CoCalc Public Fileswww / papers / thesis-old / vissha.tex
Author: William A. Stein
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1% vissha.tex
2\chapter{Application: The Birch and Swinnerton-Dyer conjecture}
3\label{chap:bsd}
4In this chapter, we study visibility of Shafarevich-Tate
5group of abelian varieties.
6
7Consider an abelian variety~$A$ and an injection
8$\iota:A\ra J$ of~$A$ into an abelian variety~$J$.
9Mazur (see~\cite{cremona-mazur})
10introduced the subgroup $H^1(\Q,A)^\circ$\label{defn:visiblepart}
11of elements of $H^1(\Q,A)$ \defn{visible inside of~$J$}.
12This is the kernel of the map $H^1(\Q,A)\ra H^1(\Q,J)$
13induced by~$\iota$.
14Of particular interest is the visible subgroup $\Sha(A/\Q)^\circ$
15consisting of elements of $\Sha(A/\Q)$ that lie in
16$H^1(\Q,A)^\circ$.
17
18In~\cite{cremona-mazur}, Cremona and Mazur asked the following
19question: if $E\subset J_0(N)$ is a elliptic curve, how much of
20$\Sha(E/\Q)$ is visible in an abelian surface inside~$J_0(N)$?
21In their numerical experiment, visibility appears to occur surprisingly
22often.  However, they warn that the conductors considered ($N\leq 5500$)
23are quite small, and the situation is likely to change as the conductors
24get larger.
25
26In~\cite{mazur:visthree}, Mazur
27proved that if~$E$ is an elliptic curve over a number field~$K$, then
28every element of order~$3$ in $\Sha(E/K)$ is visible in some abelian
29surface.
30
31In this paper,  we consider questions of visibility for abelian
32varieties~$A$ of dimension possibly greater than one.
33This extension is important for several reasons.  The numbers become
34large at much smaller conductor, so behavior that is not
35observed when considering elliptic curves becomes apparent.
36Certain moduli surfaces are of general type, so one does not
37expect visibility, in abelian surfaces,
38of large order Shafarevich-Tate group of elliptic curves; it is thus
39essential to consider visibility in higher dimensional abelian
40varieties. J.~de Jong proved in~\cite[\S3]{cremona-mazur}
41that every element of $\Sha(E/\Q)$ becomes visible in some abelian variety.
42
43%Because the conductors in our computations are small, it
44%is feasible to investigate whether an element of~$\Sha$ that
45%is invisible at level~$N$ becomes visible at level~$\ell N$
46%for~$\ell$ a small prime.
47
48\section{The Birch and Swinnerton-Dyer conjecture}
49In~\cite{tate:bsd}, Tate stated a generalization of the
50conjecture of Birch and Swinnerton-Dyer to abelian varieties over
51number fields.  We recall a special case of the conjecture.
52Consider an abelian variety~$A$ over~$\Q$ that is a quotient
53of~$J_0(N)$ for some~$N$, and let~$\cA/\Z$ be its N\'{e}ron model.
54For each prime~$p$, let $C_{A,p}$ denote the
55component group of the closed fiber $\cA_{\Fp}$.
56A basis $h_1,\ldots,h_d$ for $H^0(\cA,\Omega_{\cA/\Z})$
57defines a measure~$\mu$ on $A(\R)$.
58Under the further assumption that $L(A,1)\neq 0$,
59it is proved in \cite{kolyvagin-logachev:finiteness} that~$A(\Q)$
60and~$\Sha(A/\Q)$ are both finite groups.
61Birch, Swinnerton-Dyer, and Tate gave a conjectural relationship
62between all of these finite groups that expresses the special
63value at~$1$ of $L(A,s)$ in terms of the volumes of all of these
64groups of rational points.
65\begin{conjecture}\label{conj:bsd}
66If~$A$ is a modular abelian variety of analytic rank~$0$, then
67$$L(A,1) = \frac{\prod_{p|N} \#C_{A,p}(\Fp)} 68 {\#A(\Q)\cdot\#\Adual(\Q)} 69\cdot \#\Sha(A/\Q) \cdot \mu(A(\R)). 70$$
71\end{conjecture}
72As evidence for this conjecture, Tate proved that both the left and
73right hand sides are invariant under isogeny.  Recent work of Kato
74gives bounds on $\#\Sha(A/\Q)$ that are consistent with this conjecture.
75[[Reference the blue book with the article on this by Scholl.]]
76
77In what follows we will frequently write $c_p=\#C_{A,p}(\Fp)$
78and $\Omega_\cA=\mu(A(\R))$.
79
80\section{Tables}
81This section contains tables computed using the above algorithms
82as implemented in the author's program {\tt HECKE} (a {\tt C++} program using
83{\tt LiDIA} and {\tt NTL}), David Kohel's {\tt Magma} software,
84and {\tt PARI}.
85Each factor $A_f$ of $J_0(N)$ is denoted as follows:
86\begin{center}
87{\bf  N\, isogeny-class\, dimension}
88\end{center}
89The dimension frequently determines the factor, so it
90is included in the notation.
91We consider only the odd part of $\Sha$ so we
92only computed the odd parts of the arithmetic invariants
93of $A_f$.  Thus at this point we make the
94\begin{center}
95{\bf WARNING: ONLY ODD PARTS OF INVARIANTS ARE GIVEN!}\\
96\end{center}
97
98\noindent{\bf Tables 1-3: New Visible $\Sha$}\\
99Let $n_f$ be the largest odd square dividing the numerator
100of $L(A_f,1)/\Omega_f$.  Table 1 lists those $A_f$ such that,
101for $p|n_f$ there exists a {\em new} factor $B_g$ of $J_0(N)$,
102of positive analytic rank, and such that
103$(\Adual_f\intersect B_g^{\vee})[p]\neq 0$.
104This is necessary (and usually sufficient) for the $p$-torsion
105in the new visible part of $\Sha$ to be nonzero.
106In many cases it could be seen that there were no other appropriate
107new factors by looking at the signs of the Atkin-Lehner involutions.
108Up to level $1001$ our search was systematic.
109The two examples after level $1001$ were not found by
110systematic search (i.e., there may be a gap).
111In those cases for which $4|N$, we put $c_2=a$, as we don't know
112how to compute $c_2$ exactly when the reduction is additive.
113Table 2 contains further arithmetic information about each
114explanatory factor.
115%Table 3 gives the prime factorizations
116%of the composite levels.
117
118The explanatory factors of level $\leq 1028$ are {\em exactly} the set of
119rank $2$ elliptic curves of level $\leq 1028$.
120By \cite{brumer:rank}, the explanatory factor at level $1061$ is the
121first surface of rank $4$ (and prime level).
122
123\noindent{\bf Table 4: Component groups}\\
124Table 4 gives the quantities involved in the formula for Tamagawa numbers,
125for each of the $A_f$ from table 1.
126
127\noindent{\bf Table 5: Odd square numerator}\\
128In order to find the $A_f$, we first enumerated those
129$A_f$ for which the numerator
130of  $L(A_f,1)/\Omega_f$ is divisible by an odd square $n_f$.
131For $N<1000$, these are given in table 5.
132Any odd visible $\Sha$ coprime to
133primes dividing torsion and $c_p$ must show up as a divisor
134of the numerator, and given BSD, it must show up as a square
135divisor because the Mordell-Weil rank of the explanatory factor
136is even.  It would be interesting to compute the conjectural
137order of $\Sha$ for each abelian variety in this table, but
138not in table 1, and show (when possible) that the visible $\Sha$
139is old.
140
141\noindent{\bf Large Sha}\\
142Upon seeing an early draft of this paper,
143Armand Brumer provided us with the first quotients of $J_0(p)$
144having positive rank and dimension $3$ and $4$.  Computing further
145I've found  the following spectacular table.
146The listed orders of $\Sha$ {\em really} is the numerator
147of the $L$-function as a long computation showed.
148
149\begin{center}
150$$\begin{array}{|lcl|}\hline 151\mathbf{A_f} & \text{divisior of |\Sha|} & \mathbf{B_g}\\\hline 152\mathbf{389E20} &5 &\mathbf{389A1}\\ 153\mathbf{1061D46}&151 &\mathbf{1061B2}\\ 154\mathbf{1693B72}&1301 &\mathbf{1693A3}\\ 155\mathbf{2593B109}&67\cdot 2213 &\mathbf{2593A4}\\\hline 156\end{array}$$
157\end{center}
158
159For 1693 the odd part of the special value is $1301^2/141$.
160For 2593 it is $67^2\cdot 2213^2/3^3$.
161
162
163\begin{table}\label{table:newvis}
164\caption{New visible Shafarevich-Tate}\vspace{-2ex}
165$$\begin{array}{|lccccccl|}\hline 166\mathbf{A_f} &n_f &w_q& c_p &T&TL(1)/\Omega_f &\delta_A & \mathbf{B_g}\\\hline 167%\vspace{-2.5ex} & & &&& & & \\ 168\mathbf{389E20}&5^2&-&97&97&5^2&5&\mathbf{389A1}\\ 169\mathbf{433D16}&7^2&-&3^2&3^2&7^2&3\cdot 7\cdot 37&\mathbf{433A1}\\ 170\mathbf{446F8}&11^2&+- &1,3&3&11^2&11\cdot359353&\mathbf{446B1}\\ 171\mathbf{563E31} & 13^2 & - & 281 &281 &13^2 &13 &\mathbf{563A1}\\ 172\mathbf{571D2} & 3^2 & - & 1 &1 & 3^2 & 3^2\cdot 127&\mathbf{571B1}\\ 173\mathbf{655D13} & 3^4 &+- & 1,1 & 1 & 3^4 & 3^2\cdot 19\cdot 515741&\mathbf{655A1}\\ 174 175\mathbf{664F8}&5^2 &-+ &a,1&1& 5^2 &5 & \mathbf{664A1}\\ 176% Sha dim Wq c_p T TL/O delta B 177\mathbf{681B1}& 3^2 &+- & 1,1 & 1 & 3^2 & 3\cdot 5^3 & \mathbf{681C1}\\ 178\mathbf{707G15}& 13^2 &+- & 1,1 & 1 & 13^2 & 13\cdot 800077& \mathbf{707A1}\\ 179\mathbf{709C30}& 11^2 &- & 59 &59 & 11^2 & 11 & \mathbf{709A1}\\ 180\mathbf{718F7}& 7^2 &+- & 1,1 & 1 & 7^2 &7\cdot 151\cdot 35573 & \mathbf{718B1}\\ 181\mathbf{794G14}& 11^2 &+- & 3,1 & 3 & 11^2 &3\cdot7\cdot11\cdot47\cdot35447& \mathbf{794A1}\\ 182\mathbf{817E15}& 7^2 &+- & 1,5 & 5 & 7^2 & 7\cdot 79 & \mathbf{817A1}\\ 183\mathbf{916G9}& 11^2 &-+ & a,1 & 1 & 11^2 & 3^9\cdot 11\cdot 17\cdot 239 & \mathbf{916C1}\\ 184\mathbf{944O6}& 7^2 &+- & a,1 & 1 & 7^2 & 7 & \mathbf{944E1}\\ 185\mathbf{997H42}& 3^4 &- & 83 & 83 & 3^4 & 3^2 & \mathbf{997B1,C1}\\ 186\mathbf{1001L7}& 7^2 &+-+& 1,1,1 & 1 & 7^2 & 7\cdot19\cdot47\cdot2273&\mathbf{1001C1}\\ 187\mathbf{1028E14}&3^2\cdot 11^2&-+ & a,1 & 3 & 3^4\cdot 11^2 & 3^{13}\cdot 11 & \mathbf{1028A1}\\ 188\mathbf{1061D46} &151^2 &- &5\cdot 53&5\cdot 53&151^2& 61\cdot 151\cdot 179 &\mathbf{1061B2}\\ 189 190\mathbf{2593B109}&67^2\cdot 2213^2 & - & ? & ? & ? & ? & \mathbf{2593A4}\\\hline 191\end{array}$$
192\end{table}
193
194\begin{table}
195\caption{Explanatory factors}\vspace{-2ex}
196$$\begin{array}{|lcccccc|}\hline 197\mathbf{B_g}&\text{rank}&w_q&c_p&T&\delta_B&\text{Comments}\\\hline 198%\vspace{-2.5ex} & & && & & \\ 199\mathbf{389A1}& 2 &-&1 &1&5&\text{first curve of rank 2}\\ 200\mathbf{433A1}&2 &-&1&1&7&\\ 201\mathbf{446B1}&2 &+-&1,1& 1 &11&\text{this is \mathbf{446D} in \cite{cremona:algs}}\\ 202\mathbf{563A1}&2 &- & 1 & 1 & 13 & \\ 203\mathbf{571B1}&2 &- & 1 & 1 & 3 & \\ 204\mathbf{655A1}&2 &+-& 1,1 & 1 & 3^2 & \\ 205\mathbf{664A1}&2 &-+& 1,1 & 1 & 5 & \\ 206% RANK wq g_p Tor delta comments 207\mathbf{681C1} & 2 & +- & 1,1 & 1 & 3 & \\ 208\mathbf{707A1} & 2 & +- & 1,1 & 1 & 13 & \\ 209\mathbf{709A1} & 2 & - & 1 & 1 & 11 & \\ 210\mathbf{718B1} & 2 & +- & 1,1 & 1 & 7 & \\ 211\mathbf{794A1} & 2 & +- & 1,1 & 1 & 11 & \\ 212\mathbf{817A1} & 2 & +- & 1,1 & 1 & 7 & \\ 213\mathbf{916C1} & 2 & -+ & 3,1 & 1 & 3\cdot 11 & \\ 214\mathbf{944E1} & 2 & +- & 1,1 & 1 & 7 & \\ 215\mathbf{997B1} & 2 & - & 1 & 1 & 3 & \\ 216\mathbf{997C1} & 2 & - & 1 & 1 & 3 & \\ 217\mathbf{1001C1} & 2 & +-+& 1,3,1& 1 & 3^2\cdot 7 & \\ 218\mathbf{1028A1} & 2 & -+ & 3,1 & 1 & 3\cdot 11& \text{intersects \mathbf{1028E} mod 11}\\ 219\mathbf{1061B2}& 4 & - & 1 & 1 & 151 & \text{first prime level surface of pos. even rank \cite{brumer:rank}}\\ 220\hline\end{array}$$
221\end{table}
222
223
224\begin{table}
225\caption{Factorizations}\vspace{-1ex}
226$$\begin{array}{|llll|}\hline 227\mathbf{446}=2\cdot 223& 228\mathbf{655}=5\cdot 131& 229\mathbf{664}=2^3\cdot 83& 230\mathbf{681}=3\cdot 227\\ 231\mathbf{707}=7\cdot 101& 232\mathbf{718}=2\cdot 359& 233\mathbf{794}=2\cdot 397& 234\mathbf{817}=19\cdot 43\\ 235\mathbf{916}=2^2\cdot 229& 236\mathbf{944}=2^4\cdot 59& 237\mathbf{1001}=7\cdot 11\cdot 13& 238\mathbf{1028}=2^2\cdot 257\\ 239\hline\end{array}$$
240\end{table}
241
242\begin{table}
243\caption{Component groups}\vspace{-2ex}
244$$\begin{array}{|lccccc|}\hline 245\vspace{-2ex}&&&&&\\ 246\mathbf{A_f} & p & w_p &|\coker| &|\disc(X[I_f])| &|\Phi(\Fpbar)| \\ 247\vspace{-2ex}& & & & & \\\hline 248\mathbf{389E20}& 389&-& 97 & 5\cdot 97 & 97 \\ 249\mathbf{433D16}& 433&-& 3^2& 3^3\cdot 7\cdot 37 & 3^2 \\ 250\mathbf{446F8} & 223&-& 3 & 3\cdot 11\cdot 359353 & 3 \\ 251 & 2 &+ & 3 & 3\cdot 11& 3\cdot 359353\\ 252\mathbf{563E31}& 563&-& 281& 13\cdot 281& 281 \\ 253\mathbf{571D2} & 571&-& 1 & 3^2\cdot 127 &1 \\ 254\mathbf{655D13}& 131&-& 1 & 3^{2}\cdot19\cdot515741 & 1\\ 255 & 5&+& 1 & 3^2 & 19\cdot 515741 \\ 256\mathbf{664F8} & 83&+& 1 & 5 & 1 \\ 257\mathbf{681B1} & 227&-& 1 & 3\cdot 5^3 & 1 \\ 258 & 3&+& 1 & 3\cdot 5^2& 5 \\ 259\mathbf{707G15}& 101&-& 1 & 13\cdot800077 & 1 \\ 260 & 7&+& 1 & 13& 800077 \\ 261\mathbf{709C30}& 709&-& 59& 11\cdot 59 & 59 \\ 262\mathbf{718F7} & 359&-& 1 & 7\cdot 151\cdot 35573 &1 \\ 263 & 2 &+& 1 & 7 & 151\cdot 35573 \\ 264\mathbf{794G14}& 397&-& 3 & 3^2\cdot7\cdot11\cdot47\cdot35447 & 3 \\ 265 & 2&+& 3 & 3\cdot11& 3^2\cdot 7\cdot 47\cdot 35447 \\ 266\mathbf{817E15}& 43&- & 5 & 5\cdot7\cdot 79 & 5 \\ 267 & 19&+ &1 & 7 & 79 \\ 268\mathbf{916G9} & 229&+ &1 & 3^9\cdot 11\cdot 17\cdot 239 & 1 \\ 269 270\mathbf{944O6} & 59&-& 1 & 7 & 1\\ 271\mathbf{997H42}& 997&-& 83& 3^2\cdot 83 & 83 \\ 272\mathbf{1001L7}& 13&+& 1& 7\cdot 19\cdot 47\cdot 2273& 1\\ 273 & 11&-& 1& 7\cdot19\cdot47\cdot2273 & 1 \\ 274 & 7&+& 1& 7\cdot 19\cdot 47 & 2273 \\ 275\mathbf{1028E14}&257&+& 1 & 3^{13}\cdot 11 & 1 \\ 276\mathbf{1061D46}&1061&-& 5\cdot53 & 5\cdot 53\cdot 61\cdot 151\cdot 179 & 5\cdot 53 \\ 277\hline \end{array}$$
278\end{table}
279
280\begin{table}
281\caption{Odd square numerator}\vspace{2ex}
282\begin{tabular}{|llllll|}\hline
283$\mathbf{305D7}:3$&
284$\mathbf{309D8}:5$&
285$\mathbf{335E11}:3^2$&
286$\mathbf{389E20}:5$&
287$\mathbf{394A2}:5$&
288$\mathbf{399G5}:3^4$\\
289
290$\mathbf{433D16}:7$&
291$\mathbf{435G2}:3$&
292$\mathbf{436C4}:3$&
293$\mathbf{446E7}:3$&
294$\mathbf{446F8}:11$&
295$\mathbf{455D4}:3$\\
296
297$\mathbf{473F9}:3$&
298$\mathbf{500C4}:3$&
299$\mathbf{502E6}:11$&
300$\mathbf{506I4}:5$&
301$\mathbf{524D4}:3$&
302$\mathbf{530G4}:7$\\
303
304$\mathbf{538E7}:3$&
305$\mathbf{551H18}:3$&
306$\mathbf{553D13}:3$&
307$\mathbf{555E2}:3$&
308$\mathbf{556C7}:3$&
309$\mathbf{563E31}:13$\\
310
311$\mathbf{564C3}:3$&
312$\mathbf{571D2}:3$&
313$\mathbf{579G13}:3\cdot 5$&
314$\mathbf{597E14}:19$&
315$\mathbf{602G3}:3$&
316$\mathbf{604C6}:3$\\
317
318$\mathbf{615F6}:5$&
319$\mathbf{615G8}:7$&
320$\mathbf{620D3}:3$&
321$\mathbf{620E4}:3$&
322$\mathbf{626F12}:5$&
323$\mathbf{629G15}:3$\\
324
325$\mathbf{642D2}:3$&
326$\mathbf{644C5}:3$&
327$\mathbf{644D5}:3$&
328$\mathbf{655D13}:3^2$&
329$\mathbf{660F2}:3$&
330$\mathbf{662E10}:43$\\
331
332$\mathbf{664F8}:5$&
333$\mathbf{668B5}:3$&
334% $\mathbf{675O}:3$&   % REMOVED, since new algorithm gives bsd value of 1.
335$\mathbf{678I2}:3$&
336$\mathbf{681B1}:3$&
337$\mathbf{681I10}:3$&
338$\mathbf{682I6}:11$\\
339
340$\mathbf{707G15}:13$&
341$\mathbf{709C30}:11$&
342$\mathbf{718F7}:7$&
343$\mathbf{721F14}:3^2$&
344$\mathbf{724C8}:3$&
345$\mathbf{756G2}:3$\\
346
347$\mathbf{764A8}:3$&
348$\mathbf{765M4}:3$&
349$\mathbf{766B4}:3$&
350$\mathbf{772C9}:3$&
351$\mathbf{790H6}:3$&
352$\mathbf{794G12}:11$\\
353
354$\mathbf{794H14}:5^2$&
355$\mathbf{796C8}:3$&
356$\mathbf{817E15}:7$&
357$\mathbf{820C4}:3$&
358$\mathbf{825E2}:3$&
359$\mathbf{844C10}:3^2$\\
360
361$\mathbf{855M4}:3$&
362$\mathbf{860D4}:3$&
363$\mathbf{868E5}:3$&
364$\mathbf{876E5}:3$&
365$\mathbf{878C2}:3$&
366$\mathbf{884D6}:3$\\
367
368$\mathbf{885L9}:3^2$&
369$\mathbf{894H2}:3$&
370$\mathbf{902I5}:3$&
371$\mathbf{913G17}:3$&
372$\mathbf{916G9}:11$&
373$\mathbf{918O2}:5$\\
374
375$\mathbf{918P2}:3$&
376$\mathbf{925K7}:3$&
377$\mathbf{932B13}:3^2$&
378$\mathbf{933E14}:19$&
379$\mathbf{934I12}:7$&   %-+
380$\mathbf{944O6}:7$\\
381
382$\mathbf{946K7}:3$&
383$\mathbf{949B2}:3$&
384$\mathbf{951D19}:3$&
385$\mathbf{959D24}:3$&
386$\mathbf{964C12}:3^2$&   % -+, same as EC 964A but that has rank=0.
387$\mathbf{966J1}:3$\\
388
389$\mathbf{970I5}:3$&
390$\mathbf{980F1}:3$&
391$\mathbf{980J2}:3$&
392$\mathbf{986J7}:5$&
393$\mathbf{989E22}:5$&
394$\mathbf{993B3}:3^2$\\
395
396$\mathbf{996E4}:3$&
397$\mathbf{997H42}:3^2$&
398$\mathbf{998A2}:3$&   % ++
399$\mathbf{998H9}:3$&
400$\mathbf{999J10}:3$&  \\
401%$\mathbf{1001L7}:7$&
402%$\mathbf{1001M8}:3$\\
403\hline\end{tabular}
404\end{table}
405
406
407\section{Prime conductor}
408Using [REFS] we computed the odd part $\Shaan$ of the
409analytic order of the Shafarevich and Tate group of every
410rank zero optimal quotient of $J_0(p)$ for $p \leq 2593$.
411Those for which $\Shaan > 1$ are laid out in the
412first column of Table~\ref{table:primesha}.
413The fourth column lists the explanatory factor, if there is
414one at the same level.  If $\Sha$ is not visible we
415were sometimes able to find an explanatory factor
416at higher level; this is discussed in the next section.
417
418\begin{table}\label{table:primesha}
419\caption{Odd parts of Shafarevich-Tate groups at prime level}
420$$421\begin{array}{lccclcc} 422\mbox{\rm\bf A}& \mbox{\rm dim}& \Shaan & \mbox{\rm mod deg} & \mbox{\rm\bf B} & \mbox{\rm dim} & \mbox{\rm mod deg} \\ 423& & & & & & \vspace{-2ex} \\ 424\mbox{\rm\bf 389E}& 20 &5^{2}&5&\mbox{\rm\bf 389A}& 1 &5\\ 425\mbox{\rm\bf 433D}& 16 &7^{2}&3\cdot7\cdot37&\mbox{\rm\bf 433A}& 1 &7\\ 426\mbox{\rm\bf 563E}& 31 &13^{2}&13&\mbox{\rm\bf 563A}& 1 &13\\ 427\mbox{\rm\bf 571D}& 2 &3^{2}&3^{2}\cdot127&\mbox{\rm\bf 571B}& 1 &3\\ 428\mbox{\rm\bf 709C}& 30 &11^{2}&11&\mbox{\rm\bf 709A}& 1 &11\\ 429\mbox{\rm\bf 997H}& 42 &3^{4}&3^{2}&\mbox{\rm\bf 997B}& 1 &3\\ 430\mbox{\rm\bf 1061D}& 46 &151^{2}&61\cdot151\cdot179&\mbox{\rm\bf 1061B}& 2 &151\\ 431\mbox{\rm\bf 1091C}& 62 &7^{2}&1&\mbox{\rm NONE} & & \\ 432\mbox{\rm\bf 1171D}& 53 &11^{2}&3^{4}\cdot11&\mbox{\rm\bf 1171A}& 1 &11\\ 433\mbox{\rm\bf 1283C}& 62 &5^{2}&5\cdot41\cdot59&\mbox{\rm NONE} & & \\ 434\mbox{\rm\bf 1429B}& 64 &5^{2}&1&\mbox{\rm NONE} & & \\ 435\mbox{\rm\bf 1481C}& 71 &13^{2}&5^{2}\cdot2833&\mbox{\rm NONE} & & \\ 436\mbox{\rm\bf 1483D}& 67 &3^{2}\cdot5^{2}&3\cdot5&\mbox{\rm\bf 1483A}& 1 &3\cdot5\\ 437\mbox{\rm\bf 1531D}& 73 &3^{2}&3&\mbox{\rm\bf 1531A}& 1 &3\\ 438\mbox{\rm\bf 1559B}& 90 &11^{2}&1&\mbox{\rm NONE} & & \\ 439\mbox{\rm\bf 1567D}& 69 &7^{2}\cdot41^{2}&7\cdot41&\mbox{\rm\bf 1567B}& 3 &7\cdot41\\ 440\mbox{\rm\bf 1613D}& 75 &5^{2}&5\cdot19&\mbox{\rm\bf 1613A}& 1 &5\\ 441\mbox{\rm\bf 1621C}& 70 &17^{2}&17&\mbox{\rm\bf 1621A}& 1 &17\\ 442\mbox{\rm\bf 1627C}& 73 &3^{4}&3^{2}&\mbox{\rm\bf 1627A}& 1 &3^{2}\\ 443\mbox{\rm\bf 1693C}& 72 &1301^{2}&1301&\mbox{\rm\bf 1693A}& 3 &1301\\ 444\mbox{\rm\bf 1811D}& 98 &31^{2}&1&\mbox{\rm NONE} & & \\ 445\mbox{\rm\bf 1847B}& 98 &3^{6}&1&\mbox{\rm NONE} & & \\ 446\mbox{\rm\bf 1871C}& 98 &19^{2}&14699&\mbox{\rm NONE} & & \\ 447\mbox{\rm\bf 1877B}& 86 &7^{2}&1&\mbox{\rm NONE} & & \\ 448\mbox{\rm\bf 1907D}& 90 &7^{2}&3\cdot5\cdot7\cdot11&\mbox{\rm\bf 1907A}& 1 &7\\ 449\mbox{\rm\bf 1913B}& 1 &3^{2}&3\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\ 450\mbox{\rm\bf 1913E}& 84 &5^{4}\cdot61^{2}&5^{2}\cdot61\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\ 451\mbox{\rm\bf 1933C}& 83 &3^{2}\cdot7^{2}&3\cdot7&\mbox{\rm\bf 1933A}& 1 &3\cdot7\\ 452\mbox{\rm\bf 1997C}& 93 &17^{2}&1&\mbox{\rm NONE} & & \\ 453\mbox{\rm\bf 2027C}& 94 &29^{2}&29&\mbox{\rm\bf 2027A}& 1 &29\\ 454\mbox{\rm\bf 2029C}& 90 &5^{2}\cdot269^{2}&5\cdot269&\mbox{\rm\bf 2029A}& 2 &5\cdot269\\ 455\mbox{\rm\bf 2039F}& 99 &3^{4}\cdot5^{2}&19\cdot29\cdot7759\cdot3214201&\mbox{\rm NONE} & & \\ 456\mbox{\rm\bf 2063C}& 106 &13^{2}&61\cdot139&\mbox{\rm NONE} & & \\ 457\mbox{\rm\bf 2089J}& 91 &11^{2}&3\cdot5\cdot11\cdot19\cdot73\cdot139&\mbox{\rm\bf 2089B}& 1 &11\\ 458\mbox{\rm\bf 2099B}& 106 &3^{2}&1&\mbox{\rm NONE} & & \\ 459\mbox{\rm\bf 2111B}& 112 &211^{2}&1&\mbox{\rm NONE} & & \\ 460\mbox{\rm\bf 2113B}& 91 &7^{2}&1&\mbox{\rm NONE} & & \\ 461\mbox{\rm\bf 2161C}& 98 &23^{2}&1&\mbox{\rm NONE} & & \\ 462\mbox{\rm\bf 2333C}& 101 &83341^{2}&83341&\mbox{\rm\bf 2333A}& 4 &83341\\ 463\mbox{\rm\bf 2339C}& 114 &3^{8}&6791&\mbox{\rm NONE} & & \\ 464\mbox{\rm\bf 2411B}& 123 &11^{2}&1&\mbox{\rm NONE} & & \\ 465\end{array} 466$$
467\end{table}
468
469
470Table~\ref{table:primesha} has many interesting properties.\footnote{
471Examples {\bf 1091C} and {\bf 1429B} were
472discoverd in \cite{agashe} and
473{\bf 1913B} in \cite{cremona-mazur}.}
474There are $23$ examples in which $\Sha$ is (almost certainly)
475visible and $18$ examples in which $\Sha$ is invisible.  The largest visible
476$\Sha$ found occurs at level $2333$ and has order at least $83341^2$
477($83341$ is prime).  Believe it or not, the largest invisible $\Sha$
478occurs in a $112$ dimensional abelian variety at level
479$2111$ and has order at least $211^2$. Furthermore,
480
481\begin{enumerate}
482\item The Shafarevich-Tate group is never Eisenstein in these examples
483[[recheck, as I've added new examples!]]
484\item {\bf 1283C}. This example demonstrates that $\Shaan$ can divide the
485  modular degree, yet still be invisible.  The other factor
486  {\bf 1283A} (of dimension 2) which satisfies a $5$-congruence with
487  {\bf 1283C} has analytic rank (and hence algebraic rank) $0$.
488\end{enumerate}
489
490\comment{
491\section{Explanatory factors at higher level}
492Consider one of the items in Table~\ref{table:primesha} for which
493$\Sha$ is invisible.  It is natural to ask whether these
494elements of $\Sha$ become visible somewhere.''
495For example, Barry Mazur \cite{mazur:visthree} proved that if
496$E$ is an elliptic curve and $c\in\Sha(E)$ has order $3$ then
497there is some abelian surface $A$ and an
498injection $\iota: E\hookrightarrow A$ such that
499$\iota_*(c)=0\in H^1(\Q,A)$.  T. Klenke has proved
500a partial statement in this direction for elements of
501order $2$ as part of his Harvard Ph.D. thesis.
502Johann de Jong (see \cite[Remark 3]{cremona-mazur})
503showed that every element of the Shafarevich-Tate
504group of an elliptic curve is visible in some Jacobian.
505
506Consider an abelian variety $A$, taken
507from Table~\ref{table:primesha},  for which
508$\Sha$ has an invisible element $c$.   Thus
509$A$ sits inside $J_0(p)$ for some prime $p$,
510and we ask is there a prime $q$ such that $\delta(c)=0$
511for one of the degeneracy maps
512$\delta : J_0(p)\ra J_0(pq)$?''
513
514The author has no idea\footnote{Lo\"\i{}c Merel suspects the answer
515might be yes whereas Richard Taylor is more skeptical.}.
516To get a feeling for what might happen we consider in
517detail abelian variety $A=A_f$ at level $p=1091$ in
518which $\Shaan$ is divisible by $7$.
519
520There is a prime $\lambda$ of the ring
521$\Z[f] = \Z[\ldots a_n\ldots]$ attached to $A$.
522The Fourier coefficients of $f$ modulo $\lambda$ are
523$$\begin{array}{rcccccccccccccccc} 524p= &2 &3 &5 &7 &11 &13 &17 &19 &23 &29 &31 &37 &41 &43 &47 &53\\ 525a_p= &3 &0 &1 &5 &0 &2 &0 &5 &4 &6 &3 &3 &5 &5 &6 &5 526\end{array}$$
527These were computed by finding an eigenvector in  $H_1(X_0(N);\F_7)$.
528[[SAY MORE ABOUT THE TRICK FOR FINDING ALL RIBET $q$'s.]]
529
530According to Ribet's level raising theorem \cite{ribet:raising}
531there is a newform $g$ of level $1091\ell$ such that
532$f\con g$ modulo [[something]] if $a_\ell = \pm (\ell+1)\pmod{\lambda}$.
533Fortunately this criterion is already satisfied for $\ell=2$.
534Looking closely at level $2\cdot 1091$ (for example,
535in Cremona's online tables \cite{cremona:onlinetables})
536we find an elliptic curve $E$ whose corresponding newform
537$g=\sum b_n q^n$ has Fourier coefficients
538$$\begin{array}{rcccccccccccccccc} 539p = &2 & 3 & 5 & 7& 11& 13& 17& 19& 23& 29& 31& 37&41&43&47&53\\ 540b_p=&1 & 0 & 1 &-2& 0 & -5& 0 & -2& 4& 6& -4& -4&-9&-9&-8&-2\\ 541\end{array}$$
542
543This is convincing evidence that one of the
544two images of $A$ in $J_0(2\cdot 1091)$ shares some
545$7$-torsion with the elliptic curve \abvar{2182B}.
546This can be [[WILL BE!!, EASILY]] established by
547a direct computation with the period lattices.
548This is at first disconcerting because the rank of
549this elliptic curve is {\em not} $2$.  However, the
550rank is still positive; it is $1$ with
551Mordell-Weil group $\E(\Q)=Z$.
552
553I would not be at all surprised if your
554$7$-torsion in Sha does become visible in $J_0(2\cdot1091)$.
555The curve \abvar{2182B}, which shares 7-torsion with $A$ is
556
557\begin{verbatim}
558   e=ellinit([1,-1,1,-67,67]);
559The Tamagawa number c_2 is 14   (!!)
560  ? elllocalred(e,2)
561  %2 = [1, 18, [1, 0, 0, 0], 14]
562The Tamagawa number c_1091 is 1.
563  ? elllocalred(e,1091)
564  %3 = [1, 5, [1, 0, 0, 0], 1]
565
566I have this feeling that the right statement about congruence
567and mordell-Weil is really something like
568    congruence ==> "Selmer + Comp group"'s are identified.
569Anyway, the extra component group of order 7 may perhaps
570account for the other nontrivial element of Sha.  This might
571just be wild speculation.
572
573Good luck.
574
575william
576            Barry,
577
578Amod asked me to investigate whether his element of order 7
579in the winding quotient J_e at level 1091 becomes visible at
580higher level.  Luckily, Ribet's level raising theorem predicts the
581existence of a form at level 2*1091 congruent mod a prime over
5827 to the form corresponding to J_e. Even more luckily, one of the
583two rational newforms does the trick.  Thus an image of J_e in
584J_0(2*1091) shares 7-torsion with an elliptic curve E (2182B
585in Cremona's tables).  This elliptic curve has:
586
587     E(Q) = Z
588     Sha(E/Q) = 0
589     c_2 = 14,   c_1091 = 1
590     L^(1)(f,1)/r! = 4.27332686791516
591
592
593So there is reasonable hope that the elements of order 7 in
594Sha(J_e) are visible at this higher level, even though they
595are invisible even in J_1(1091).
596
597Best,
598William
599
600
601  Dear William,
602    This is terrific. I assume that you will be showing that for J_e the
603winding
604    (not quite quotient, but more conveniently sub-thing) in J_0(1091),
605the image of
606
607                 Sha(J_e)  --->  Sha(J_0(2*1091))
608
609     just dies?  Since our working hope, I think, is that for any N there
610is an
611     M so that
612                      Sha(J_0(N)) ---. Sha(J_0(N.M))
613
614
615     dies, this suggests returning to the (mod 3) N=2849 example, where I
616"know" that there must exist such an M  (because all three-torsion in Sha
617on elliptic curves is visible in some appropriate abelian surface which is
618isogenous to a prouct of two elliptic curves, and therefore, is abelian
619surface is probably "modular").  But I don't know a specific M.
620
621Barry
622
623\end{verbatim}
624}
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