1% \documentclass{amsart}
2\documentclass[12pt]{amsart}
3\usepackage{amscd}
4\newfont{\cyr}{wncyr10 scaled \magstep1}
5\newcommand{\Sh}{\hbox{\cyr Sh}}
6\newcommand{\C}{{\mathbf C}}
7\newcommand{\Q}{{\mathbf Q}}
8\newcommand{\Qbar}{\overline{\Q}}
9%\newcommand{\GalQ}{{\Gal}(\Qbar/\Q)}
10\newcommand{\CC}{{\mathcal C}}
11\newcommand{\Z}{{\mathbf Z}}
12\newcommand{\R}{{\mathbf R}}
13\newcommand{\F}{{\mathbf F}}
14\newcommand{\G}{{\mathbf G}}
15\newcommand{\OO}{{\mathcal O}}
16\newcommand{\JJ}{{\mathcal J}}
17\newcommand{\DD}{{\mathcal D}}
18\newcommand{\aaa}{{\mathfrak a}}
19\newcommand{\PP}{{\mathbf P}}
20\newcommand{\tors}{_{\text{tors}}}
21\newcommand{\unr}{^{\text{unr}}}
22\newcommand{\nichts}{{\left.\right.}}
23
24
25\DeclareMathOperator{\Gal}{Gal}
26\DeclareMathOperator{\Norm}{Norm}
27\DeclareMathOperator{\Sel}{Sel}
28\DeclareMathOperator{\Tr}{Tr}
29
30\newtheorem{theorem}{Theorem}[section]
31\newtheorem{lemma}[theorem]{Lemma}
32\newtheorem{cor}[theorem]{Corollary}
33\newtheorem{prop}[theorem]{Proposition}
34
35\theoremstyle{definition}
36\newtheorem{question}{Question}
37\newtheorem{conj}{Conjecture}
38
39\theoremstyle{remark}
40\newtheorem{rem}{Remark$\!\!$}		\renewcommand{\therem}{}
41\newtheorem{rems}{Remarks$\!\!$}	\renewcommand{\therems}{}
42
43\topmargin -0.3in
44\headsep 0.3in
45\oddsidemargin 0in
46\evensidemargin 0in
47\textwidth 6.5in
48\textheight 9in
49
50%%%\renewcommand{\baselinestretch}{2}
51
52\begin{document}
53\noindent{\bf Tables to make it clear what we computed using modular methods.}
54Pages 1--2 contain a table which interleaves the modular and non-modular computations.
55Page 3 contains just the modular computations.\vspace{2ex}
56
57\noindent{\bf Modular computations for optimal quotients. }
58The rows labeled with $N$ were computed by Flynn, Lepr\'{e}vost, Schaefer,
59Stoll, and Wetherell and refer to the curves of Table 1 of {\em Empirical evidence}.
60The rows labeled in bold below $A$ were computed by me (Stein)
61using modular algorithms,'' and refer to the corresponding
62optimal quotients of $J_0(N)$.   There are exactly four cases for
63which our computations definitely do not agree: these are the four
64Hasegawa  (but not Wang) curves.  This is fine because these Hasegawa
65curves are not optimal.
66The notation $a|T|b$ means that $a$ divides $T$ and $T$ divides $b$.
67\vspace{3ex}
68
69\newcommand{\mcc}[1]{\multicolumn{1}{|c|}{#1}}
70\newcommand{\mcd}[1]{\multicolumn{2}{|c|}{#1}}
71\begin{center}
72\begin{tabular}{|l|l|c|r@{.}l|r@{.}l|l|l|c|c|}
73\hline
74\mcc{$A$} & \mcc{$N$} & $r$
75& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
76& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & T & \mcc{disagreement}
77\\ \hline\hline
78%  A        N        r   L(1)       Om       Reg         cp's   Tor
79           & 23    & 0 & 0&24843 &  2&7328 & 1         & 11    & 11 & \\
80{\bf 23A}  &       &   & 0&24843 &  2&7328 & 1         & 11    & 11 & \\
81           & 29    & 0 & 0&29152 &  2&0407 & 1         & 7     &  7 & \\
82{\bf 29A}  &       & - & 0&29152 &  2&0407 & -         & 7     &  7 & \\
83           & 31    & 0 & 0&44929 &  2&2464 & 1         & 5     &  5 &  \\
84{\bf 31A}  &       & - & 0&44929 &  2&2464 & -         & 5     &  5 & \\
85\hline
86           & 35    & 0 & 0&37275 &  2&9820 & 1         & 16,2  & 16  & \\
87{\bf 35B}  &       & - & 0&37275 &  2&9820 & -         & 8,$2|c_7 | 4$   & $8|T|16$   & *\\
88           & 39    & 0 & 0&38204 & 10&697  & 1         & 28,1  & 28 &  \\
89{\bf 39B}  &       & - & 0&38204 & 10&697 & -         & 14,2  & $14|T|28$  &  *\\
90           & 63    & 0 & 0&75328 &  4&5197 & 1         & 2,3   &  6  &  \\
91{\bf 63B}  &       & - & 0&75328 &  4&5197 & -         & ?,3   &  $6|T|12$ &\\
92           & 65,A  & 0 & 0&45207 &  6&3289 & 1         & 7,1   & 14 &  \\
93{\bf 65C}  &       & - & 0&45207 &  6&3289 & -         & 7,1   & $7|T|14$&   \\
94           & 65,B  & 0 & 0&91225 &  5&4735 & 1         & 1,3   &  6  & \\
95{\bf 65B}  &       & - & 0&91225 &  5&4735 & -         & 1,3   & $3|T|6$ &   \\
96\hline
97           & 67    & 2 & 0&23410 & 20&465  & 0.011439  & 1     &  1  & \\
98{\bf 67B}  &       &   &  &      & 20&465  & -         & 1     &  1  & \\
99           & 73    & 2 & 0&25812 & 24&093  & 0.010713  & 1     &  1  &  \\
100{\bf 73B}  &       &   &  &      & 24&093  & -         & 1     &  1 & \\
101           & 85    & 2 & 0&34334 &  9&1728 & 0.018715  & 4,2   &  2  &  \\
102{\bf 85B}  &       &   &  &      & 18&3455 & -         & 2,1   &  $1|T|2$ & *\\
103           & 87    & 0 & 1&4323  &  7&1617 & 1         & 5,1   &  5  &  \\
104{\bf 87A}  &       &   & 1&4323  &  7&1617 & -         & 5,1   &  5  & \\
105\hline
106           & 93    & 2 & 0&33996 & 18&142  & 0.0046847 & 4,1   &  1  & \\
107{\bf 93A}  &       &   &  &      & 18&142  & -         & $2|c_3|4$,1 &  1 & \\
108           & 103   & 2 & 0&37585 & 16&855  & 0.022299  & 1     &  1  &  \\
109{\bf 103A} &       &   &  &      & 16&855  & -         & 1     &  1  & \\
110           & 107   & 2 & 0&53438 & 11&883  & 0.044970  & 1     &  1  &  \\
111{\bf 107A} &       &   &  &      & 11&883  & -         & 1     &  1  & \\
112           & 115   & 2 & 0&41693 & 10&678  & 0.0097618 & 4,1             &  1  &  \\
113{\bf 115B} &       &   &  &      & 10&678  & -         & $2|c_5|4$, 1    & 1  & \\
114\hline
115\end{tabular}
116\vfill
117\mbox{}
118
119\begin{tabular}{|l|l|c|r@{.}l|r@{.}l|l|l|c|c|}
120\hline
121\mcc{$A$} & \mcc{$C$} & $r$
122& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
123& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & $|\text{T}|$ & \mcc{disagreement}
124\\ \hline\hline
125%  A        N        r   L(1)       Om       Reg         cp's   Tor
126            & 117,A & 0 & 1&0985  &  3&2954 & 1         & 4,3   &  6  &  \\
127{\bf 117B}  &       &   & 1&0985  &  3&2954 &           & ?,3   & $3|T|12$  & \\
128            & 117,B & 0 & 1&9510  &  1&9510 & 1         & 4,1   &  2  & \\
129{\bf 117C}  &       &   & 1&9510  &  1&9510 &           & ?,1   & $1|T|2^2$ & \\
130            & 125,A & 2 & 0&62996 & 13&026  & 0.048361  & 1     &  1  &  \\
131{\bf 125A}  &       &   &  &      & 13&026  & -         & ?     &  1 & \\
132            & 125,B & 0 & 2&0842  &  2&6052 & 1         & 5     &  5  &  \\
133{\bf 125B}  &       &   & 2&0842  &  2&6052 & -         & ?     &  5 & \\
134\hline
135            & 133,A & 0 & 2&2265  &  2&7832 & 1         & 5,1   &  5  &  \\
136{\bf 133B}  &       &   & 2&2265  &  2&7832 & -         & 5,1   &  5  & \\
137            & 133,B & 2 & 0&43884 & 15&318  & 0.028648  & 1,1   &  1  &  \\
138{\bf 133A}  &       &   &  &      & 15&318  &           & 1,1   &  1 & \\
139            & 135   & 0 & 1&5110  &  4&5331 & 1         & 3,1   &  3  &  \\
140{\bf 135D}  &       &   & 1&5110  &  4&5331 & -         & ?,1   &  3 & \\
141            & 147   & 2 & 0&61816 & 13&616  & 0.045400  & 2,2   &  2  &  \\
142{\bf 147D}  &       &   &  &      & 13&616  & -         & ?,2   &  $1|T|2$ & \\
143\hline
144            & 161   & 2 & 0&82364 & 11&871  & 0.017345  & 4,1   &  1  &  \\
145{\bf 161B}  &       &   &  &      & 11&871  & -         & $2|T|4$,1  & 1   & \\
146            & 165   & 2 & 0&68650 &  9&5431 & 0.071936  & 4,2,2 &  4  &  \\
147{\bf 165A}  &       &   &  &      &  9&5431 & -         & 2,1,1 &  $1|T|4$ & *\\
148            & 167   & 2 & 0&91530 &  7&3327 & 0.12482   & 1     &  1  &  \\
149{\bf 167A}  &       &   &  &      &  7&3327 & -         & 1     &  1  & \\
150            & 175   & 0 & 0&97209 &  4&8605 & 1         & 1,5   &  5  &  \\
151{\bf 175E}  &       &   & 0&972   &  4&86     &           & ?,5   &  5  & \\
152\hline
153            & 177   & 2 & 0&90451 & 13&742  & 0.065821  & 1,1   &  1 &  \\
154{\bf 177A}  &       &   &  &      & 13&742  & -         & 1,1    & 1 & \\
155            & 188   & 2 & 1&1708  & 11&519  & 0.011293  & 9,1   &  1 &  \\
156 {\bf 188B} &       &   &  &      & 11&519  & -         & ?,1      & 1  & \\
157            & 189   & 0 & 1&2982  &  3&8946 & 1         & 1,3   &  3 &  \\
158 {\bf 189E} &       &   & 1&2982  &  3&894  & -          &?,3    & 3  & \\
159            & 191   & 2 & 0&95958 & 17&357  & 0.055286  & 1     &  1 &  \\
160 {\bf 191A} &       &   &  &      & 17&35   &           & 1     &  1 & \\
161\hline
162\end{tabular}
163\end{center}
164
165\newpage
166\noindent
167{\bf Part of table computed using modular algorithms.}
168For clarity, here is just the part of the table which can be computed using only
169modular algorithms.  Vanishing or not of $L(J,1)$ was also
170computed and agreed in all cases.  I have no information about
171the $c_p$ when $p^2\mid N$.  My understanding is that the value
172$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$
173was also computed by Joe (?) using modular symbols.  I haven't checked
174this computation when $r>0$ because I haven't implimented the algorithm.
175In all but 6 of the optimal cases, the exact torsion subgroup of the optimal
176quotient could be determined using Hecke operators and modular symbols
177(we don't mention this in the paper).
178\vspace{2ex}
179
180\begin{center}
181\begin{tabular}{|l|c|r@{.}l|r@{.}l|l|l|c|c|}
182\hline
183\mcc{$A$} & $r$
184& \mcd{$\lim\limits_{s\rightarrow 1}\frac{L(J,s)}{(s-1)^{r}}$}
185& \mcd{$\Omega$} & \mcc{Reg} & \mcc{$c_{p}$'s} & T
186\\ \hline\hline
187%  A               r   L(1)       Om       Reg         cp's   Tor
188{\bf 23A}         &   & 0&24843 &  2&7328 & 1         & 11    & 11 \\
189{\bf 29A}         & - & 0&29152 &  2&0407 & -         & 7     &  7 \\
190{\bf 31A}         & - & 0&44929 &  2&2464 & -         & 5     &  5 \\
191\hline
192{\bf 35B}         & - & 0&37275 &  2&9820 & -         & 8,$2|c_7 | 4$   & $8|T|16$  \\
193{\bf 39B}         & - & 0&38204 & 10&697 & -         & 14,2  & $14|T|28$ \\
194{\bf 63B}         & - & 0&75328 &  4&5197 & -         & ?,3   &  $6|T|12$\\
195{\bf 65C}         & - & 0&45207 &  6&3289 & -         & 7,1   & $7|T|14$\\
196{\bf 65B}         & - & 0&91225 &  5&4735 & -         & 1,3   & $3|T|6$ \\
197\hline
198{\bf 67B}         &   &  &      & 20&465  & -         & 1     &  1  \\
199{\bf 73B}         &   &  &      & 24&093  & -         & 1     &  1 \\
200{\bf 85B}         &   &  &      & 18&3455 & -         & 2,1   &  $1|T|2$ \\
201{\bf 87A}         &   & 1&4323  &  7&1617 & -         & 5,1   &  5  \\
202\hline
203{\bf 93A}        &   &  &      & 18&142  & -         & $2|c_3|4$,1 &  1 \\
204{\bf 103A}        &   &  &      & 16&855  & -         & 1     &  1  \\
205{\bf 107A}        &   &  &      & 11&883  & -         & 1     &  1  \\
206{\bf 115B}        &   &  &      & 10&678  & -         & $2|c_5|4$, 1    & 1  \\
207\hline
208{\bf 117B}         &   & 1&0985  &  3&2954 &           & ?,3   & $3|T|12$  \\
209{\bf 117C}         &   & 1&9510  &  1&9510 &           & ?,1   & $1|T|2^2$ \\
210{\bf 125A}         &   &  &      & 13&026  & -         & ?     &  1 \\
211{\bf 125B}         &   & 2&0842  &  2&6052 & -         & ?     &  5 \\
212\hline
213{\bf 133B}         &   & 2&2265  &  2&7832 & -         & 5,1   &  5  \\
214{\bf 133A}         &   &  &      & 15&318  &           & 1,1   &  1 \\
215{\bf 135D}         &   & 1&5110  &  4&5331 & -         & ?,1   &  3  \\
216{\bf 147D}         &   &  &      & 13&616  & -         & ?,2   &  $1|T|2$ \\
217\hline
218{\bf 161B}         &   &  &      & 11&871  & -         & $2|T|4$,1  & 1   \\
219{\bf 165A}         &   &  &      &  9&5431 & -         & 2,1,1 &  $1|T|4$ \\
220{\bf 167A}         &   &  &      &  7&3327 & -         & 1     &  1  \\
221{\bf 175E}         &   & 0&972   &  4&86     &           & ?,5   &  5   \\
222\hline
223{\bf 177A}         &   &  &      & 13&742  & -         & 1,1    & 1\\
224 {\bf 188B}        &   &  &      & 11&519  & -         & ?,1      & 1  \\
225 {\bf 189E}        &   & 1&2982  &  3&894  & -          &?,3    & 3  \\
226 {\bf 191A}        &   &  &      & 17&35   &           & 1     &  1  \\
227\hline
228\end{tabular}
229\end{center}
230
231
232\end{document}
233
234
235
236
237
238
239