CoCalc Public Fileswww / papers / evidence / evidence2.texOpen with one click!
Author: William A. Stein
Compute Environment: Ubuntu 18.04 (Deprecated)
1
% see end of document for change notes.c
2
3
\documentclass{mcom-l}
4
%\documentclass[12pt]{amsart}
5
\usepackage{amscd}
6
\newfont{\cyr}{wncyr10 scaled \magstep1}
7
\newcommand{\Sh}{\hbox{\cyr Sh}}
8
\newcommand{\C}{{\mathbf C}}
9
\newcommand{\Q}{{\mathbf Q}}
10
\newcommand{\Qbar}{\overline{\Q}}
11
%\newcommand{\GalQ}{{\Gal}(\Qbar/\Q)}
12
\newcommand{\CC}{{\mathcal C}}
13
\newcommand{\Z}{{\mathbf Z}}
14
\newcommand{\R}{{\mathbf R}}
15
\newcommand{\F}{{\mathbf F}}
16
\newcommand{\G}{{\mathbf G}}
17
\newcommand{\OO}{{\mathcal O}}
18
\newcommand{\JJ}{{\mathcal J}}
19
\newcommand{\DD}{{\mathcal D}}
20
\newcommand{\aaa}{{\mathfrak a}}
21
\newcommand{\PP}{{\mathbf P}}
22
\newcommand{\tors}{_{\text{tors}}}
23
\newcommand{\unr}{^{\text{unr}}}
24
\newcommand{\nichts}{{\left.\right.}}
25
26
27
28
\DeclareMathOperator{\Gal}{Gal}
29
\DeclareMathOperator{\Norm}{Norm}
30
\DeclareMathOperator{\Sel}{Sel}
31
\DeclareMathOperator{\Tr}{Tr}
32
33
\newtheorem{theorem}{Theorem}[section]
34
\newtheorem{lemma}[theorem]{Lemma}
35
\newtheorem{cor}[theorem]{Corollary}
36
\newtheorem{prop}[theorem]{Proposition}
37
38
\theoremstyle{definition}
39
\newtheorem{question}{Question}
40
\newtheorem{conj}{Conjecture}
41
42
43
\theoremstyle{remark}
44
\newtheorem{remark}[theorem]{Remark}
45
46
\numberwithin{equation}{section}
47
48
49
50
51
\topmargin -0.3in
52
\headsep 0.3in
53
\oddsidemargin 0in
54
\evensidemargin 0in
55
\textwidth 6.5in
56
\textheight 9in
57
58
%%%\renewcommand{\baselinestretch}{2}
59
60
\begin{document}
61
62
\title[Modular Jacobians]{Empirical evidence for the Birch and
63
Swinnerton-Dyer conjectures for
64
modular Jacobians of genus~2 curves}
65
66
\author{E.\ Victor Flynn}
67
\address{Department of Mathematical Sciences, University of
68
Liverpool, P.O.Box 147,
69
Liverpool 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
74
Martin 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,
79
Santa 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,
84
Cambridge, MA 02138, USA}
85
\email{was@math.berkeley.edu}
86
87
\author{Michael Stoll}
88
\address{Mathematisches Institut, Universit\"{a}tsstr.\ 1, D-40225
89
D\"{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,
94
1042 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
99
abelian variety}
100
101
\thanks{The first author thanks the Nuffield Foundation
102
(Grant SCI/180/96/71/G) for financial support.
103
The second author did some of the research at
104
the Max-Planck Institut f\"ur Mathematik and
105
the Technische Universit\"at Berlin.
106
The third author thanks the National Security Agency (Grant
107
MDA904-99-1-0013).
108
The fourth author was supported by a Sarah M. Hallam fellowship.
109
The fifth author did some of the research at
110
the Max-Planck-Institut f\"ur Mathematik.
111
The sixth author thanks the National Science Foundation
112
(Grant DMS-9705959).
113
The authors had useful conversations with John Cremona, Qing Liu,
114
Karl Rubin and
115
Peter Swinnerton-Dyer and are grateful to Xiangdong Wang and Michael
116
M\"{u}ller for making data available to them.}
117
118
\date{June 5, 2000}
119
120
\begin{abstract}
121
This paper provides empirical evidence for the Birch and
122
Swinnerton-Dyer conjectures for modular Jacobians of genus~2 curves.
123
The second of these conjectures relates six quantities associated to
124
a Jacobian over the rational numbers. One of these
125
six quantities is
126
the size of the Shafarevich-Tate group.
127
Unable to compute that, we
128
computed the five other quantities and solved for the last one. In
129
all 32~cases, the result is very close to an integer that is a power
130
of~2. In addition, this power of~2 agrees with the size of the
131
2-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
147
The conjectures of Birch and Swinnerton-Dyer, originally stated
148
for elliptic curves over~$\Q$, have been a constant source of
149
motivation for the study of elliptic curves, with the ultimate
150
goal being to find a proof.
151
This has resulted not only in a better
152
theoretical understanding, but also in the development of better
153
algorithms for computing the analytic and arithmetic
154
invariants that are so intriguingly related by them. We now know
155
that the first and, up to a non-zero rational factor, the
156
second conjecture hold for modular elliptic curves over~$\Q$
157
\footnote{It has recently been announced by
158
Brueil, Conrad, Diamond and Taylor that they have extended Wiles'
159
results and shown
160
that all elliptic curves over~$\Q$ are modular.}
161
in the
162
analytic rank~0 and~1 cases (see \cite{GZ,Ko,Wal1,Wal2}).
163
Furthermore,
164
a number of people have provided numerical evidence for the
165
conjectures for a large number of elliptic curves; see
166
for example~\cite{BGZ,BSD,Ca,Cr}.
167
168
By now, our theoretical and algorithmic knowledge of curves of
169
genus~2 and their Jacobians has reached a state that makes it
170
possible to conduct similar investigations. The Birch and
171
Swinnerton-Dyer conjectures have been generalized to arbitrary
172
abelian varieties over number fields by Tate~\cite{Ta}. If
173
$J$ is the Jacobian of a genus~2 curve over $\Q$,
174
then the first conjecture
175
states 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
177
Jacobian. 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}
183
In 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
185
integral over $J(\R)$ of a particular differential 2-form; the
186
precise choice of this differential is described in
187
Section~\ref{Omega}. ${\rm Reg}$ is the regulator of $J(\Q)$. For
188
primes $p$, we use $c_{p}$ to denote the size of $J(\Q_p)/J^0(\Q_p)$,
189
where $J^0(\Q_p)$ 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
193
As in the case of elliptic curves, the first conjecture assumes
194
that the $L$-series can be analytically continued to $s = 1$,
195
and the second conjecture additionally assumes that the
196
Shafarevich-Tate group is finite. Neither of these assumptions is
197
known to hold for arbitrary genus~2 curves. The analytic
198
continuation of the $L$-series, however, is known to exist for
199
modular abelian varieties over~$\Q$, where an abelian
200
variety is called {\em modular} if it is a quotient of the Jacobian~$J_0(N)$
201
of the modular curve~$X_0(N)$ for some level~$N$. For simplicity,
202
we will also call a genus~2 curve {\em modular} when its Jacobian is
203
modular in this sense. So it is certainly a good idea to look
204
at modular genus~2 curves over~$\Q$, since we then at least know that the
205
statement of the first conjecture makes sense. Moreover, for many modular
206
abelian varieties it is also known that the Shafarevich-Tate group
207
is finite, therefore the statement of the second conjecture also
208
makes sense. As it turns out, all of our examples belong to this
209
class.
210
An additional benefit of choosing modular genus~2 curves is
211
that one can find lists of such curves in the literature.
212
In fact, we have no way of verifying the Birch and Swinnerton-Dyer
213
conjectures for Jacobians of genus 2 curves that are not modular, since
214
there is no known way of computing the analytic rank or
215
the leading coefficient of the $L$-series at $s=1$.
216
217
218
In this article, we provide empirical evidence for the Birch and
219
Swinnerton-Dyer conjectures for such modular genus~2 curves. Since there
220
is no known effective way of computing the size of the Shafarevich-Tate
221
group, we computed the other five terms in equation~\eqref{eqn1}
222
(in two different ways, if possible). This required several different
223
algorithms, some of which were developed or improved while we were
224
working on this paper. If one of these algorithms
225
is already well described in the literature, then we simply cite it.
226
Otherwise, we describe it here in some detail (in particular,
227
algorithms for computing $\Omega$ and
228
$c_p$).
229
230
For modular abelian varieties associated to newforms whose
231
$L$-series have analytic rank~0 or~1, the first Birch and Swinnerton-Dyer
232
conjecture has been proven. In such cases, the
233
Shafarevich-Tate group is also known to be finite and the second conjecture
234
has been proven, up to a non-zero rational factor. This all
235
follows {}from results in
236
\cite{GZ,KL,Wal1,Wal2}.
237
In our examples, all of the analytic
238
ranks are either~0 or~1. Thus we already know that the first
239
conjecture holds. Since the Jacobians we consider are associated to a
240
quadratic conjugate pair of newforms, the analytic rank of the
241
Jacobian is twice the analytic rank of either newform (see \cite{GZ}).
242
243
The second Birch and Swinnerton-Dyer conjecture has not been proven
244
for the cases we consider. In order to verify equation~\eqref{eqn1},
245
we 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
247
accuracy of our calculations. This number is a power of~2, which
248
coincides with the independently computed size of the 2-torsion
249
subgroup of~$\Sh(J,\Q)$. Hence, we have verified the second
250
Birch and Swinnerton-Dyer conjecture for our curves at least
251
numerically, if we assume that the Shafarevich-Tate group consists
252
of 2-torsion only. (This is an ad hoc assumption based only
253
on the fact that we do not know better.) See Section~\ref{Shah} for
254
circumstances under which the verification is exact.
255
256
The curves are listed in Table~\ref{table1},
257
and the numerical results can be found in Table~\ref{table2}.
258
259
260
\section{The Curves}
261
\label{curves}
262
263
Each 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
265
one of these genus~2 curves arises {}from a given level $N$, then we
266
denote 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
268
of, say, $C_N$ to $J_0(N)$ depends on the source. Briefly, {}from
269
Hasegawa \cite{Hs} we obtain quotients of $X_0(N)$ and {}from Wang
270
\cite{Wan} we obtain curves whose Jacobians are quotients of $J_0(N)$.
271
In both cases the Jacobian $J_N$ of $C_N$ is isogenous to a
272
2-dimensional factor of $J_0(N)$. (When not referring to a specific
273
curve, we will typically drop the subscript $N$ {}from $J$.)
274
In this way we can also associate
275
$C_N$ with a 2-dimensional subspace of $S_2(N)$, the space of cusp
276
forms of weight~2 for $\Gamma_0(N)$.
277
278
We now discuss the precise source of the genus~2 curves we will
279
consider. Hasegawa \cite{Hs} has provided exact equations for all
280
genus~2 curves which are quotients of $X_0(N)$ by a subgroup of the
281
Atkin-Lehner involutions. There are 142 such curves. We are
282
particularly interested in those where the Jacobian corresponds to a
283
subspace of $S_2(N)$ spanned by a quadratic conjugate pair of
284
newforms. There are 21 of these with level $N \leq 200$. For these
285
curves we will provide evidence for the second conjecture. There are
286
seven more such curves with $N > 200$. We can classify the other
287
2-dimensional subspaces into four types. There are
288
2-dimensional subspaces of oldforms that are irreducible under the
289
action of the Hecke algebra. There are also 2-dimensional subspaces
290
that are reducible under the action of the Hecke algebra and are
291
spanned by two oldforms, two newforms or one of each. The Jacobians
292
corresponding to the latter three kinds are always isogenous, over
293
$\Q$, to the product of two elliptic curves. Given the small levels,
294
these are elliptic curves for which Cremona \cite{Cr} has already
295
provided evidence for the Birch and Swinnerton-Dyer conjectures. In
296
Table~\ref{Hasegawa}, we describe the kind of cusp forms spanning the
297
2-dimensional subspace and the signs of their functional equations
298
{}from the level at which they are newforms. The analytic and
299
Mordell-Weil ranks were always the smallest possible given those signs.
300
301
The second set of curves was created by Wang \cite{Wan} and is further
302
discussed in \cite{FM}. This set consists of 28 curves that were
303
constructed by considering the spaces $S_2(N)$ with $N \leq 200$.
304
Whenever a subspace spanned by a pair of quadratic conjugate newforms
305
was found, these newforms were integrated to produce a quotient
306
abelian variety~$A$ of $J_0(N)$. These quotients are {\em optimal} in the
307
sense of \cite{Ma}, in that the kernel of the quotient map is
308
connected.
309
310
The period matrix for~$A$ was created using certain intersection
311
numbers. When all of the intersection numbers have the same value,
312
then the polarization on~$A$ induced {}from the canonical polarization
313
of~$J_0(N)$ is equivalent to a principal polarization. (Two
314
polarizations are {\em equivalent} if they differ by an integer multiple.)
315
Conversely, every 2-dimensional optimal quotient of $J_0(N)$ in which
316
the induced polarization is equivalent to a principal polarization is
317
found in this way.
318
319
Using theta functions, numerical approximations were found for the
320
Igusa invariants of the abelian surfaces. These numbers coincide with
321
rational numbers of fairly small height within the limits of the
322
precision used for the computations. Wang then constructed curves
323
defined over~$\Q$ whose Igusa invariants are the rational numbers
324
found. (There is one abelian surface at level $N = 177$ for which Wang
325
was not able to find a curve.) If we assume that these rational
326
numbers are the true Igusa invariants of the abelian surfaces, then it
327
follows that Wang's curves have Jacobians isomorphic, over~$\Qbar$, to
328
the principally polarized abelian surfaces in his list. Since the
329
classification given by these invariants is only up to isomorphism
330
over~$\Qbar$, the Jacobians of Wang's curves are not necessarily
331
isomorphic to, but can be twists of, the optimal quotients
332
of~$J_0(N)$ over~$\Q$ (see below).
333
334
There are four curves in Hasegawa's list which do not show up in
335
Wang's list (they are listed in Table~\ref{table1} with an $H$ in the
336
last column). Their Jacobians are quotients of~$J_0(N)$, but are not
337
optimal quotients. It is likely that there are modular genus~2 curves
338
which neither are Atkin-Lehner quotients of~$X_0(N)$ (in Hasegawa's
339
sense) nor have Jacobians that are optimal quotients. These curves
340
could be found by looking at the optimal quotient abelian surfaces and
341
checking whether they are isogenous to a principally polarized abelian
342
surface over $\Q$.
343
344
For 17 of the curves in Wang's list, the 2-dimensional subspace
345
spanned by the newforms is the same as that giving one of Hasegawa's
346
curves. In all of those cases, the curve given by Wang's equation is
347
isomorphic, over $\Q$, to that given by Hasegawa. This verifies Wang's
348
equations for these 17 curves. They are listed in Table~\ref{table1}
349
with $HW$ in the last column.
350
351
The remaining eleven curves (listed in Table~\ref{table1} with a
352
$W$ in the last column) derive from the other eleven optimal
353
quotients in Wang's list. These are described in more detail in
354
Section~\ref{bad11} below.
355
356
With the exception of curves $C_{63}$, $C_{117,A}$ and $C_{189}$, the
357
Jacobians of all of our curves are absolutely simple, and the
358
canonically polarized Jacobians have automorphism groups of size two.
359
We showed that these Jacobians are absolutely simple using an argument
360
like those in \cite{Le,Sto1}. The automorphism group of the
361
canonically polarized Jacobian of a hyperelliptic curve is isomorphic
362
to the automorphism group of the curve (see \cite[Thm.\
363
12.1]{Mi2}). Each automorphism of a hyperelliptic curve is induced by
364
a linear fractional transformation on $x$-coordinates (see \cite[p.\
365
1]{CF}). Each automorphism also permutes the six Weierstrass
366
points. Once we believed we had found all of the automorphisms, we
367
were able to show that there are no more by considering all linear
368
fractional transformations sending three fixed Weierstrass points to
369
any three Weierstrass points. In each case, we worked with sufficient
370
accuracy to show that other linear fractional transformations did not
371
permute the Weierstrass points.
372
373
Let $\zeta_{3}$ denote a primitive third root of unity. The
374
Jacobians of curves $C_{63}$, $C_{117,A}$ and $C_{189}$ are each
375
isogenous to the product of two elliptic curves over $\Q(\zeta_3)$,
376
though not over $\Q$, where they are simple. These genus~2 curves
377
have automorphism groups of size 12. In the following table we list
378
the curve at the left. On the right we give one of the elliptic
379
curves which is a factor of its Jacobian. The second factor is the
380
conjugate.
381
\[
382
\begin{array}{ll}
383
C_{63}: & y^2 = x(x^2 + (9 - 12\zeta_{3})x - 48\zeta_{3}) \\
384
C_{117,A}: & y^2 = x(x^2 - (12 + 27\zeta_{3})x - (48 + 48\zeta_{3})) \\
385
C_{189}: & y^2 = x^3 + (66 - 3\zeta_{3})x^2 + (342 + 81\zeta_{3})x
386
+ 105 + 21\zeta_{3}
387
\end{array}
388
\]
389
Note that these three Jacobians are examples of abelian varieties
390
`with extra twist' as discussed in~\cite{Cr2}, where they can be
391
found in the tables on page~409.
392
393
\subsection{Models for the Wang-only curves}
394
\label{bad11}
395
396
As we have already noted, a modular genus~2 curve may be found by
397
either, both, or neither of Wang's and
398
Hasegawa's techniques.
399
Hasegawa's method allows for the exact determination, over $\Q$, of
400
the equation of any modular genus~2 curve it has found. On the other
401
hand, if Wang's technique detects a modular genus~2 curve $C_N$, his
402
method produces real approximations to a curve $C'_N$ which is defined
403
over $\Q$ and is isomorphic to $C_N$ over $\Qbar$. We will call
404
$C'_N$ a {\em twisted modular genus~2 curve}.
405
406
In this section we attempt to determine equations for the eleven
407
modular genus~2 curves detected by Wang but not by Hasegawa. If we
408
assume that Wang's equations for the twisted modular genus~2 curves
409
are correct, we find that we are able to determine the twists. In
410
turn, this gives us strong evidence that Wang's equations for the
411
twisted curves were correct. Undoing the twist, we determine probable
412
equations for the modular genus~2 curves. We end by providing further
413
evidence for the correctness of these equations.
414
415
In what follows, we will use the notation of~\cite{Cr} and recommend
416
it as a reference on the general results that we assume here and in
417
Section~\ref{modular} and the appendix.
418
Fix a level~$N$ and let
419
$f(z) \in S_2(N)$. Then $f$ has a Fourier expansion
420
\[ f(z) = \sum\limits_{n=1}^{\infty} a_{n} e^{2 \pi i n z}\,. \]
421
For a newform~$f$, we have $a_1 \neq 0$; so we can normalize it by
422
setting $a_1 = 1$. In our cases, the $a_n$'s are integers in a real
423
quadratic field. For each prime~$p$ not dividing~$N$, the
424
corresponding 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
426
norm and trace of~$a_p$. The product of this Euler factor and its
427
conjugate is
428
$1 - Tr(a_p)\,p^{-s} + (N(a_p) + 2p)\,p^{-2s}
429
- p\,Tr(a_p)\,p^{-3s} + p^2\,p^{-4s}$.
430
Therefore, the characteristic
431
polynomial of the $p$-Frobenius on the corresponding abelian variety
432
over $\F_{p}$ is
433
$x^4 - Tr(a_p)\,x^3 + (N(a_p) + 2p)\,x^2 - p\,Tr(a_p)\,x + p^2$.
434
Let $C$ be a curve, over $\Q$, whose Jacobian, over $\Q$, comes {}from
435
the space spanned by $f$ and its conjugate. Then we know that
436
$p+1 - \#C(\F_{p}) = Tr(a_p)$ and
437
$\frac{1}{2}(\#C(\F_{p})^{2} + \#C(\F_{p^2})) - (p+1)\# C(\F_{p}) - p =
438
N(a_p)$ (see \cite[Lemma 3]{MS}).
439
For 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
441
Wang's equations. {}From these we could compute the characteristic
442
polynomials of Frobenius and see if they agreed with those predicted
443
by the $a_p$'s of the newforms.
444
445
Of the eleven curves, the characteristic polynomials agreed for only
446
four. In each of the remaining seven cases we found a twist of Wang's
447
curve whose characteristic polynomials agreed with those predicted by
448
the newform for all odd primes less than 200 not dividing $N$. Four
449
of these twists were quadratic and three were of higher degree. It
450
is these twists that appear in Table~\ref{table1}.
451
452
We can provide further evidence that these equations are correct.
453
For each curve given in Table~\ref{table1}, it is easy to determine
454
the primes of singular reduction. In Section~\ref{Tamagawa} we will
455
provide techniques for determining which of those primes divides the
456
conductor of its Jacobian. In each case, the primes dividing the
457
conductor of the Jacobian of the curve are exactly the primes
458
dividing the level $N$; this is necessary. With the exception of
459
curve $C_{188}$, 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
462
conductor of the curve. In each case (other than curve $C_{188}$),
463
the conductor is the square of the level; this is also necessary. For
464
curve $C_{188}$, the odd part of the conductor of the curve is the
465
square of the odd part of the level.
466
467
In addition, since the Jacobians of the Wang curves are optimal
468
quotients, we can compute~$k\cdot\Omega$ (where $k$ is the Manin constant,
469
conjectured to be 1)
470
using the newforms.
471
In each case, these agree (to within the accuracy of our computations)
472
with the $\Omega$'s computed using the equations for the curves.
473
We can also compute the value of~$c_p$ for optimal quotients from
474
the newforms, when $p$ exactly divides~$N$ and the eigenvalue of the
475
$p$th Atkin-Lehner involution is $-1$. When $p$ exactly divides~$N$
476
and the eigenvalue of the $p$th Atkin-Lehner involution is~$+1$, the
477
component group is either $0$, $\Z/2\Z$, or~$(\Z/2\Z)^2$. These results
478
are always in agreement with the values computed using the equations
479
for the curves. The algorithms based on the newforms are
480
described in Section~\ref{modular}, those based on the
481
equations of the curves are described in Section~\ref{algms}.
482
483
Lastly, we were able to compute the Mordell-Weil ranks of the Jacobians
484
of the curves given by ten of these eleven equations. In
485
each case it agrees with the analytic rank of the Jacobian,
486
as deduced {}from the newforms.
487
488
It should be noted that curve~$C_{125,B}$ is the $\sqrt{5}$-twist of
489
curve~$C_{125,A}$; the corresponding statement holds for the associated
490
2-dimensional subspaces of~$S_2(125)$. Since curve~$C_{125,A}$ is
491
a Hasegawa curve, this proves that the equation given in Table~\ref{table1}
492
for curve~$C_{125,B}$ is correct.
493
494
The $a_p$'s and other information concerning Wang's curves are
495
currently kept in a database at the Institut f\"{u}r experimentelle
496
Mathematik in Essen, Germany. Most recently, this database was under
497
the care of Michael M\"{u}ller. William Stein also keeps a database
498
of~$a_p$'s for newforms.
499
500
\begin{remark}
501
For the remainder of this paper we will assume that the equations for
502
the curves given in Table~\ref{table1} are correct; that is, that
503
they are equations for the curves whose Jacobians are isogenous
504
to a factor of~$J_0(N)$ in the way described above.
505
Some of the quantities can be computed either {}from the newform
506
or {}from the equation for the curve. We performed both computations
507
whenever possible, and view this duplicate effort as an attempt to
508
verify our implementation of the algorithms rather than an attempt
509
to verify the equations in Table~\ref{table1}. For most quantities,
510
one method or the other is not guaranteed to produce a value; in this
511
case, we simply quote the value {}from whichever method did succeed.
512
The reader who is disturbed by this philosophy should
513
ignore the Wang-only curves, since the equations for the Hasegawa
514
curves can be proven to be correct.
515
\end{remark}
516
517
518
\section{Algorithms for genus~2 curves}
519
\label{algms}
520
521
In this section, we describe the algorithms that are based on the
522
given models for the curves. We give algorithms that compute all
523
terms on the right hand side of equation~\eqref{eqn1}, with the
524
exception of the size of the Shafarevich-Tate group. We describe,
525
however, how to find the size of its 2-torsion subgroup. Note that these
526
algorithms are for general genus 2 curves and do not depend on modularity.
527
528
\subsection{Torsion Subgroup}
529
\label{torsion}
530
531
The computation of the torsion subgroup of~$J(\Q)$ is straightforward.
532
We used the technique described in~\cite[pp.~78--82]{CF}.
533
This technique is not always effective, however. For an algorithm working
534
in all cases see~\cite{Sto3}.
535
536
\subsection{Mordell-Weil rank and $\Sh(J,\Q)[2]$}
537
\label{MW}
538
539
The group $J(\Q)$ is a finitely generated abelian group and so is
540
isomorphic to $\Z^{r} \oplus J(\Q)\tors$ for some $r$ called the
541
Mordell-Weil rank.
542
As noted above (see Section~\ref{intro}), we justifiably use
543
$r$ to denote both the analytic and Mordell-Weil ranks since they
544
agree for all curves in Table~\ref{table1}.
545
546
We used the algorithm described in \cite{FPS} to compute ${\rm
547
Sel}^{2}_{\rm fake}(J,\Q)$ (notation {}from \cite{PSc}), which is a
548
quotient of the 2-Selmer group ${\rm Sel}^{2}(J,\Q)$. More details
549
on this algorithm can be found in \cite{Sto2}. Theorem 13.2 of
550
\cite{PSc} explains how to get ${\rm Sel}^{2}(J,\Q)$ {}from ${\rm
551
Sel}^{2}_{\rm fake}(J,\Q)$. Let $M[2]$ denote the 2-torsion of an
552
abelian group $M$ and let dim$V$ denote the dimension of an $\F_{2}$
553
vector space $V$. We have
554
$\dim {\rm Sel}^{2}(J,\Q) = r + \dim J(\Q)[2] + \dim \Sh(J,\Q)[2]$.
555
In other words,
556
\[ \dim\, \Sh (J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2]. \]
557
558
It is interesting to note that in all 30 cases where
559
$\dim \Sh(J,\Q)[2] \le 1$, we were able to compute the Mordell-Weil rank
560
independently from the analytic rank.
561
The
562
cases where $\dim \Sh(J,\Q)[2] = 1$ are discussed in more
563
detail in Section~\ref{Shah}.
564
For both of the remaining cases we have $\dim \Sh(J,\Q)[2]=2$.
565
One of these cases is
566
$C_{125,B}$. For this curve we computed
567
${\rm Sel}^{\sqrt{5}}(J_{125,B},\Q)$
568
using the technique described in
569
\cite{Sc}. {}From this, we were able to determine that the Mordell-Weil
570
rank is 0 independently from the analytic rank.
571
For the other case,
572
$C_{133,A}$,
573
we could show that $r$ had to be either~0
574
or~2 {}from the equation, but we needed the analytic computation to
575
show that $r=0$.
576
577
\subsection{Regulator}
578
\label{reg}
579
580
When the Mordell-Weil rank is~0, then the regulator is~1. When the
581
Mordell-Weil rank is positive, then to compute the regulator, we
582
first need to find generators for $J(\Q)/J(\Q)\tors$. The regulator
583
is the determinant of the canonical height pairing matrix on this set
584
of generators. An algorithm for computing the generators and
585
canonical heights is given in~\cite{FS}; it was used to find
586
generators for $J(\Q)/J(\Q)\tors$ and to compute the regulators. In
587
that article, the algorithm for computing height constants at the
588
infinite prime is not clearly explained and there are some errors in
589
the examples. A clear algorithm for computing infinite height
590
constants is given in~\cite{Sto3}. In~\cite{Sto4}, some improvements of
591
the results and algorithms in~\cite{FS} and~\cite{Sto3} are discussed.
592
The regulators in Table~\ref{table2} have been double-checked using
593
these improved algorithms.
594
595
\subsection{Tamagawa Numbers}
596
\label{Tamagawa}
597
598
Let $\OO$ be the integer ring in~$K$ which will be $\Q_{p}$ or
599
$\Q_{p}\unr$ (the maximal unramified extension of $\Q_{p})$.
600
Let $\JJ$ be the N\'{e}ron model of~$J$ over~$\OO$.
601
Define $\JJ^{0}$ to be the open subgroup scheme of~$\JJ$ whose
602
generic fiber is isomorphic to~$J$ over~$K$ and whose special fiber
603
is the identity component of the closed fiber of~$\JJ$.
604
The group $\JJ^{0}(\OO)$ is isomorphic to a subgroup of~$J(K)$ which
605
we denote $J^{0}(K)$. The group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is
606
the component group of~$\JJ$ over~$\OO_{\Q_{p}\unr}$. We are
607
interested in computing $c_p = \#J(\Q_{p})/J^{0}(\Q_{p})$, which is
608
sometimes called the Tamagawa number.
609
Since N\'{e}ron models are stable under unramified base extension,
610
the $\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of
611
$J^{0}(\Q_{p}\unr)$ is~$J^{0}(\Q_{p})$.
612
Since $H^1(\Gal(\Q_{p}\unr/\Q_{p}), J^{0}(\Q_{p}\unr))$
613
is trivial (see~\cite[p.\ 58]{Mi1}) we see that the
614
$\Gal(\Q_{p}\unr/\Q_{p})$-invariant subgroup of
615
$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is $ J(\Q_{p})/J^{0}(\Q_{p})$.
616
617
There exist several discussions in the literature on constructing the
618
group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ starting with an integral
619
model of the underlying curve. For our purposes, we especially
620
recommend Silverman's book~\cite{Si}, Chapter~IV, Sections 4 and~7.
621
For a more detailed treatment, see~\cite[chap.\ 9]{BLR} and~\cite[\S 2]{Ed2}.
622
One can find justifications for what we will do in these sources. While
623
constructing such groups, we ran into a number of difficulties that
624
we did not find described anywhere. For that reason, we will present
625
examples of such difficulties that arose, as well as our methods of
626
resolution. We do not claim that we will describe all situations
627
that could arise.
628
629
When computing $c_p$ we need a proper, regular model~$\CC$ for~$C$
630
over~$\Z_p$. Let $\Z_p\unr$ denote the ring of integers of~$\Q_p\unr$
631
and note that $\Z_p\unr$ is a pro-\'etale Galois extension
632
of~$\Z_p$ with Galois group
633
$\Gal(\Z_p\unr/\Z_p) = \Gal(\Q_p\unr/\Q_p)$.
634
It follows that giving a model for~$C$ over~$\Z_p$ is equivalent to
635
giving a model for~$C$ over~$\Z_p\unr$ that
636
is equipped with a Galois action. We have found it convenient to
637
always work with the latter description. Thus for us, giving a model
638
over~$\Z_p$ will always mean giving a model over~$\Z_p\unr$ together
639
with a Galois action.
640
641
In order to find a proper, regular model for~$C$ over~$\Z_p$,
642
we start with the models in Table~\ref{table1}. Technically, we
643
consider the curves to be the two affine pieces $y^2+g(x)y=f(x)$ and
644
$v^2 + u^3 g(1/u)v = u^6 f(1/u)$, glued together by $ux=1$, $v=u^3y$.
645
We blow them up at all points that are not regular until we have a
646
regular model. (A point is {\em regular} if the cotangent space there has
647
two generators.) These curves are all proper, and this is not
648
affected by blowing up.
649
650
Let $\CC_p$ denote the special fiber of~$\CC$ over~$\Z_p\unr$. The
651
group $J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is isomorphic to a quotient
652
of the degree~0 part of the free group on the irreducible components
653
of~$\CC_{p}$. Let the irreducible components be denoted $\DD_{i}$ for
654
$1\leq i\leq n$, and let the multiplicity of~$\DD_{i}$ in~$\CC_p$ be
655
$d_{i}$. Then the degree~0 part of the free group has the form
656
\[ L = \{ \sum\limits_{i=1}^{n} \alpha_{i}\DD_{i} \mid
657
\sum\limits_{i=1}^{n} d_{i}\alpha_{i} = 0 \}\,. \]
658
659
In order to describe the group that we quotient out by, we must
660
discuss the intersection pairing. For components $\DD_{i}$ and~$\DD_{j}$
661
of the special fiber, let $\DD_{i} \cdot \DD_{j}$ denote
662
their intersection pairing. In all of the special fibers that arise
663
in our examples, distinct components intersect transversally. Thus,
664
if $i \neq j$, then $\DD_{i} \cdot \DD_{j}$ equals the number of points
665
at which $\DD_{i}$ and $\DD_{j}$ intersect. The case of
666
self-intersection ($i=j$) is discussed below.
667
668
The kernel of the map {}from~$L$ to
669
$J(\Q_{p}\unr)/J^{0}(\Q_{p}\unr)$ is generated by
670
divisors of the form
671
\[ [\DD_j] = \sum\limits_{i=1}^{n} (\DD_{j} \cdot \DD_{i}) \DD_{i} \]
672
for each component~$\DD_j$. We can deduce $\DD_{j} \cdot \DD_{j}$ by
673
noting that $[\DD_j]$ must be contained in the group~$L$. This follows
674
{}from the fact that the intersection pairing of
675
$\CC_{p} = \sum d_i\DD_{i}$ with any irreducible component is 0.
676
677
\vspace{1mm}
678
\noindent
679
{\bf Example 1.} Curve $C_{65,B}$ over $\Z_{2}$.
680
681
The Jacobian
682
of $C_{65,B}$
683
is a quotient of the Jacobian of~$X_0(65)$.
684
Since 65 is odd, $J_0(65)$ has good reducation at~2; however,
685
$C_{65,B}$ has singular
686
reduction at~2. Since the equation for this curve
687
is conjectural (it is a Wang-only curve), it will be nice to verify
688
that 2 does not divide the conductor of its Jacobian, i.e.\ that the
689
Jacobian has good reduction at~2. In addition, we will need a
690
proper, regular model for this curve in order to find~$\Omega$.
691
692
We start with the arithmetic surface over~$\Z_{2}\unr$ given by the
693
two pieces
694
$y^2 = f(x) = -x^6 + 10x^5 - 32x^4 + 20x^3 + 40x^2 + 6x - 1$ and
695
$v^2 = u^6 f(1/u)$. (Here and in the following we will not specify the
696
gluing maps.) This arithmetic surface is regular at $u=0$ so we
697
focus our attention on the first affine piece. The special fiber of
698
$y^2 = f(x)$ over~$\Z_{2}\unr$
699
is given by
700
$(y + x^3 + 1)^2 = 0 \pmod 2$; this is a genus~0 curve of multiplicity~2
701
that we denote~$A$. This model is not regular at the two points
702
$(x-\alpha, y, 2)$, where $\alpha$ is a root of $x^2 - 3x - 1$.
703
The current special fiber is in Figure~\ref{special2} and is labelled
704
{\it Fiber~1}.
705
706
We fix $\alpha$ and move $(x - \alpha, y, 2)$ to the origin using the
707
substitution $x_0 = x-\alpha$. We get
708
\[ y^2 = -x_0^6 + (-6\alpha + 10)x_0^5 + (5\alpha - 47)x_0^4
709
+ (-28\alpha + 60)x_0^3 + (-11\alpha - 2)x_0^2
710
+ (-24\alpha - 16)x_0
711
\]
712
which we rewrite as the pair of equations
713
\begin{align*}
714
g_{1}(x_{0},y,p)
715
&= -x_0^6 + (-3\alpha + 5) p x_0^5 + (5\alpha - 47) x_0^4
716
+ (-7\alpha + 15) p^2 x_0^3 \\
717
& \qquad {} + (-11\alpha - 2) x_0^2 + (-3\alpha - 2) p^3 x_0 - y^2
718
\\
719
&= 0,\\
720
p &= 2.
721
\end{align*}
722
To blow up at $(x_0,y,p)$, we introduce projective coordinates
723
$(x_1,y_1,p_1)$ with $x_{0} y_1 = x_{1} y$, $x_{0} p_{1} = x_{1} p$, and
724
$y p_1 = y_{1} p$. We look in three affine pieces that cover the blow-up
725
of $g_1(x_{0},y,p)=0,$ $p=2$
726
and check for regularity.
727
728
\begin{description}
729
\item[$x_{1} = 1$] We have $y = x_{0} y_{1}$, $p = x_{0} p_{1}$. We get
730
$g_2(x_{0},y_{1},p_{1}) = 0$, $x_{0} p_{1} = 2$, where
731
\begin{align*}
732
g_2(x_{0},y_{1},p_{1}) &= x_{0}^{-2}g_{1}(x_{0},x_{0}y_{1},x_{0}p_{1}) \\
733
&= -x_0^4 + (-3\alpha + 5) p_1 x_0^4 + (5\alpha - 47) x_0^2
734
+ (-7\alpha + 15) p_1^2 x_0^3 \\
735
& \qquad{} + (-11\alpha - 2) + (-3\alpha - 2) p_1^3 x_0^2 - y_1^2 \,.
736
\end{align*}
737
In the reduction we have either $x_{0} = 0$ or $p_1 = 0$.
738
\begin{description}
739
\item[$x_{0} = 0$] $(y_{1} + \alpha + 1)^2 = 0$.
740
This is a new component which we denote $B$. It has genus~0 and
741
multiplicity~2. We check regularity along~$B$ at
742
$(x_{0}, y_{1} + \alpha + 1, p_{1}-t, 2)$, with $t$ in $\Z_2\unr$, and
743
find that $B$ is nowhere regular.
744
\item[$p_{1} = 0$]
745
$(y_{1} + x_{0}^2 + \alpha x_{0} + (\alpha + 1))^2 = 0$.
746
Using the gluing maps, we see that this is~$A$.
747
\end{description}
748
749
\item[$y_{1} = 1$] We get no new information {}from this affine piece.
750
751
\item[$p_{1} = 1$] We have $x_{0} = x_{1} p$, $y = y_{1} p$. We get
752
$g_{3}(x_{1},y_{1},p) = p^{-2} g_{1}(x_{1}p,y_{1}p,p) = 0$, $p = 2$.
753
In the reduction we have
754
\begin{description}
755
\item[$p=0$] $(y_1 + (\alpha+1)x_1)^2 = 0$. Using the gluing maps, we
756
see that this is~$B$. It is nowhere regular.
757
\end{description}
758
\end{description}
759
760
The current special fiber is in
761
Figure~\ref{special2} and is labelled {\it Fiber~2}. It is not regular
762
along~$B$ and at the other point on~$A$ which we have not yet blown up.
763
The component $B$ does not lie entirely in any one affine piece
764
so we will blow up the affine pieces $x_1 = 1$ and $p_1 = 1$ along~$B$.
765
766
To blow up $x_1 = 1$ along~$B$ we make the substitution
767
$y_2 = y_1 + \alpha + 1$ and replace each factor of~2 in a coefficient
768
by~$x_0 p_1$. We have $g_{4}(x_0,y_2,p_1) = 0$ and $x_0 p_1 = 2$, and we
769
want to blow up along the line $(x_0, y_2, 2)$. Blowing up along a line
770
is similar to blowing up at a point: since we are blowing up at
771
$(x_0, y_2, 2) = (x_0, y_2)$, we introduce projective
772
coordinates $x_3, y_3$ together with the relation $x_0 y_3 = x_3 y_2$. We
773
consider two affine pieces that cover the blow-up of $x_1 = 1$.
774
775
\begin{description}
776
\item[$x_3 = 1$] We have $y_2 = y_{3} x_{0}$. We get
777
$g_{5}(x_{0},y_{3},p_{1}) = x_{0}^{-2} g_{4}(x_{0},y_{3}x_{0},p_1) = 0$
778
and $x_{0} p_{1} = 2$. In the reduction we have
779
\begin{description}
780
\item[$x_{0} = 0$]
781
$y_{3}^2 + (\alpha + 1) y_{3} p_{1} + \alpha p_{1}^3 + p_{1}^2
782
+ \alpha + 1 = 0$.
783
This is~$B$. It is now a non-singular genus~1 curve.
784
\item[$p_{1} = 0$] $(x_0 + y_3 + \alpha)^2 = 0$. This is~$A$. The point
785
where $B$ meets~$A$ transversally is regular.
786
\end{description}
787
788
\item[$y_3 = 1$] We get no new information {}from this affine piece.
789
\end{description}
790
791
When we blow up $p_1 = 1$ along~$B$ we get essentially the same thing and
792
all points are again regular.
793
794
The other non-regular point on~$A$ is the conjugate of the one we
795
blew up. Therefore, after performing the conjugate blow ups, it too
796
will be a genus~1 component crossing~$A$ transversally. We denote
797
this component $D$; it is conjugate to~$B$.
798
799
800
\begin{figure}
801
\caption{Special fibers of curve $C_{65,B}$ over $\Z_{2}$;
802
points not regular are thick}
803
\label{special2}
804
\begin{picture}(400,130)
805
\put(20,5){\begin{picture}(100,125)
806
\thinlines
807
\put(20,55){\line(1,0){60}}
808
\put(85,55){\makebox(0,0){A}}
809
\put(75,62){\makebox(0,0){2}}
810
\put(40,55){\circle*{5}}
811
\put(60,55){\circle*{5}}
812
\put(50,5){\makebox(0,0){Fiber 1}}
813
\end{picture}}
814
\put(145,5){\begin{picture}(100,125)
815
\thinlines
816
\put(50,5){\makebox(0,0){Fiber 2}}
817
\put(20,55){\line(1,0){60}}
818
\put(85,55){\makebox(0,0){A}}
819
\put(75,62){\makebox(0,0){2}}
820
\put(60,55){\circle*{5}}
821
\put(40,15){\line(0,1){80}}
822
\put(40.5,15){\line(0,1){80}}
823
\put(39.5,15){\line(0,1){80}}
824
\put(39,15){\line(0,1){80}}
825
\put(41,15){\line(0,1){80}}
826
\put(40,105){\makebox(0,0){B}}
827
\put(34,90){\makebox(0,0){2}}
828
\end{picture}}
829
\put(270,5){\begin{picture}(100,125)
830
\thinlines
831
\put(20,55){\line(1,0){60}}
832
\put(85,55){\makebox(0,0){A}}
833
\put(75,62){\makebox(0,0){2}}
834
\put(40,15){\line(0,1){80}}
835
\put(40,105){\makebox(0,0){B}}
836
\put(60,15){\line(0,1){80}}
837
\put(60,105){\makebox(0,0){D}}
838
\put(50,5){\makebox(0,0){Fiber 3}}
839
\end{picture}}
840
\end{picture}
841
\end{figure}
842
843
We now have a proper, regular model~$\CC$ of~$C$ over~$\Z_2$.
844
Let $\CC_2$ be the special fiber of this model; a
845
diagram of~$\CC_2$ is in Figure~\ref{special2} and is labelled
846
{\it Fiber~3}. We can use $\CC$ to show that the
847
N\'eron model $\JJ$ of the Jacobian $J = J_{65,B}$ has good
848
reduction at~2.
849
850
We know that the reduction of~$\JJ^0$ is the extension of an abelian
851
variety by a connected linear group. Since $\CC$ is regular and
852
proper, the abelian variety part of the reduction is the product of
853
the Jacobians of the normalizations of the components of~$\CC_2$ (see
854
\cite[9.3/11 and 9.5/4]{BLR}). Thus, the abelian variety part is the
855
product of the Jacobians of~$B$ and~$D$. Since this is
856
2-dimensional, the reduction of~$\JJ^0$ is an abelian variety. In
857
other words, since the sum of the genera of the components of the
858
special fiber is equal to the dimension of~$J$, the reduction is an
859
abelian variety. It follows that $\JJ$ has good reduction at~2, that
860
the conductor of~$J$ is odd, and that $c_2 = 1$. As noted above, this
861
gives further evidence that the equation given in Table~\ref{table1}
862
is correct.
863
864
865
\vspace{1mm}
866
\noindent
867
{\bf Example 2.} Curve $C_{63}$ over $\Z_{3}$.
868
869
The Tamagawa number is often found using the intersection matrix and
870
sub-determinants. This is not entirely satisfactory for cases where
871
the special fiber has several components and a non-trivial Galois
872
action. Here is an example of how to resolve this (see also~\cite{BL}).
873
874
When we blow up curve~$C_{63}$ over~$\Z_{3}\unr$, we get
875
the special fiber shown in Figure~\ref{special1}.
876
Elements of $\Gal(\Q_{3}\unr/\Q_{3})$
877
that do not fix the quadratic unramified extension of~$\Q_{3}$
878
switch $H$ and~$I$. The other components are defined over~$\Q_{3}$.
879
All components have genus~0. The group $J(\Q_{3}\unr)/J^{0}(\Q_{3}\unr)$
880
is isomorphic to a quotient of
881
\begin{align*}
882
L = \{ \alpha A + \beta B + \delta D + \epsilon E + \phi F + \gamma G
883
&+ \eta H + \iota I \\
884
&\mid \alpha + \beta + 2\delta + 2\epsilon + 4\phi + 2\gamma
885
+ 2\eta + 2\iota = 0 \} \,.
886
\end{align*}
887
888
The 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
930
When 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*}
937
At this point, it is straightforward to simplify the representation by
938
elimination. Note that we projected away {}from~$A$, which is
939
Galois-invariant. It is best to continue eliminating Galois-invariant
940
elements 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
943
extension 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
950
By an {\em integral differential} (or {\em integral form}) on $J$ we mean the
951
pullback to $J$ of a global relative differential form on the N\'eron
952
model of $J$ over $\Z$. The set of integral $n$-forms on $J$ is a
953
full-rank lattice in the vector space of global holomorphic $n$-forms
954
on $J$. Since $J$ is an abelian variety of dimension 2, the integral
955
1-forms are a free $\Z$-module of rank 2 and the integral 2-forms are
956
a free $\Z$-module of rank 1. Moreover, the wedge of a basis for the
957
integral 1-forms is a generator for the integral 2-forms. The
958
quantity $\Omega$ is the integral, over the real points of $J$, of a
959
generator for the integral 2-forms. (We choose the generator that
960
leads to a positive integral.)
961
962
We now translate this into a computation on the curve $C$. Let
963
$\{\omega_1, \omega_2\}$ be a $\Q$-basis for the holomorphic
964
differentials 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
967
integrating 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
969
traces {}from the complex matrix. The columns of $M_{\R}$ generate a
970
lattice $\Lambda$ in $\R^2$. If we make the standard identification
971
between the holomorphic 1-forms on $J$ and the holomorphic
972
differentials on $C$ (see \cite{Mi2}), then the notation
973
$\int_{J(\R)} |\omega_1 \wedge \omega_2|$ makes sense and its value
974
can be computed as the area of a fundamental domain for $\Lambda$.
975
976
If $\{\omega_1, \omega_2\}$ is a basis for the integral 1-forms on
977
$J$, then $\int_{J(\R)} |\omega_1 \wedge \omega_2| = \Omega$. On the
978
other 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
981
simple change-of-basis calculation. This assumes, of course, that we
982
know how to express a basis for the integral 1-forms in terms of the
983
basis $\{\omega_1, \omega_2\}$; this is addressed in more detail
984
below.
985
986
It 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
989
fixed by complex conjugation (call this the real homology).
990
Integrating would give us a $2\times 2$ real matrix $M'_{\R}$ and the
991
determinant of $M'_{\R}$ would equal the integral of $\omega_1
992
\wedge \omega_2$ over the connected component of $J(\R)$.
993
In other words, the number of real connected components of $J$ is
994
equal to the index of the $\C/\R$-traces in the real homology.
995
996
We now come to the question of determining the differentials on $C$
997
which correspond to the integral 1-forms on $J$. Call these the
998
integral differentials on $C$. This computation can be done one
999
prime at a time. At each prime $p$ this is equivalent to determining
1000
a $\Z_p\unr$-basis for the global relative differentials on any
1001
proper, regular model for $C$ over $\Z_p\unr$. In fact a more
1002
general class of models can be used; see the discussion of models
1003
with rational singularities in \cite[\S 6.7]{BLR} and \cite[\S
1004
4.1]{Li}.
1005
1006
We start with the model $y^2 + g(x)y=f(x)$ given in
1007
Table~\ref{table1}. Note that the substitution $X=x$ and $Y=2y+g(x)$
1008
gives us a model of the form $Y^2=F(X)$. For integration purposes,
1009
our 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
1011
non-singular reduction for the $y^2 + g(x)y=f(x)$ model, these
1012
differentials will generate the integral 1-forms. For each prime $p$
1013
of singular reduction we give the following algorithm. All steps
1014
take 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
1057
Techniques for explicitly computing a proper, regular model are
1058
discussed in Section~\ref{Tamagawa}. A configuration diagram should
1059
include the genus, multiplicity and self-intersection number of
1060
each component and the number and type of intersections between
1061
components. Note that when an exceptional component is blown down,
1062
all of the self-intersection numbers of the components intersecting
1063
it will go up (towards 0). In particular, components which were not
1064
exceptional before may become exceptional in the new configuration.
1065
1066
Steps 3 and 4 are intended to make this algorithm more efficient for
1067
a human. They are entirely optional. For a computer implementation
1068
it may be easier to simply check every component than to worry about
1069
manipulating configurations.
1070
1071
The curves in Table~\ref{table1} are given as $y^2 + g(x)y=f(x)$. We
1072
assumed, at first, that $dx/(2y+g(x))$ and $x\,dx/(2y+g(x))$ generate
1073
the integral differentials. We integrated these differentials around
1074
each of the four paths generating the complex homology and found a
1075
provisional $\Omega$. Then we checked the proper, regular models to
1076
determine if these differentials really do generate the integral
1077
differentials and adjusted $\Omega$ when necessary. There were
1078
three curves where we needed to adjust $\Omega$. We describe the
1079
adjustment 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
1088
The primes of singular reduction for curve $C_{65,B}$ are 2, 5 and
1089
13. In Example 1 of Section~\ref{Tamagawa}, we found a proper,
1090
regular model $\CC$ for $C$ over $\Z_2\unr$. The configuration for
1091
the special fiber of $\CC$ is sketched in Figure~\ref{special2} under
1092
the label {\it Fiber 3}. Component $A$ is exceptional and can be
1093
blown down to produce a model in which $B$ and $D$ cross
1094
transversally. Since $B$ and $D$ both have genus 1, we cannot
1095
eliminate either of these components. Furthermore, it suffices to
1096
check $B$, since $D$ is its Galois conjugate.
1097
1098
To get {}from the equation of the curve listed in Table~\ref{table1}
1099
to an affine containing an open subset of $B$ we need to make the
1100
substitutions $x=x_0 - \alpha$ and $y=x_0 (y_{3}x_0 - \alpha - 1)$.
1101
We also have $x_{0}p_{1}=2$. Using the substitutions and the
1102
relation $dx_{0}/x_0 = -dp_{1}/p_1$, we get
1103
\[ \frac{dx}{2y} = \frac{-dp_1}{2p_1(y_3 x_0 - \alpha - 1)}
1104
\text{\quad and\quad}
1105
\frac{x\,dx}{2y}
1106
= \frac{-(x_0 + \alpha)\,dp_1}{2p_1(y_3 x_0 - \alpha - 1)} \,.
1107
\]
1108
Note that $p_1 - t$ is a uniformizer at $p_1 = t$ almost everywhere
1109
on~$B$. When we multiply each differential by~2, then the
1110
denominator of each is almost everywhere non-zero; thus, $dx/y$ and
1111
$x\,dx/y$ are integral at~$2$. Moreover, although the linear
1112
combination $(x-\alpha)\,dx/y$ is identically zero on~$B$, it is not
1113
identically zero on~$D$ (its Galois conjugate is not identically zero
1114
on~$B$). Thus, our new basis is correct at~2. We multiply the
1115
provisional $\Omega$ by~4 to get a new provisional $\Omega$ which is
1116
correct at~$2$.
1117
1118
Similar (but somewhat simpler) computations at the primes $5$ and~$13$
1119
show that no adjustment is needed at these primes. Thus, $dx/y$
1120
and $x\,dx/y$ form a basis for the integral differentials of curve
1121
$C_{65,B}$, and the correct value of $\Omega$ is 4 times our original
1122
guess.
1123
1124
\section{Modular algorithms}
1125
\label{modular}
1126
1127
In this section, we describe the algorithms that were used to compute
1128
some of the data from the newforms. This includes the analytic rank
1129
and leading coefficient of the $L$-series. For optimal quotients,
1130
the value of~$k\cdot\Omega$ can also be found ($k$ is the Manin constant),
1131
as well as partial information
1132
on 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
1137
Fix a Jacobian~$J$ corresponding to the 2-dimensional subspace of
1138
$S_2(N)$ spanned by quadratic conjugate, normalized newforms~$f$
1139
and~$\overline{f}$. Let $W_N$ be the Fricke involution. The newforms~$f$
1140
and~$\overline{f}$ have the same eigenvalue~$\epsilon_N$ with respect
1141
to~$W_N$, namely $+1$ or~$-1$. In the notation of Section~\ref{curves}, let
1142
\[ L(f,s) = \sum\limits_{n=1}^{\infty} \frac{a_n}{n^s} \]
1143
be the $L$-series of~$f$; then $L(\overline{f},s)$ is the Dirichlet
1144
series whose coefficients are the conjugates of the
1145
coefficients of~$L(f,s)$. (Recall that the~$a_n$ are integers in some
1146
real quadratic field.) The order of~$L(f,s)$ at~$s = 1$ is even
1147
when $\epsilon_N = -1$ 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
1149
modulo~4 when $\epsilon_N = -1$ and 2 modulo~4 when $\epsilon_N = +1$.
1150
We found that the ranks were all 0 or~2. To prove that the analytic
1151
rank 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
1153
prove that the analytic rank is~2, we need to show that $L'(f,1) \neq 0$
1154
and $L'(\overline{f},1) \neq 0$. When $\epsilon_N = -1$, we can
1155
evaluate $L(f,1)$ as in~\cite[\S 2.11]{Cr}. When $\epsilon_N = +1$, we
1156
can 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
1158
approximations were all less than~$10^{-67}$. In this way we
1159
determined the analytic ranks, which we denote~$r$. As noted in the
1160
introduction, the analytic rank equals the Mordell-Weil rank if $r = 0$
1161
or~$r = 2$. Thus, we can simply call $r$ the rank, without fear of
1162
ambiguity.
1163
1164
To compute the leading coefficient of~$L(J,s)$ at~$s = 1$, we note that
1165
$\lim_{s \to 1} L(J,s)/(s-1)^r = L^{(r)}(J,1)/r!$.
1166
In the $r=0$ case, we simply have $L(J,1) = L(f,1)L(\overline{f},1)$.
1167
In the $r=2$ case, we have
1168
$L''(J,s)
1169
= L''(f,s)L(\overline{f},s) + 2L'(f,s)L'(\overline{f},s)
1170
+ L(f,s)L''(\overline{f},s)$.
1171
Evaluating both sides
1172
at $s=1$ we get $\frac{1}{2}L''(J,1) = L'(f,1)L'(\overline{f},1)$.
1173
1174
\subsection{Computing $k\cdot\Omega$}\label{modomega}
1175
Let $J$, $f$ and $\overline{f}$ be as in Section~\ref{l} and assume
1176
$J$ is an optimal quotient. Let $V$ be the 2-dimensional space
1177
spanned by $f$ and $\overline{f}$. Choose a basis
1178
$\{\omega_1,\omega_2\}$ for the subgroup of $V$ consisting of forms
1179
whose $q$-expansion coefficients lie in $\Z$. Let $k\cdot\Omega$ be
1180
the volume of the real points of the quotient of $\C\times\C$ by the
1181
lattice of period integrals $(\int_\gamma \omega_1,
1182
\int_\gamma\omega_2)$ with $\gamma$ in the integral homology
1183
$H_1(X_0(N),\Z)$.
1184
The rational number $k$
1185
is called
1186
the {\em Manin constant}. In practice we compute $k\cdot\Omega$
1187
using modular symbols and a generalization to dimension 2~of the
1188
algorithm 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
1190
Sections~\ref{l} and~\ref{bsdratio} can also be used to compute
1191
$k\cdot\Omega$.
1192
1193
A slight generalization of the argument of
1194
Proposition 2 of \cite{Ed1} proves that $k$ is, in fact, an integer.
1195
This generalization can be found in \cite{AS2}, where
1196
one also finds a conjecture that~$k$ must equal~$1$ for all optimal quotients of
1197
Jacobians of modular curves, which generalizes the longstanding conjecture of Manin
1198
that~$k$ equals~$1$ for all optimal elliptic curves. In unpublished work, Edixhoven
1199
has partially proven Manin's conjecture.
1200
1201
The computations of the present paper verify that $k$ equals~$1$ for the
1202
optimal quotients that we are considering. Specifically, we computed
1203
$k\cdot\Omega$ as above and $\Omega$ as described in Section~\ref{Omega}.
1204
The 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}
1207
We compute the rational number $L(J,1)/(k\cdot\Omega)$, for optimal
1208
quotients,
1209
using the algorithm in \cite{AS1}.
1210
This algorithm generalizes the algorithm described in
1211
\cite[\S2.8]{Cr} to dimension greater than 1.
1212
1213
\subsection{Tamagawa numbers}
1214
In this section we assume that $p$ is a prime which
1215
exactly divides the conductor $N$ of $J$.
1216
Under these conditions, Grothendieck \cite{Gr} gave a
1217
description of the component group of $J$ in
1218
terms of a monodromy pairing on certain character groups.
1219
(For more details, see Ribet \cite[\S2]{Ri}.)
1220
If, in addition, $J$ is a new optimal quotient of $J_0(N)$, one
1221
deduces the following. When
1222
the 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 $-1$,
1225
the algorithm described in \cite{Ste} can be used to compute
1226
the value of~$c_p$.
1227
1228
\subsection{Torsion subgroup}
1229
\label{modtors}
1230
1231
To compute an integer divisible by the order of the
1232
torsion subgroup of $J$ we make use of the following two observations.
1233
First, it is a consequence of the Eichler-Shimura relation
1234
\cite[\S7.9]{Sh} that if $p$ is a prime not dividing the
1235
conductor $N$ of $J$ and $f(T)$ is the characteristic polynomial
1236
of the endomorphism $T_p$
1237
of $J$, then $\#J(\F_p) = f(p+1)$ (see \cite[\S2.4]{Cr}
1238
for an algorithm to compute $f(T)$).
1239
Second, if $p$ is an odd prime at which $J$ has good reduction,
1240
then 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
1242
optimal quotient.
1243
To obtain a lower bound on the torsion subgroup for optimal quotients,
1244
we use modular symbols and the Abel-Jacobi theorem \cite[IV.2]{La}
1245
to 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
1251
In Table~\ref{table1}, we list the 32 curves described in
1252
Section~\ref{curves}. We give the level $N$ {}from which each curve
1253
arose, an integral model for the curve, and list the source(s) {}from
1254
which it came ($H$ for Hasegawa \cite{Ha}, $W$ for Wang \cite{Wan}).
1255
Throughout 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
1263
23 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1264
$-2 x^5 - 3 x^2 + 2 x - 2$ & HW \\
1265
29 & $y^2 + (x^3 + 1)y$ & $=$ &
1266
$-x^5 - 3 x^4 + 2 x^2 + 2 x - 2$ & HW \\
1267
31 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1268
$-x^5 - 5 x^4 - 5 x^3 + 3 x^2 + 2 x - 3$ & HW \\
1269
35 & $y^2 + (x^3 + x)y$ & $=$ &
1270
$-x^5 - 8 x^3 - 7 x^2 - 16 x - 19$ & H \\ \hline
1271
39 & $y^2 + (x^3 + 1)y$ & $=$ &
1272
$-5 x^4 - 2 x^3 + 16 x^2 - 12 x + 2$ & H \\
1273
63 & $y^2 + (x^3 - 1)y$ & $=$ &
1274
$14 x^3 - 7$ & W \\
1275
65,A & $y^2 + (x^3 + 1)y$ & $=$ &
1276
$-4 x^6 + 9 x^4 + 7 x^3 + 18 x^2 - 10$ & W \\
1277
65,B & $y^2$ & $=$ &
1278
$-x^6 + 10 x^5 - 32 x^4 + 20 x^3 + 40 x^2 + 6 x - 1$ & W \\ \hline
1279
67 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1280
$x^5 - x$ & HW \\
1281
73 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1282
$-x^5 - 2 x^3 + x$ & HW \\
1283
85 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1284
$x^4 + x^3 + 3 x^2 - 2 x + 1$ & H \\
1285
87 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1286
$-x^4 + x^3 - 3 x^2 + x - 1$ & HW \\ \hline
1287
93 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1288
$-2 x^5 + x^4 + x^3$ & HW \\
1289
103 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1290
$x^5 + x^4$ & HW \\
1291
107 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1292
$x^4 - x^2 - x - 1$ & HW \\
1293
115 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1294
$2 x^3 + x^2 + x$ & HW \\ \hline
1295
117,A & $y^2 + (x^3 - 1)y$ & $=$ &
1296
$3 x^3 - 7$ & W \\
1297
117,B & $y^2 + (x^3 + 1)y$ & $=$ &
1298
$-x^6 - 3 x^4 - 5 x^3 - 12 x^2 - 9 x - 7$ & W \\
1299
125,A & $y^2 + (x^3 + x + 1)y$ & $=$ &
1300
$x^5 + 2 x^4 + 2 x^3 + x^2 - x - 1$ & HW \\
1301
125,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
1303
133,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 \\
1305
133,B & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1306
$-x^5 + x^4 - 2 x^3 + 2 x^2 - 2 x$ & HW \\
1307
135 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1308
$x^4 - 3 x^3 + 2 x^2 - 8 x - 3$ & W \\
1309
147 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1310
$x^5 + 2 x^4 + x^3 + x^2 + 1$ & HW \\ \hline
1311
161 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1312
$x^3 + 4 x^2 + 4 x + 1$ & HW \\
1313
165 & $y^2 + (x^3 + x^2 + x)y$ & $=$ &
1314
$x^5 + 2 x^4 + 3 x^3 + x^2 - 3 x$ & H \\
1315
167 & $y^2 + (x^3 + x + 1)y$ & $=$ &
1316
$-x^5 - x^3 - x^2 - 1$ & HW \\
1317
175 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1318
$-x^5 - x^4 - 2 x^3 - 4 x^2 - 2 x - 1$ & W \\ \hline
1319
177 & $y^2 + (x^3 + x^2 + 1)y$ & $=$ &
1320
$x^5 + x^4 + x^3$ & HW \\
1321
188 & $y^2$ & $=$ &
1322
$x^5 - x^4 + x^3 + x^2 - 2 x + 1$ & W \\
1323
189 & $y^2 + (x^3 - 1)y$ & $=$ &
1324
$x^3 - 7$ & W \\
1325
191 & $y^2 + (x^3 + x + 1)y$ & $=$ &
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
1333
In Table~\ref{table2}, we list the curve~$C_N$ simply by~$N$, the
1334
level {}from which it arose. Let $r$ denote the rank. We
1335
list ${\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)$ where $L(J,s)$ is the
1336
$L$-series for the Jacobian $J$ of~$C_N$ and round off the results to
1337
five digits. The symbol $\Omega$ was defined in Section~\ref{Omega}
1338
and is also rounded to five digits. Let Reg denote the regulator,
1339
also rounded to five digits. We list the $c_{p}$'s by primes of
1340
increasing order dividing the level~$N$. We denote $J(\Q)\tors = \Phi$
1341
and list its size. We use $\Sh ?$ to denote the size of
1342
$({\lim}_{s\rightarrow 1}(s-1)^{-r}L(J,s)) \cdot
1343
(\#J(\Q)\tors)^2/(\Omega\cdot {\rm Reg} \cdot \prod c_{p})$,
1344
rounded to the nearest integer. We will refer to this as the {\em conjectured
1345
size of} $\Sh(J,\Q)$.
1346
For rank~$0$ optimal quotients this integer equals the (a priori)
1347
rational number $(L(J,1)/(k\cdot\Omega))\cdot((\#J(\Q)\tors)^2/\prod c_{p})$;
1348
of course there is no rounding error in this computation. For all other cases
1349
the last column gives a bound on the accuracy of the
1350
computations; all values of $\Sh ?$ were at least this close to the
1351
nearest 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
1364
23 & 0 & 0&24843 & 2&7328 & 1 & 11 & 11 & 1 & $ $ \\
1365
29 & 0 & 0&29152 & 2&0407 & 1 & 7 & 7 & 1 & $ $ \\
1366
31 & 0 & 0&44929 & 2&2464 & 1 & 5 & 5 & 1 & $ $ \\
1367
35 & 0 & 0&37275 & 2&9820 & 1 & 16,2 & 16 & 1 & $ < 10^{-25} $ \\
1368
\hline
1369
39 & 0 & 0&38204 & 10&697 & 1 & 28,1 & 28 & 1 & $ < 10^{-25} $ \\
1370
63 & 0 & 0&75328 & 4&5197 & 1 & 2,3 & 6 & 1 & $ $ \\
1371
65,A & 0 & 0&45207 & 6&3289 & 1 & 7,1 & 14 & 2 & $ $ \\
1372
65,B & 0 & 0&91225 & 5&4735 & 1 & 1,3 & 6 & 2 & $ $ \\
1373
\hline
1374
67 & 2 & 0&23410 & 20&465 & 0.011439 & 1 & 1 & 1 & $ < 10^{-50} $ \\
1375
73 & 2 & 0&25812 & 24&093 & 0.010713 & 1 & 1 & 1 & $ < 10^{-49} $ \\
1376
85 & 2 & 0&34334 & 9&1728 & 0.018715 & 4,2 & 2 & 1 & $ < 10^{-26} $ \\
1377
87 & 0 & 1&4323 & 7&1617 & 1 & 5,1 & 5 & 1 & $ $ \\
1378
\hline
1379
93 & 2 & 0&33996 & 18&142 & 0.0046847 & 4,1 & 1 & 1 & $ < 10^{-49} $ \\
1380
103 & 2 & 0&37585 & 16&855 & 0.022299 & 1 & 1 & 1 & $ < 10^{-49} $ \\
1381
107 & 2 & 0&53438 & 11&883 & 0.044970 & 1 & 1 & 1 & $ < 10^{-49} $ \\
1382
115 & 2 & 0&41693 & 10&678 & 0.0097618 & 4,1 & 1 & 1 & $ < 10^{-50} $ \\
1383
\hline
1384
117,A & 0 & 1&0985 & 3&2954 & 1 & 4,3 & 6 & 1 & $ $ \\
1385
117,B & 0 & 1&9510 & 1&9510 & 1 & 4,1 & 2 & 1 & $ $ \\
1386
125,A & 2 & 0&62996 & 13&026 & 0.048361 & 1 & 1 & 1 & $ < 10^{-50} $ \\
1387
125,B & 0 & 2&0842 & 2&6052 & 1 & 5 & 5 & 4 & $ $ \\
1388
\hline
1389
133,A & 0 & 2&2265 & 2&7832 & 1 & 5,1 & 5 & 4 & $ $ \\
1390
133,B & 2 & 0&43884 & 15&318 & 0.028648 & 1,1 & 1 & 1 & $ < 10^{-49} $ \\
1391
135 & 0 & 1&5110 & 4&5331 & 1 & 3,1 & 3 & 1 & $ $ \\
1392
147 & 2 & 0&61816 & 13&616 & 0.045400 & 2,2 & 2 & 1 & $ < 10^{-50} $ \\
1393
\hline
1394
161 & 2 & 0&82364 & 11&871 & 0.017345 & 4,1 & 1 & 1 & $ < 10^{-47} $ \\
1395
165 & 2 & 0&68650 & 9&5431 & 0.071936 & 4,2,2 & 4 & 1 & $ < 10^{-26} $ \\
1396
167 & 2 & 0&91530 & 7&3327 & 0.12482 & 1 & 1 & 1 & $ < 10^{-47} $ \\
1397
175 & 0 & 0&97209 & 4&8605 & 1 & 1,5 & 5 & 1 & $ $ \\
1398
\hline
1399
177 & 2 & 0&90451 & 13&742 & 0.065821 & 1,1 & 1 & 1 & $ < 10^{-45} $ \\
1400
188 & 2 & 1&1708 & 11&519 & 0.011293 & 9,1 & 1 & 1 & $ < 10^{-44} $ \\
1401
189 & 0 & 1&2982 & 3&8946 & 1 & 1,3 & 3 & 1 & $ $ \\
1402
191 & 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
1410
In Table~\ref{table3} are generators of $J(\Q)/J(\Q)\tors$ for the
1411
curves whose Jacobians have Mordell-Weil rank~2. The generators are
1412
given as divisor classes. Whenever possible, we have chosen
1413
generators of the form $[P - Q]$ where $P$ and~$Q$ are rational
1414
points on the curve. Curve~167 is the only example where this is not
1415
the case, since the degree zero divisors supported on the (known)
1416
rational points on~$C_{167}$ generate a subgroup of index two in the
1417
full Mordell-Weil group.
1418
Affine points are given by their $x$ and $y$ coordinates in the model
1419
given in Table~\ref{table1}. There are two points at infinity in the
1420
normalization of the curves described by our equations, with the
1421
exception 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.
1423
The (only) point at infinity on curve~$C_{188}$ is simply
1424
denoted~$\infty$.
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
1439
103 & $ [(0, 0) - \infty_{-1}]$ &
1440
$ [(0, -1) - (0,0)] $ \\
1441
107 & $ [\infty_{-1} - \infty_{0}]$ &
1442
$ [(-1, -1) - \infty_{-1}] $ \\
1443
115 & $ [(1, -4) - \infty_{0}] $ &
1444
$ [(1, 1) - (-2, 2)] $ \\
1445
125,A & $ [\infty_{-1} - \infty_{0}] $ &
1446
$ [(-1, 0) - \infty_{-1}] $ \\ \hline
1447
133,B & $ [\infty_{-1} - \infty_{0}] $ &
1448
$ [(0, -1) - \infty_{-1}] $ \\
1449
147 & $ [\infty_{-1} - \infty_{0}] $ &
1450
$ [(-1, -1) - \infty_{0}] $ \\
1451
161 & $ [(1, 2) - (-1, 1)] $ &
1452
$ [(\frac{1}{2}, -3) - (1, 2)] $ \\
1453
165 & $ [(1, 1) - \infty_{-1}] $ &
1454
$ [(0, 0) - \infty_{0} ] $ \\ \hline
1455
167 & $ [(-1 ,1) - \infty_{0}] $ &
1456
$ [(i, 0) + (-i, 0) - \infty_{0} - \infty_{-1}] $ \\
1457
177 & $ [(0, -1) - \infty_{0}] $ &
1458
$ [(0, 0) - (0, -1)] $ \\
1459
188 & $ [(0, -1) - \infty] $ &
1460
$ [(0, 1) - (1, -2)] $ \\
1461
191 & $ [\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
1470
In Table~\ref{table4} are the reduction types, {}from the
1471
classification of~\cite{NU}, of the special fibers of the minimal,
1472
proper, regular models of the curves for each of the primes of
1473
singular reduction for the curve. They are the same as the primes
1474
dividing the level except that curve~$C_{65,A}$ has singular
1475
reduction at the prime~3 and curve~$C_{65,B}$ has singular reduction
1476
at 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
1485
23 & 23 & ${\rm I}_{3-2-1}$ & & &
1486
117,A & 3 & ${\rm III}-{\rm III}^{\ast}-0$
1487
& 13 & ${\rm I}_{1-1-1}$ \\
1488
29 & 29 & ${\rm I}_{3-1-1}$ & & &
1489
117,B & 3 & ${\rm I}_{3-1-1}^{\ast}$
1490
& 13 & ${\rm I}_{1-1-0}$ \\
1491
31 & 31 & ${\rm I}_{2-1-1}$ & & &
1492
125,A & 5 & ${\rm VIII}-1$ & & \\
1493
35 & 5 & ${\rm I}_{3-2-2}$
1494
& 7 & ${\rm I}_{2-1-0}$ &
1495
125,B & 5 & ${\rm IX}-3$ & & \\ \hline
1496
39 & 3 & ${\rm I}_{6-2-2}$
1497
& 13 & ${\rm I}_{1-1-0}$ &
1498
133,A & 7 & ${\rm I}_{2-1-1}$
1499
& 19 & ${\rm I}_{1-1-0}$ \\
1500
63 & 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 & ${\rm I}_{1-1-0}$ \\
1504
65,A & 3 & ${\rm I}_{0}-{\rm I}_{0}-1$
1505
& 5 & ${\rm I}_{3-1-1}$ &
1506
135 & 3 & III
1507
& 5 & ${\rm I}_{3-1-0}$ \\
1508
65,A & 13 & ${\rm I}_{1-1-0}$ & & &
1509
147 & 3 & ${\rm I}_{2-1-0}$
1510
& 7 & VII \\ \hline
1511
65,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 & ${\rm I}_{1-1-0}$ \\
1515
65,B & 13 & ${\rm I}_{1-1-1}$ & & &
1516
165 & 3 & ${\rm I}_{2-2-0}$
1517
& 5 & ${\rm I}_{2-1-0}$ \\
1518
67 & 67 & ${\rm I}_{1-1-0}$ & & &
1519
165 & 11 & ${\rm I}_{2-1-0}$ & & \\
1520
73 & 73 & ${\rm I}_{1-1-0}$ & & &
1521
167 & 167 & ${\rm I}_{1-1-0}$ & & \\ \hline
1522
85 & 5 & ${\rm I}_{2-2-0}$
1523
& 17 & ${\rm I}_{2-1-0}$ &
1524
175 & 5 & ${\rm II}-{\rm II}-0$
1525
& 7 & ${\rm I}_{2-1-1}$ \\
1526
87 & 3 & ${\rm I}_{2-1-1}$
1527
& 29 & ${\rm I}_{1-1-0}$ &
1528
177 & 3 & ${\rm I}_{1-1-0}$
1529
& 59 & ${\rm I}_{1-1-0}$ \\
1530
93 & 3 & ${\rm I}_{2-2-0}$
1531
& 31 & ${\rm I}_{1-1-0}$ &
1532
188 & 2 & ${\rm IV}-{\rm IV}-0$
1533
& 47 & ${\rm I}_{1-1-0}$ \\
1534
103 & 103 & ${\rm I}_{1-1-0}$ & & &
1535
189 & 3 & ${\rm II}-{\rm IV}^{\ast}-0$
1536
& 7 & ${\rm I}_{1-1-1}$ \\ \hline
1537
107 & 107 & ${\rm I}_{1-1-0}$ & & &
1538
191 & 191 & ${\rm I}_{1-1-0}$ & & \\
1539
115 & 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
1549
second conjecture}
1550
\label{Shah}
1551
1552
{}From Section~\ref{MW} we have
1553
$\dim \Sh(J,\Q)[2] = \dim {\rm Sel}^{2}(J,\Q) - r - \dim J(\Q)[2]$.
1554
With 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
1557
size of $\Sh(J,\Q)$ is~1. For curves $C_{65,A}$, $C_{65,B}$,
1558
$C_{125,B}$ and $C_{133,A}$ we have $\dim \Sh(J,\Q)[2] = 1, 1, 2$
1559
and~2 and the conjectured size of $\Sh(J,\Q) = 2, 2, 4$ and~4,
1560
respectively. We see that in each case, the (conjectured) size of
1561
the odd part of $\Sh(J,\Q)$ is~1 and the 2-part is accounted for by
1562
its 2-torsion.
1563
1564
Recall that for rank 0 optimal quotients we are able to exactly
1565
determine the value which the second Birch and Swinnerton-Dyer
1566
conjecture predicts for $\Sh(J,\Q)$. From the previous paragraph,
1567
we then see that equation~\eqref{eqn1} holds if and only if
1568
$\Sh(J,\Q)$ is killed by $2$.
1569
1570
It is also interesting to consider deficient primes. A prime $p$ is
1571
deficient with respect to a curve $C$ of genus~2, if $C$ has no
1572
degree 1 rational divisor over~$\Q_{p}$. {}From~\cite{PSt}, the
1573
number of deficient primes has the same parity as $\dim \Sh(J,\Q)[2]$.
1574
Curve $C_{65,A}$ has one deficient prime~$3$. Curve
1575
$C_{65,B}$ has one deficient prime~$2$. Curve $C_{117,B}$ has two
1576
deficient primes $3$ and~$\infty$. The rest of the curves have no
1577
deficient primes.
1578
1579
Since we have found $r$ (analytic rank) independent points on each
1580
Jacobian, we have a direct proof that the Mordell-Weil rank must
1581
equal the analytic rank if $\dim \Sh(J,\Q)[2] = 0$. For
1582
curves $C_{65,A}$ and $C_{65,B}$, the presence of an odd number of
1583
deficient primes gives us a
1584
similar result. For $C_{125,B}$ we used a $\sqrt{5}$-Selmer group
1585
to get a similar result.
1586
Thus, we have an independent proof of equality
1587
between analytic and Mordell-Weil ranks for all curves except
1588
$C_{133,A}$.
1589
1590
The 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
1592
pair, the Mordell-Weil rank is~2 for one curve and the 2-torsion of
1593
the Shafarevich-Tate group has dimension~2 for the other. In
1594
addition, the two Jacobians, when canonically embedded into~$J_0(N)$,
1595
intersect in their 2-torsion subgroups, and one can check that their
1596
2-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
1598
of the 2-torsion subgroups. Thus these are examples of the principle
1599
of a `visible part of a Shafarevich-Tate group' as discussed
1600
in~\cite{CM}.
1601
1602
\vspace{5mm}
1603
\begin{center}
1604
{\sc Appendix: Other Hasegawa curves}
1605
\end{center}
1606
1607
In Table~\ref{Hasegawa} is data concerning all 142 of Hasegawa's
1608
curves in the order presented in his paper. Let us explain the
1609
entries. The first column in each set of three columns gives the
1610
level, $N$. The second column gives a classification of the cusp
1611
forms spanning the 2-dimensional subspace of $S_2(N)$ corresponding
1612
to the Jacobian. When that subspace is irreducible with respect to
1613
the action of the Hecke algebra and is spanned by two newforms or two
1614
oldforms, we write $2n$ or $2o$, respectively. When that subspace is
1615
reducible and is spanned by two oldforms, two newforms or one of
1616
each, we write $oo$, $nn$ and $on$, respectively. The third column
1617
contains the sign of the functional equation at the level $M$ at
1618
which the cusp form is a newform. This is the negative of
1619
$\epsilon_M$ (described in Section~\ref{l}). The order of the two
1620
signs in the third column agrees with that of the forms listed in the
1621
second column. We include this information for those who would like
1622
to further study these curves. The curves with $N<200$ classified as
1623
$2n$ appeared already in Table~\ref{table1}.
1624
1625
The smallest possible Mordell-Weil ranks corresponding to $++$, $+-$,
1626
$-+$ and $--$, predicted by the first Birch and Swinnerton-Dyer
1627
conjecture, are $0$, $1$, $1$ and $2$ respectively. In all cases,
1628
those were, in fact, the Mordell-Weil ranks. This was determined by
1629
computing 2-Selmer groups with a computer program based on
1630
\cite{Sto2}. Of course, these are cases where the first Birch and
1631
Swinnerton-Dyer conjecture is already known to hold. In the cases
1632
where the Mordell-Weil rank is positive, the Mordell-Weil group has a
1633
subgroup of finite index generated by degree zero divisors supported
1634
on rational points with $x$-coordinates with numerators bounded by 7
1635
(in absolute value) and denominators by 12 with one exception. On
1636
the second curve with $N=138$, the divisor class
1637
$[(3+2\sqrt{2},80+56\sqrt{2}) + (3-2\sqrt{2},80-56\sqrt{2})-2\infty]$
1638
generates 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
1646
22 & $oo$ & $++$ & 58 & $nn$ & $+-$ & 87 & $2o$ & $++$ & 129 & $on$ & $--$ &
1647
198 & $2o$ & $+-$ \\
1648
23 & $2n$ & $++$ & 60 & $oo$ & $++$ & 88 & $on$ & $+-$ & 130 & $on$ & $-+$ &
1649
204 & $2o$ & $+-$ \\
1650
26 & $nn$ & $++$ & 60 & $2o$ & $++$ & 90 & $on$ & $++$ & 132 & $oo$ & $++$ &
1651
205 & $2n$ & $--$ \\
1652
28 & $oo$ & $++$ & 60 & $2o$ & $++$ & 90 & $oo$ & $++$ & 133 & $2n$ & $--$ &
1653
206 & $2o$ & $--$ \\
1654
29 & $2n$ & $++$ & 62 & $2o$ & $++$ & 90 & $oo$ & $++$ & 134 & $2o$ & $--$ &
1655
209 & $2n$ & $--$ \\
1656
30 & $on$ & $++$ & 66 & $nn$ & $++$ & 90 & $oo$ & $++$ & 135 & $on$ & $+-$ &
1657
210 & $on$ & $+-$ \\
1658
30 & $oo$ & $++$ & 66 & $2o$ & $++$ & 91 & $nn$ & $--$ & 138 & $nn$ & $+-$ &
1659
213 & $2n$ & $--$ \\
1660
30 & $on$ & $++$ & 66 & $2o$ & $++$ & 93 & $2n$ & $--$ & 138 & $on$ & $+-$ &
1661
215 & $on$ & $--$ \\
1662
31 & $2n$ & $++$ & 66 & $on$ & $++$ & 98 & $oo$ & $++$ & 140 & $oo$ & $++$ &
1663
221 & $2n$ & $--$ \\
1664
33 & $on$ & $++$ & 67 & $2n$ & $--$ & 100 & $oo$ & $++$ & 142 & $nn$ & $+-$
1665
& 230 & $2o$ & $--$ \\ \hline
1666
35 & $2n$ & $++$ & 68 & $oo$ & $++$ & 102 & $on$ & $+-$ & 143 & $on$ & $+-$
1667
& 255 & $2o$ & $--$ \\
1668
37 & $nn$ & $+-$ & 69 & $2o$ & $++$ & 102 & $on$ & $+-$ & 146 & $2o$ & $--$
1669
& 266 & $2o$ & $--$ \\
1670
38 & $on$ & $++$ & 70 & $on$ & $++$ & 103 & $2n$ & $--$ & 147 & $2n$ & $--$
1671
& 276 & $2o$ & $+-$ \\
1672
39 & $2n$ & $++$ & 70 & $2o$ & $++$ & 104 & $2o$ & $++$ & 150 & $on$ & $++$
1673
& 284 & $2o$ & $+-$ \\
1674
40 & $on$ & $++$ & 70 & $2o$ & $++$ & 106 & $on$ & $--$ & 153 & $on$ & $+-$
1675
& 285 & $on$ & $--$ \\
1676
40 & $oo$ & $++$ & 70 & $2o$ & $++$ & 107 & $2n$ & $--$ & 154 & $on$ & $--$
1677
& 286 & $on$ & $--$ \\
1678
42 & $on$ & $++$ & 72 & $on$ & $++$ & 110 & $on$ & $++$ & 156 & $oo$ & $++$
1679
& 287 & $2n$ & $--$ \\
1680
42 & $oo$ & $++$ & 72 & $oo$ & $++$ & 111 & $oo$ & $+-$ & 158 & $on$ & $--$
1681
& 299 & $2n$ & $--$ \\
1682
42 & $on$ & $++$ & 73 & $2n$ & $--$ & 112 & $on$ & $+-$ & 161 & $2n$ & $--$
1683
& 330 & $2o$ & $--$ \\
1684
42 & $oo$ & $++$ & 74 & $oo$ & $+-$ & 114 & $oo$ & $+-$ & 165 & $2n$ & $--$
1685
& 357 & $2n$ & $--$ \\ \hline
1686
44 & $2o$ & $++$ & 77 & $on$ & $+-$ & 115 & $2n$ & $--$ & 166 & $on$ & $--$
1687
& 380 & $2o$ & $+-$ \\
1688
46 & $2o$ & $++$ & 78 & $oo$ & $++$ & 116 & $2o$ & $+-$ & 167 & $2n$ & $--$
1689
& 390 & $on$ & $--$ \\
1690
48 & $on$ & $++$ & 78 & $2o$ & $++$ & 117 & $2o$ & $++$ & 168 & $2o$ & $++$
1691
& & & \\
1692
48 & $oo$ & $++$ & 80 & $oo$ & $++$ & 120 & $oo$ & $++$ & 170 & $2o$ & $--$
1693
& & & \\
1694
50 & $nn$ & $++$ & 84 & $oo$ & $++$ & 120 & $on$ & $++$ & 177 & $2n$ & $--$
1695
& & & \\
1696
52 & $oo$ & $++$ & 84 & $oo$ & $++$ & 121 & $on$ & $+-$ & 180 & $2o$ & $++$
1697
& & & \\
1698
52 & $oo$ & $++$ & 84 & $oo$ & $++$ & 122 & $on$ & $--$ & 184 & $on$ & $+-$
1699
& & & \\
1700
54 & $on$ & $++$ & 84 & $oo$ & $++$ & 125 & $2n$ & $--$ & 186 & $2o$ & $--$
1701
& & & \\
1702
57 & $on$ & $+-$ & 85 & $2n$ & $--$ & 126 & $oo$ & $++$ & 190 & $on$ & $+-$
1703
& & & \\
1704
57 & $on$ & $+-$ & 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
1714
\begin{thebibliography}{99}
1715
1716
\bibitem[AS1]{AS1}
1717
A.\ Agash\'{e} and W.A.\ Stein, \textit{Some abelian varieties with visible
1718
Shafarevich-Tate groups}, preprint, 2000.
1719
\bibitem[AS2]{AS2}
1720
A.\ Agash\'{e}, and W.A.\ Stein, \textit{The
1721
generalized Manin constant, congruence
1722
primes, and the modular degree}, in preparation, 2000.
1723
\bibitem[BSD]{BSD}
1724
B.\ Birch and H.P.F.\ Swinnerton-Dyer, \textit{Notes on elliptic curves.
1725
II}, J. reine angew. Math., \textbf{ 218} (1965), 79--108. MR 31
1726
\#3419
1727
\bibitem[BL]{BL}
1728
S.\ Bosch and Q.\ Liu, \textit{Rational points of the group of components
1729
of a N\'{e}ron model}, Manuscripta Math, \textbf{ 98} (1999), 275--293.
1730
\bibitem[BLR]{BLR}
1731
S.\ Bosch, W.\ L\"{u}tkebohmert and M.\ Raynaud, \textit{N\'{e}ron models},
1732
Springer-Verlag, Berlin, 1990. MR \textbf{ 91i}:14034
1733
\bibitem[BGZ]{BGZ}
1734
J.\ Buhler, B.H.\ Gross and D.B.\ Zagier, \textit{On the conjecture of
1735
Birch and Swinnerton-Dyer for an elliptic curve of rank $3$}.
1736
Math. Comp., \textbf{ 44} (1985), 473--481. MR \textbf{ 86g}:11037
1737
\bibitem[Ca]{Ca}
1738
J.W.S.\ Cassels, \textit{Arithmetic on curves of genus 1. VIII. On conjectures
1739
of Birch and Swinnerton-Dyer.},
1740
J. reine angew. Math., \textbf{ 217} (1965), 180--199.
1741
MR 31 \#3420
1742
\bibitem[CF]{CF}
1743
J.W.S.\ Cassels and E.V.\ Flynn, \textit{Prolegomena to a middlebrow
1744
arithmetic of curves of genus~2}, London Math. Soc., Lecture Note Series
1745
230,
1746
Cambridge Univ. Press, Cambridge, 1996. MR \textbf{ 97i}:11071
1747
\bibitem[Cr1]{Cr2}
1748
J.E.\ Cremona, \textit{Abelian varieties with extra twist, cusp forms, and
1749
elliptic curves over imaginary quadratic fields},
1750
J. London Math.\ Soc.\ (2), \textbf{ 45} (1992), 404--416.
1751
MR \textbf{ 93h}:11056
1752
\bibitem[Cr2]{Cr}
1753
J.E.\ Cremona, \textit{Algorithms for modular elliptic curves. 2nd edition},
1754
Cambridge Univ. Press, Cambridge, 1997. MR \textbf{ 93m}:11053
1755
\bibitem[CM]{CM}
1756
J.E.\ Cremona and B.\ Mazur,
1757
\textit{Visualizing elements in the Shafarevich-Tate group},
1758
to appear in {\it Experiment.\ Math.}
1759
\bibitem[Ed1]{Ed1}
1760
B.\ Edixhoven, \textit{On
1761
the Manin constants of modular elliptic curves}, Arithmetic
1762
algebraic geometry (Texel, 1989), Progr. Math., 89, Birkhauser Boston,
1763
Boston, MA, 1991, pp.\ 25--39.
1764
\bibitem[Ed2]{Ed2}
1765
B.\ Edixhoven, \textit{L'action de l'alg\`{e}bre de Hecke sur les groupes de
1766
composantes des jacobiennes des courbes modulaires est ``Eisenstein''},
1767
Ast\'{e}risque, No.\ 196--197 (1992), 159--170. MR \textbf{ 92k}:11059
1768
\bibitem[FPS]{FPS}
1769
E.V.\ Flynn, B.\ Poonen and E.F.\ Schaefer, \textit{Cycles of quadratic
1770
polynomials and rational points on a genus-two curve}, Duke Math.\ J.,
1771
\textbf{ 90} (1997), 435--463. MR \textbf{ 98j}:11048
1772
\bibitem[FS]{FS}
1773
E.V.\ Flynn and N.P.\ Smart, \textit{Canonical heights on the Jacobians
1774
of curves of genus~2 and the infinite descent}, Acta Arith.,
1775
\textbf{ 79} (1997), 333--352. MR \textbf{ 98f}:11066
1776
\bibitem[FM]{FM}
1777
G.\ Frey and M.\ M\"{u}ller, \textit{Arithmetic
1778
of modular curves and applications},
1779
in {\it Algorithmic algebra and number theory}, Ed.\ Matzat et al.,
1780
Springer-Verlag, Berlin, 1999, pp.\ 11--48. MR \textbf{ 00a}:11095
1781
\bibitem[GZ]{GZ}
1782
B.H.\ Gross and D.B.\ Zagier,
1783
\textit{Heegner points and derivatives of $L$-series},
1784
Invent. Math.,
1785
\textbf{ 84} (1986), 225--320. MR \textbf{ 87j}:11057
1786
\bibitem[Gr]{Gr}
1787
A.\ Grothendieck, \textit{Groupes de monodromie en g\'eom\'etrie alg\'ebrique},
1788
SGA 7 I, Expos\'{e} IX, Lecture Notes in Math.\ vol.\
1789
288, Springer, Berlin--Heidelberg--New York, 1972, pp.\ 313--523.
1790
MR 50 \#7134
1791
\bibitem[Ha]{Ha}
1792
R.\ Hartshorne, \textit{Algebraic geometry}, Grad.\ Texts in Math.\ 52,
1793
Springer-Verlag, New York, 1977.
1794
MR 57 \#3116
1795
\bibitem[Hs]{Hs}
1796
Y.\ Hasegawa, \textit{Table of quotient curves of modular curves $X_0(N)$
1797
with genus~2}, Proc.\ Japan.\ Acad., \textbf{ 71} (1995), 235--239.
1798
MR \textbf{ 97e}:11071
1799
\bibitem[Ko]{Ko}
1800
V.A.\ Kolyvagin, \textit{Finiteness of $E(\Q)$ and $\Sh (E,\Q)$ for a subclass
1801
of Weil curves}, Izv.\ Akad.\ Nauk SSSR Ser.\ Mat., \textbf{ 52} (1988),
1802
522--540. MR \textbf{ 89m}:11056
1803
\bibitem[KL]{KL}
1804
V.A.\ Kolyvagin and D.Y.\ Logachev, \textit{Finiteness of the Shafarevich-Tate
1805
group and the group of rational points for some modular abelian varieties},
1806
Leningrad Math J., \textbf{ 1} (1990), 1229--1253. MR \textbf{ 91c}:11032
1807
\bibitem[La]{La}
1808
S.\ Lang, \textit{Introduction to modular forms}, Springer-Verlag, Berlin, 1976.
1809
MR 55 \#2751
1810
\bibitem[Le]{Le}
1811
F.\ Lepr\'{e}vost, \textit{Jacobiennes de certaines courbes de genre 2:
1812
torsion et simplicit\'e}, J. Th\'eor. Nombres Bordeaux, \textbf{ 7} (1995),
1813
283--306. MR \textbf{ 98a}:11078
1814
\bibitem[Li]{Li}
1815
Q.\ Liu, \textit{Conducteur et discriminant minimal de courbes de genre 2},
1816
Compos.\ Math., \textbf{ 94} (1994), 51--79. MR \textbf{ 96b}:14038
1817
\bibitem[Ma]{Ma}
1818
B.\ Mazur, \textit{Rational isogenies of prime degree (with an appendix by D.
1819
Goldfeld)}, Invent.\ Math., ~\textbf{ 44} (1978), 129--162.
1820
MR \textbf{ 80h}:14022
1821
\bibitem[MS]{MS}
1822
J.R.\ Merriman and N.P.\ Smart, \textit{Curves of genus~2 with good reduction
1823
away {}from 2 with a rational Weierstrass point}, Math.\ Proc.\ Cambridge
1824
Philos.\ Soc., \textbf{114} (1993), 203--214. MR \textbf{ 94h}:14031
1825
\bibitem[Mi1]{Mi1}
1826
J.S.\ Milne, \textit{Arithmetic duality theorems}, Academic Press, Boston, 1986.
1827
MR \textbf{ 88e}:14028
1828
\bibitem[Mi2]{Mi2}
1829
J.S.\ Milne, \textit{Jacobian varieties}, in: {\it Arithmetic geometry},
1830
Ed.\ G.\ Cornell, G. and J.H.\ Silverman, Springer-Verlag, New York, 1986,
1831
pp.\ 167--212. MR \textbf{ 89b}:14029
1832
\bibitem[NU]{NU}
1833
Y.\ Namikawa and K.\ Ueno, \textit{The complete classification of fibres in
1834
pencils of curves of genus two}, Manuscripta Math., \textbf{ 9} (1973),
1835
143--186. MR 51 \#5595
1836
\bibitem[PSc]{PSc}
1837
B.\ Poonen and E.F.\ Schaefer, \textit{Explicit descent for Jacobians of
1838
cyclic covers of the projective line}, J. reine angew. Math.,
1839
\textbf{ 488} (1997), 141--188. MR \textbf{ 98k}:11087
1840
\bibitem[PSt]{PSt}
1841
B.\ Poonen and M.\ Stoll, \textit{The Cassels-Tate pairing on polarized
1842
abelian varieties}, Ann.\ of Math. (2), \textbf{150} (1999), 1109--1149.
1843
\bibitem[Ri]{Ri}
1844
K.\ Ribet, \textit{On modular representations of $\Gal(\Qbar/\Q)$
1845
arising from modular forms}, Invent. math.,
1846
\textbf{100} (1990), 431--476. MR \textbf{ 91g}:11066
1847
\bibitem[Sc]{Sc}
1848
E.F.\ Schaefer, \textit{Computing a Selmer group of a Jacobian using functions
1849
on the curve}, Math.\ Ann., \textbf{310} (1998), 447-471.
1850
MR \textbf{ 99h}:11063
1851
\bibitem[Sh]{Sh}
1852
G.\ Shimura, \textit{Introduction to the arithmetic theory of
1853
automorphic functions}, Princeton University Press, 1994.
1854
MR \textbf{ 95e}:11048
1855
\bibitem[Si]{Si}
1856
J.H.\ Silverman, \textit{Advanced topics in the arithmetic of elliptic curves},
1857
Grad.\ Texts in Math.\ 151, Springer-Verlag, New York,
1858
1994. MR \textbf{ 96b}:11074
1859
\bibitem[Ste]{Ste}
1860
W.A.\ Stein, \textit{Component groups of optimal quotients of Jacobians},
1861
preprint, 2000.
1862
\bibitem[Sto1]{Sto1}
1863
M.\ Stoll, \textit{Two simple 2-dimensional abelian varieties
1864
defined over~$\Q$ with Mordell-Weil rank at least~$19$},
1865
C. R. Acad. Sci. Paris, S\'erie I, \textbf{321} (1995), 1341--1344.
1866
MR \textbf{ 96j}:11084
1867
\bibitem[Sto2]{Sto2}
1868
M.\ Stoll, \textit{Implementing 2-descent in genus~2}, preprint, 1999.
1869
\bibitem[Sto3]{Sto3}
1870
M.\ Stoll, \textit{On the height constant for curves of genus two},
1871
Acta Arith, \textbf{90} (1999), 183--201.
1872
\bibitem[Sto4]{Sto4}
1873
M.\ Stoll, \textit{On the height constant for curves of genus two, II},
1874
in preparation.
1875
\bibitem[Ta]{Ta}
1876
J.\ Tate, \textit{On the conjectures of Birch and Swinneron-Dyer and a geometric
1877
analog}. S\'{e}minaire Bourbaki, \textbf{ 306} 1965/1966