I}_{2-1-0}$ &
1524  175   &  5 & ${\rm II}-{\rm II}-0$
1525        &  7 & ${\rm I}_{2-1-1}$ \\
152687   &  3 & ${\rm I}_{2-1-1}$
1527     & 29 & ${\rm I}_{1-1-0}$ &
1528  177   &  3 & ${\rm I}_{1-1-0}$
1529        & 59 & ${\rm I}_{1-1-0}$ \\
153093   &  3 & ${\rm I}_{2-2-0}$
1531     & 31 & ${\rm I}_{1-1-0}$ &
1532  188   &  2 & ${\rm IV}-{\rm IV}-0$
1533        & 47 & ${\rm I}_{1-1-0}$ \\
1534103  & 103 & ${\rm I}_{1-1-0}$ & & &
1535  189   &  3 & ${\rm II}-{\rm IV}^{\ast}-0$
1536        &  7 & ${\rm I}_{1-1-1}$ \\ \hline
1537107  & 107 & ${\rm I}_{1-1-0}$ & & &
1538  191   & 191 & ${\rm I}_{1-1-0}$ & & \\
1539115  &  5 & ${\rm I}_{2-2-0}$
1540     & 23 & ${\rm I}_{1-1-0}$ & & & & & \\ \hline
1541\end{tabular}
1542\end{center}
1543\caption{Namikawa and Ueno classification of special fibers}
1544\label{table4}
1545\end{table}
1546
1547
1548\section{Discussion of Shafarevich-Tate groups and evidence for the
1549second conjecture}
1550\label{Shah}
1551
1552{}From Section~\ref{MW} we have
1553$\dim \Sh(J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2]$.
1554With the exception of curves $C_{65,A}$, $C_{65,B}$, $C_{125,B}$, and
1555$C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 0$. Thus we expect
1556$\#\Sh(J,\Q)$ to be an odd square. In each case, the conjectured
1557size of $\Sh(J,\Q)$ is~1.  For curves $C_{65,A}$, $C_{65,B}$,
1558$C_{125,B}$ and $C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 1, 1, 2$
1559and~2 and the conjectured size of $\Sh(J,\Q) = 2, 2, 4$ and~4,
1560respectively.  We see that in each case, the (conjectured) size of
1561the odd part of $\Sh(J,\Q)$ is~1 and the 2-part is accounted for by
1562its 2-torsion.
1563
1564Recall that for rank 0 optimal quotients we are able to exactly
1565determine the value which the second Birch and Swinnerton-Dyer
1566conjecture predicts for $\Sh(J,\Q)$. From the previous paragraph,
1567we then see that equation~\eqref{eqn1} holds if and only if
1568$\Sh(J,\Q)$ is killed by $2$.
1569
1570It is also interesting to consider deficient primes.  A prime $p$ is
1571deficient with respect to a curve $C$ of genus~2, if $C$ has no
1572degree 1 rational divisor over~$\Q_{p}$.  {}From~\cite{PSt}, the
1573number of deficient primes has the same parity as $\dim \Sh(J,\Q)[2]$.
1574Curve $C_{65,A}$ has one deficient prime~$3$. Curve
1575$C_{65,B}$ has one deficient prime~$2$. Curve $C_{117,B}$ has two
1576deficient primes $3$ and~$\infty$.  The rest of the curves have no
1577deficient primes.
1578
1579Since we have found $r$ (analytic rank) independent points on each
1580Jacobian, we have a direct proof that the Mordell-Weil rank must
1581equal the analytic rank if $\dim \Sh(J,\Q)[2] = 0$.  For
1582curves $C_{65,A}$ and $C_{65,B}$, the presence of an odd number of
1583deficient primes gives us a
1584similar result.  For $C_{125,B}$ we used a $\sqrt{5}$-Selmer group
1585to get a similar result.
1586Thus, we have an independent proof of equality
1587between analytic and Mordell-Weil ranks for all curves except
1588$C_{133,A}$.
1589
1590The 2-Selmer groups have the same dimensions for the pairs
1591$C_{125,A}$, $C_{125,B}$ and $C_{133,A}$, $C_{133,B}$.  For each
1592pair, the Mordell-Weil rank is~2 for one curve and the 2-torsion of
1593the Shafarevich-Tate group has dimension~2 for the other. In
1594addition, the two Jacobians, when canonically embedded into~$J_0(N)$,
1595intersect in their 2-torsion subgroups, and one can check that their
15962-Selmer groups become equal under the identification of
1597$H^1(\Q, J_{N,A}[2])$ with $H^1(\Q, J_{N,B}[2])$ induced by the identification
1598of the 2-torsion subgroups.  Thus these are examples of the principle
1599of a visible part of a Shafarevich-Tate group' as discussed
1600in~\cite{CM}.
1601
1602\vspace{5mm}
1603\begin{center}
1604{\sc Appendix: Other Hasegawa curves}
1605\end{center}
1606
1607In Table~\ref{Hasegawa} is data concerning all 142 of Hasegawa's
1608curves in the order presented in his paper. Let us explain the
1609entries.  The first column in each set of three columns gives the
1610level, $N$. The second column gives a classification of the cusp
1611forms spanning the 2-dimensional subspace of $S_2(N)$ corresponding
1612to the Jacobian.  When that subspace is irreducible with respect to
1613the action of the Hecke algebra and is spanned by two newforms or two
1614oldforms, we write $2n$ or $2o$, respectively.  When that subspace is
1615reducible and is spanned by two oldforms, two newforms or one of
1616each, we write $oo$, $nn$ and $on$, respectively. The third column
1617contains the sign of the functional equation at the level $M$ at
1618which the cusp form is a newform. This is the negative of
1619$\epsilon_M$ (described in Section~\ref{l}).  The order of the two
1620signs in the third column agrees with that of the forms listed in the
1621second column.  We include this information for those who would like
1622to further study these curves.  The curves with $N<200$ classified as
1623$2n$ appeared already in Table~\ref{table1}.
1624
1625The smallest possible Mordell-Weil ranks corresponding to $++$, $+-$,
1626$-+$ and $--$, predicted by the first Birch and Swinnerton-Dyer
1627conjecture, are $0$, $1$, $1$ and $2$ respectively. In all cases,
1628those were, in fact, the Mordell-Weil ranks. This was determined by
1629computing 2-Selmer groups with a computer program based on
1630\cite{Sto2}.  Of course, these are cases where the first Birch and
1631Swinnerton-Dyer conjecture is already known to hold.  In the cases
1632where the Mordell-Weil rank is positive, the Mordell-Weil group has a
1633subgroup of finite index generated by degree zero divisors supported
1634on rational points with $x$-coordinates with numerators bounded by 7
1635(in absolute value) and denominators by 12 with one exception.  On
1636the second curve with $N=138$, the divisor class
1637$[(3+2\sqrt{2},80+56\sqrt{2}) + (3-2\sqrt{2},80-56\sqrt{2})-2\infty]$
1638generates a subgroup of finite index in the Mordell-Weil group.
1639
1640\vfill
1641
1642\begin{table}
1643\begin{center}
1644\begin{tabular}{|c|c|c||c|c|c||c|c|c||c|c|c||c|c|c|}
1645\hline
164622 & $oo$ & $++$ & 58 & $nn$ & $+-$ & 87 & $2o$ & $++$ & 129 & $on$ & $--$ &
1647198 & $2o$ & $+-$  \\
164823 & $2n$ & $++$ & 60 & $oo$ & $++$ & 88 & $on$ & $+-$ & 130 & $on$ & $-+$ &
1649204 & $2o$ & $+-$  \\
165026 & $nn$ & $++$ & 60 & $2o$ & $++$ & 90 & $on$ & $++$ & 132 & $oo$ & $++$ &
1651205 & $2n$ & $--$  \\
165228 & $oo$ & $++$ & 60 & $2o$ & $++$ & 90 & $oo$ & $++$ & 133 & $2n$ & $--$ &
1653206 & $2o$ & $--$  \\
165429 & $2n$ & $++$ & 62 & $2o$ & $++$ & 90 & $oo$ & $++$ & 134 & $2o$ & $--$ &
1655209 & $2n$ & $--$  \\
165630 & $on$ & $++$ & 66 & $nn$ & $++$ & 90 & $oo$ & $++$ & 135 & $on$ & $+-$ &
1657210 & $on$ & $+-$  \\
165830 & $oo$ & $++$ & 66 & $2o$ & $++$ & 91 & $nn$ & $--$ & 138 & $nn$ & $+-$ &
1659213 & $2n$ & $--$  \\
166030 & $on$ & $++$ & 66 & $2o$ & $++$ & 93 & $2n$ & $--$ & 138 & $on$ & $+-$ &
1661215 & $on$ & $--$  \\
166231 & $2n$ & $++$ & 66 & $on$ & $++$ & 98 & $oo$ & $++$ & 140 & $oo$ & $++$ &
1663221 & $2n$ & $--$  \\
166433 & $on$ & $++$ & 67 & $2n$ & $--$ & 100 & $oo$ & $++$ & 142 & $nn$ & $+-$
1665& 230 & $2o$ & $--$  \\ \hline
166635 & $2n$ & $++$ & 68 & $oo$ & $++$ & 102 & $on$ & $+-$ & 143 & $on$ & $+-$
1667& 255 & $2o$ & $--$ \\
166837 & $nn$ & $+-$ & 69 & $2o$ & $++$ & 102 & $on$ & $+-$ & 146 & $2o$ & $--$
1669& 266 & $2o$ & $--$ \\
167038 & $on$ & $++$ & 70 & $on$ & $++$ & 103 & $2n$ & $--$ & 147 & $2n$ & $--$
1671& 276 & $2o$ & $+-$ \\
167239 & $2n$ & $++$ & 70 & $2o$ & $++$ & 104 & $2o$ & $++$ & 150 & $on$ & $++$
1673& 284 & $2o$ & $+-$ \\
167440 & $on$ & $++$ & 70 & $2o$ & $++$ & 106 & $on$ & $--$ & 153 & $on$ & $+-$
1675& 285 & $on$ & $--$ \\
167640 & $oo$ & $++$ & 70 & $2o$ & $++$ & 107 & $2n$ & $--$ & 154 & $on$ & $--$
1677& 286 & $on$ & $--$ \\
167842 & $on$ & $++$ & 72 & $on$ & $++$ & 110 & $on$ & $++$ & 156 & $oo$ & $++$
1679& 287 & $2n$ & $--$ \\
168042 & $oo$ & $++$ & 72 & $oo$ & $++$ & 111 & $oo$ & $+-$ & 158 & $on$ & $--$
1681& 299 & $2n$ & $--$ \\
168242 & $on$ & $++$ & 73 & $2n$ & $--$ & 112 & $on$ & $+-$ & 161 & $2n$ & $--$
1683& 330 & $2o$ & $--$ \\
168442 & $oo$ & $++$ & 74 & $oo$ & $+-$ & 114 & $oo$ & $+-$ & 165 & $2n$ & $--$
1685& 357 & $2n$ & $--$ \\ \hline
168644 & $2o$ & $++$ & 77 & $on$ & $+-$ & 115 & $2n$ & $--$ & 166 & $on$ & $--$
1687& 380 & $2o$ & $+-$ \\
168846 & $2o$ & $++$ & 78 & $oo$ & $++$ & 116 & $2o$ & $+-$ & 167 & $2n$ & $--$
1689& 390 & $on$ & $--$ \\
169048 & $on$ & $++$ & 78 & $2o$ & $++$ & 117 & $2o$ & $++$ & 168 & $2o$ & $++$
1691& & & \\
169248 & $oo$ & $++$ & 80 & $oo$ & $++$ & 120 & $oo$ & $++$ & 170 & $2o$ & $--$
1693& & & \\
169450 & $nn$ & $++$ & 84 & $oo$ & $++$ & 120 & $on$ & $++$ & 177 & $2n$ & $--$
1695& & & \\
169652 & $oo$ & $++$ & 84 & $oo$ & $++$ & 121 & $on$ & $+-$ & 180 & $2o$ & $++$
1697& & & \\
169852 & $oo$ & $++$ & 84 & $oo$ & $++$ & 122 & $on$ & $--$ & 184 & $on$ & $+-$
1699& & & \\
170054 & $on$ & $++$ & 84 & $oo$ & $++$ & 125 & $2n$ & $--$ & 186 & $2o$ & $--$
1701& & & \\
170257 & $on$ & $+-$ & 85 & $2n$ & $--$ & 126 & $oo$ & $++$ & 190 & $on$ & $+-$
1703& & & \\
170457 & $on$ & $+-$ & 87 & $2n$ & $++$ & 126 & $on$ & $++$ & 191 & $2n$ & $--$
1705& & & \\
1706 \hline
1707\end{tabular}
1708\end{center}
1709\caption{Spaces of cusp forms associated to Hasegawa's curves}
1710\label{Hasegawa}
1711\end{table}
1712
1713\pagebreak
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