CoCalc Public Fileswww / talks / 2004-05-22-UIUC / visibility.tex
Author: William A. Stein
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55%opening
56\title{\Huge\bf\dblue Modularity of Shafarevich-Tate Groups}
57\author{\rd{\LARGE William Stein}\\
58{\tt http://modular.fas.harvard.edu}}
59\date{May 22, 2004\vspace{1ex}\\
60\includegraphics[width=0.7\textwidth]{sha_fractal.eps}}
61
62\begin{document}
63
64\maketitle
65
66
67\begin{slide}
68\h{\rd{Goal}}
69
70The goal of this 40-minute talk is to explain the meaning of
71the following conjecture and give evidence for it.
72
73{\Large{\bf Conjecture (--).} If $A$ is a modular abelian
74variety, then the Shafarevich-Tate
75group \rd{$\Sha(A)$} is \blue{modular}.}
76
77
78\end{slide}
79
80\begin{slide}
82
83\begin{enumerate}
84\item Elliptic Curves and \rd{Modular Abelian Varieties}
85\item The \rd{Birch and Swinnerton-Dyer} Conjecture
86\item \mbox{}\rd{Visibility} of Shafarevich-Tate groups
87\item \mbox{}\rd{Modularity} of Shafarevich-Tate groups
88\item Some \rd{Data}
89\end{enumerate}
90\end{slide}
91
92\begin{slide}
93\h{1. Elliptic Curves and Abelian Varieties}
94\begin{tabular}{ll}
95\hspace{-.8ex}\begin{minipage}{6in}
96\vspace{-2in}
97\rd{Elliptic curve over $\Q$:} $y^2=x^3+ax+b$
98with $a,b\in\Q$
99and $\Delta=-16(4a^3+27b^2)\neq 0$
100\end{minipage}&
101\hspace{0.5in}
102\includegraphics[width=3in]{elliptic6_floating_2.eps}
103\end{tabular}\vspace{-.7in}
104
105\rd{Abelian variety:} Any complete group variety.
106
107\rd{Examples:} Jacobians of curves.  Elliptic curves
108are the abelian varieties of dimension one.
109\rd{Modular abelian varieties}.
110\end{slide}
111
112\begin{slide}
113\h{Modular Curves, Modular Forms}
114
115\rd{Congruence Subgroup:}
116$$117\Gamma_0(N) = \left\lbrace 118\mtwo{a}{b}{c}{d} \in \SL_2(\Z) \text{ such that } N \mid c 119\right\rbrace. 120$$
121\vspace{-1ex}
122
123\rd{Modular Curve:}
124$X_0(N) = \Gamma_0(N) \setminus \text{(upper half plane)} \cup \text{(cusps)}$
125\vspace{-1ex}
126
127\rd{Example:}{\dblue  $X_0(39)$}
128\begin{center}\vspace{-.4in}
129\includegraphics[width=0.6\textwidth]{torus39.eps}
130\end{center}
131\vspace{-4ex}
132\rd{Algebraic structure over $\Q$:}
133$$134 y^2 = (x^4-7x^3+11x^2-7x+1)(x^4+x^3-x^2+x+1). 135$$
136\end{slide}
137
138\begin{slide}
139\h{Modular Abelian Varieties}
140\rd{Modular Jacobian: } $J_0(N) = \Jac(X_0(N))$
141
142\rd{Modular Abelian Variety:} Any abelian variety quotient of $J_0(N)$ (or
143of $J_1(N)$, where $J_1(N)$ is defined using $\Gamma_1(N)$).
144
145\rd{Theorem (Wiles et al.):} All elliptic curves over $\Q$ are modular.
146
147\rd{Cusp Forms: }
148$S_2(\Gamma_0(N))\isom \H^0(X_0(N)_\C,\Omega^1)$
149
150\rd{Hecke Algebra:} $\T = \Z[T_1,T_2,T_3, \ldots] \subset \End(S_2(\Gamma_0(N)))$
151
152\rd{Shimura:}
153$\T$-eigenform $f \in S_2(\Gamma_0(N))$
154gives $A_f = J_0(N)/I_f J_0(N)$, where $I_f=\Ann_\T(f)$.
155We have $\dim A_f = [\Q(a_2(f),\ldots):\Q]$.
156
157\end{slide}
158
159\begin{slide}
160\h{2. The Birch and Swinnerton-Dyer Conjecture}
161
162Let $A$ be an abelian variety over $\Q$ (e.g., $A=A_f$ modular).
163\vspace{-1in}
164\begin{center}
165\begin{minipage}{7in}
166\rd{\LARGE Conjecture of \includegraphics[height=1.1in]{bsd.eps}:}\\
167\begin{enumerate}
168\item $r = \ds \rank A(\Q) \ce \ord_{s=1}L(A,s)$, and\\
169\item
170$\ds 171\frac{L^{(r)}(A,1)}{r!} = 172\frac{\prod c_p \cdot \Omega_A \cdot \Reg(A) 173\cdot \#\Sha(A)} 174{\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}}. 175$
176\end{enumerate}
177\end{minipage}
178\end{center}
179Results: Kolyvagin, Kato, Rubin, etc.
180\end{slide}
181
182\begin{slide}
183\h{Shafarevich-Tate Group \includegraphics[width=2.2in]{sha_fractal2.eps}}
184\rd{Definition:}
185$$186\Sha(A) = \Ker\left(\H^1(K,A) \to \bigoplus_{\text{all }v} \H^1(K_v,A)\right). 187$$
188
189$\H^1(K,A)$ is \rd{Galois cohomology}.
190Interpret geometrically as the \rd{Weil-Chatalet group}:
191$$192\text{WC}(A/K) = \{\,\text{principal homogenous spaces }X\text{ for }A\,\}/\sim. 193$$
194
195$\Sha(A)$ is the subgroup of {\dblue locally trivial classes}.
196\rd{Example:}
197$$3x^3+4y^3+5z^3=0 198\in \Sha(x^3+y^3+60z^3=0)[3].$$
199\end{slide}
200
201\begin{slide}
202\h{3. Visibility of Sha \includegraphics[width=1.4in,angle=-10]{mazur.eps}}
203\vspace{-1in}
204
205{\LARGE$$0 \to A\xrightarrow{i} B \to C \to 0$$}
206
2071998 -- Barry Mazur introduced \rd{Visibility}:
208\begin{align*}
209\Vis_B(\H^1(K,A)) &= \Ker\left(\H^1(K,A) \to \H^1(K,B)\right)\\
210   &\isom \Coker(B(K) \to C(K))\\
211  & \\
212\Vis_B(\Sha(A)) &= \Ker(\Sha(A)\to \Sha(B)).
213\end{align*}
214
215\end{slide}
216
217\begin{slide}
218%\h{\begin{tabular}{lcr}
219%\begin{minipage}{2in}\mbox{}\\Visibility\end{minipage}
220%&&\includegraphics[width=3in]{eye.eps}
221%\end{tabular}}
222% The eye graphic is from
223%  http://www.littlebeast.com/ images/eye.jpg
224% and was found using the google image search.
225\begin{center}
226\pspicture(-1,-1)(5,1)
227\rput(-4,0){\h{Visibility \hspace{2in}\mbox{}}}
228\rput(20,3){\includegraphics[width=3in]{eye.eps}}
229\endpspicture
230\end{center}
231
232
233\vspace{-.7in}
234
235$$236\xymatrix{ 237&&X=\pi^{-1}(P)\[email protected]{-}[d]\ar[r]&P\[email protected]{-}[d]\[email protected]{|->}[r]&c\[email protected]{-}[d]\\ 2380\ar[r]&A(K)\ar[r]&B(K)\ar[r]^{\pi}&C(K) \ar[r]&{\Vis_B(\H^1(K,A))}\ar[r]&0 239} 240$$
241
242\rd{Why?} To write down $X$ using equations is terrifying;
243to give $P$ is just to give a rational point.  Visibility concisely
244encodes connections between Mordell-Weil and Shafarevich-Tate groups.
245
246Give nonzero $c\in \Sha(A)[5]$ with $\dim(A)=20$ by giving
247   $$(0,0)\in [y^2 + y = x^3 + x^2 - 2x]$$
248\end{slide}
249
250\begin{slide}
251\h{Everything is Visible Somewhere}
252
253\rd{Theorem (--):}
254If $c\in \H^1(K,A)$ then there exists
255$B$
256such that $i:A\hra B$ and $c\in\Vis_B(\H^1(K,A))$.
257
258
259\rd{Proof.}  Let $L$ be such that $\res_{L/K}(c)=0$. Then
260$$261\xymatrix{ 262{c} \ar\[email protected]{|->}[rr] && {\res_{L/K}(c)=0}\\ 263{\H^1(K,A)}\ar[rr]\ar[ddrr] && {\H^1(K,\rd{\Res_{L/K}(A_L)})}\ar[dd]\\ 264 & \\ 265 && {\H^1(L,A)} 266}$$
267
268Note: If $A/\Q$ is modular and $L$ is abelian,
269then $B$ is modular.
270\end{slide}
271
272
273\begin{slide}
274\h{4. Modularity of Sha}
275\rd{\bf Definition (Modular):} An element $c\in \Sha(A)$ is
276{\em modular} if it is visible in a modular abelian variety.
277I.e., if there is a factor $B$ of some $J_0(N)$ (or $J_1(N)$)
278and an inclusion $i:A\hra B$ such that $i_*(c)=0$.
279Torsor corresponding to $c$ is modular in usual''
280sense.
281
282\rd{Modularity Conjecture (--):}\\
283\mbox{}\hspace{.6in}{\dblue If $A$ is modular, then every element of
284  $\Sha(A)$ is modular.}
285
286%{\red\bf Should you believe my conjecture?}
287
288\rd{Theorem (Klenke, Mazur, Stein):} If $c\in \Sha(E)[p]$ with $p=2,3$ and $E$
289an elliptic curve, then $c$ is modular.
290
291\rd{Related questions:}\\
2921. Levels $N$ such that such that $c$
293is modular of level $N$?\\
2942. Which genus one curves are modular?
295%2. If an element
296%$c\in \Sha(A)$ is modular, does that imply that
297%$A$ is modular?
298\end{slide}
299
300\begin{slide}
302
303$\bullet$ It could yield results about {\dblue
304structure of $\Sha(A)$}, e.g., finiteness.
305
306$\bullet$ It would give a nice explanation'' of
307{\dblue where all $\Sha(A)$
308comes from''} --- it all comes from Mordell-Weil groups.
309
310$\bullet$ It {\dblue motivates proving new results} about the arithmetic
311of modular abelian varieties.
312
313$\bullet$ It provides {\dblue powerful computational tools}
314for explicitly working with Shafarevich-Tate groups.
315
316\end{slide}
317
318\begin{slide}
319\h{Visibility Construction}
320\rd{Theorem (Agashe, --):} {\em Suppose $A, B\subset J_0(N)$,
321that $B[p]\subset A$, and other technical hypotheses.
322Then
323  $$B(\Q)/p B(\Q) \hra \Vis_{J_0(N)}(\Sha(A)).$$}
324
325\rd{Proof.} Use the following diagram, chase some exact sequences, and
326apply subtle properties of N\'eron models.  (Also more general
327Hecke-equivariant version, proved recently with Jetchev.)
328$$329{\dgreen\[email protected]=0.9in{ 330{B[p]}\ar[r]\ar[d]& B \ar[r]^p\ar[d] & B \ar[d]\\ 331{A}\ar[r] & {J_0(N)} \ar[r] & C.} 332}$$
333\end{slide}
334
335\begin{slide}
336%\h{5. Some Data}
337\begin{center}
338\pspicture(-1,-1)(5,1)
339\rput(-1,0){\h{\hspace{-2in}5. Some Data}}
340\rput(20,3){\includegraphics[width=4in]{tables.eps}}
341\endpspicture
342\end{center}\vspace{-1.5in}
343
344{\LARGE $$\text{Suppose } A \subset J_0(N)$$}
345
346\begin{itemize}
347\item[(a)] Visibility of $\Sha(A)$ in $J_0(N)$, when $A$ is an elliptic
348curve.
349\item[(b)] Visibility of $\Sha(A)$ in $J_0(N)$, general $A$.
350\item[(c)] Visibility of $\Sha(A)$ in $J_0(Np)$, general $A$.  (Modularity)
351\end{itemize}
352\end{slide}
353
354\begin{slide}
355%\voffset=-1in
356\h{(a) Elliptic Curve tables from Cremona-Mazur}
357
358\begin{center}\Large
359Visibility of $\Sha(A)$ in $J_0(N)$, when $A=E$ is an \rd{elliptic
360curve}.
361\end{center}
362\begin{center}
363\includegraphics[width=2in]{cremona_table_1.eps}
364$\qquad$\includegraphics[width=2in]{cremona_table_1b.eps}
365$\qquad$\includegraphics[width=2in]{cremona_table_1c.eps}
366\includegraphics[width=1in,angle=-10]{cremona5.eps}
367\end{center}
368\end{slide}
369
370\begin{slide}
371\voffset=-1in
372\begin{center}
373\includegraphics[height=1.3\textheight]{cm1.ps}
374\end{center}
375
376\begin{center}
377\includegraphics[height=1.3\textheight]{cm2.ps}
378\end{center}
379\end{slide}
380
381\begin{slide}
382\voffset=-1in
383\h{(b) Abelian Variety Tables from Agashe-Stein}
384
385\begin{center}\Large
386 Visibility of $\Sha(A)$ in $J_0(N)$, general $A$.
387\end{center}
388
389\begin{center}
390\includegraphics[height=1.5\textheight]{as1.ps}
391\end{center}
392
393\begin{center}
394\includegraphics[height=1.5\textheight]{as2.ps}
395\end{center}
396
397\begin{center}
398\includegraphics[height=1.5\textheight]{as3.ps}
399\end{center}
400
401\begin{center}
402\includegraphics[height=1.5\textheight]{as4.ps}
403\end{center}
404
405\end{slide}
406
407\begin{slide}
408\h{(c) Visibility at Higher Level -- Evidence for Modularity Conjecture}
409
410\begin{center}\Large
411 Visibility of $\Sha(A)$ in $J_0(Np)$, general $A$.
412\end{center}
413Recall Ribet's theorem...
414\end{slide}
415
416\begin{slide}
417\vspace{-0.5in}
418\h{Ribet Level Raising \includegraphics[width=1.5in,angle=-5]{ribet.eps}}
419
420\vspace{-0.1in}Suppose\vspace{-0.4in}
421\begin{itemize}\setlength{\itemsep}{0in}
422\item $f=\sum a_n q^n\in S_2(\Gamma_0(N))$ a newform
423\item $\lambda\subset \Z[a_1,a_2,\ldots]$ a nonzero prime ideal
424s.t. $A_f[\lambda]$ irreducible.
425\end{itemize}\vspace{-0.3in}
426
427\rd{\bf Theorem:} {\dblue\large $428 a_p + p + 1 \con 0\pmod{\lambda} \implies 429$}
430there exists a $p$-newform $g\in S_2(\Gamma_0(Np))$ such that
431\vspace{-0.4in}
432\begin{itemize}\setlength{\itemsep}{0in}
433\item $i(A_f[\lambda]) = A_g[\lambda]$ some
434$i:J_0(N)\to J_0(N p)$, and
435\item sign of functional equations for $L(f,s)$ and $L(g,s)$
436same.
437\end{itemize}
438
439%{\small Note: If instead $a_p - (p+1) \con 0\pmod{\lambda}$
440%then there is such a $g$, but the sign of the functional
441%equation changes, and the new Tamagawa number of
442%$A_g$ at $p$ are divisible by $\lambda$.}
443\end{slide}
444
445\begin{slide}
446  \rd{\bf Big Computation:} For every level $N$ up to $5000$ (and
447  more), use my modular forms package in MAGMA to provably compute:
448\begin{enumerate}
449\item Each newform $f=\sum a_n q^n \in S_2(\Gamma_0(N))$.
450\item Whether or not $L(f,1)=0$.
451\item Whether or not $\ord_{s=1} L(f,s)$ is even.
452\item Characteristic polynomials of $a_2, a_3, a_5, \ldots, a_{19}$.
453\end{enumerate}
454(I hope to redo this computation
455using only open-source software that I'm currently
456writing.)
457\end{slide}
458
459\begin{slide}
460\h{Probable Modularity}
461$\bullet$ Two forms $f=\sum a_n q^n$ and $g=\sum b_n q^n$
462are {\em\dred probably congruent mod $\ell$} (away from level) if
463for $p<20$ with $p\nmid N_f N_g$ we have
464$$465 \ell \mid \text{resultant}(\charpoly(a_p),\charpoly(b_p)). 466$$
467%(Congruence would imply this, but not conversely in general.)
468
469$\bullet$ If $A=A_g\subset J_0(N)$, then there is
470{\em\dred probably a nonzero element in
471$\Sha(A)[\ell]$ modular of level $N p$} if
472there is $f$ of level $N p$ such that:\vspace{-.4in}
473\begin{enumerate}\setlength{\itemsep}{0in}
474\item $f$ and $g$ are probably congruent modulo $\ell$, and
475\item $\ord_{s=1} L(f,s)$ is positive and even.
476\end{enumerate}
477\end{slide}
478
479
480\begin{slide}
481\voffset=-1.3in
482
483\begin{center}
484\includegraphics[height=1.7\textheight]{higher.ps}
485\end{center}
486\end{slide}
487
488\begin{slide}
489\h{Questions}
490\begin{center}
491\includegraphics[width=3in]{questions.eps}
492%\Huge {\dred ?}
493\end{center}
494
495\end{document}
496
497\begin{slide}
498
499Let $N$ be a positive integer and consider the congruence
500subgroup
501$$502\Gamma_0(N) = \left\lbrace 503\mtwo{a}{b}{c}{d} \in \SL_2(\Z) \text{ such that } N \mid c 504\right\rbrace. 505$$
506(Almost everything in this talk also makes sense with $\Gamma_0(N)$
507replaced by $\Gamma_1(N)$.)
508The \textit{modular curve}
509$$510X_0(N) = \Gamma_0(N) \setminus \left(\left\lbrace 511z\in \C : \Im(z) > 0\right\rbrace 512\cup \Q \cup \left\lbrace \infty\right\rbrace \right) 513$$
514is a Riemann surface that is the set of complex
515points of an algebraic curve over $\Q$.
516We will not use that
517$$518 X_0(N)(\C) = \left\lbrace 519 \text{ isomorphism classes of }(E,C)\,\, 520 \right\rbrace 521 \cup 522 \left\lbrace 523 \text{ cusps } 524 \right\rbrace. 525$$
526Our primary interest is the Jacobian
527$$528 J_0(N) = \Jac(X_0(N)) 529$$
530which is an abelian variety over $\Q$ of dimension equal
531to the genus of $X_0(N)$.  The points on the Jacobian
532parametrize, in a natural way, the divisor classes
533of degree $0$ on $X_0(N)$.
534
535Let
536$537 S_2(\Gamma_0(N)) 538$
539be the cusp forms of weight $2$ for $\Gamma_0(N)$.
540This is the finite-dimensional complex vector space
541of holomorphic functions on the upper half plane
542such that
543$$f(z)dz = f(\gamma(z))d(\gamma(z))$$
544for all $\gamma\in\Gamma_0(N)$, and which vanish
545at the cusps''.
546The map $f(z) \mapsto f(z)dz$ induces
547$$548 S_2(\Gamma_0(N))\isom \H^0(X_0(N)_\C,\Omega^1) 549$$
550so $S_2(\Gamma_0(N))$ has dimension the genus
551of $X_0(N)$.
552
553The \textit{Hecke algebra} is a commutative ring
554$$555 \T = \Z[T_1,T_2,T_3, \ldots] 556$$
557which acts on $S_2(\Gamma_0(N))$ and $J_0(N)$.
558A \textit{newform}
559$$560 f = \sum_{n=1}^{\infty} a_n q^n \in S_2(\Gamma_0(N)) 561$$
562is an eigenvector for every element of $\T$ normalized
563so $a_1 = 1$, which does not come from'' any lower level.
564Attached to $f$ there is an ideal
565$$566 I_f = \Ann_{\T}(f) = \Ker(\T \to \Z[a_1,a_2,\ldots]), 567$$
568and (following Shimura) to this ideal we attach an abelian variety $A_f$ and an $L$-function $L(A_f,s)$.
569
570Let
571  $$572 A_f = J_0(N)[I_f]^0 = \left( \bigcap_{\vphi \in I_f} \Ker(\vphi) \right)^0 573$$
574be the connected component of the intersections of the kernels
575of elements of $I_f$.
576Then $A_f$ has dimension $[K_f:\Q] = [\Q(a_1,a_2,\ldots):\Q)]$, and
577is define over $\Q$.
578
579Let
580$$581 L(A_f,s) = \prod_{i=1}^d L(f_i,s) 582$$
583where $d=[K_f:\Q]$ and the $f_i$ are the Galois conjugates
584of $f$.  Also,
585$$586 L(f,s) = \sum_{n=1}^\infty \dfrac{a_n}{n^s}. 587$$
588Hecke proved that $L(f,s)$ is entire and satisfies
589a functional equation.
590
591The abelian varieties $A_f$ are a rich class of abelian
592varieties.  The elliptic curves over~$\Q$ are
593all isogenous to some $A_f$ (the Wiles-Breuil-Conrad-Diamond-Taylor
594modularity theorem).
595
596\section{The Birch and Swinnerton-Dyer Conjecture}
597\subsection{Conjecture}
598\begin{conjecture}[Birch and Swinnerton-Dyer]\mbox{}\vspace{-4ex}\\
599\begin{enumerate}
600\item $\rank A_f(\Q) = \ord_{s=1}L(A_f,s)$
601\item
602$\ds 603\frac{L^{(r)}(A_f,1)}{r!} = 604\frac{\prod c_p \cdot \Omega_{A_f} \cdot \Reg_{A_f} 605\cdot \#\Sha(A_f)} 606{\#A_f(\Q)_{\tor}\cdot \#A_f^{\vee}(\Q)_{\tor}}. 607$
608\end{enumerate}
609\end{conjecture}
610Remarks: Part of the conjecture is that $\Sha(A_f)$ is finite.
611There is also a conjecture for arbitrary abelian varieties
612over global fields.
613Clay Math Problem: \$1000000 prize for proof of (1) in case$\dim(A_f)=1$614 615Here: 616\begin{itemize} 617\item$c_p$is the \textit{Tamagawa number} at the prime$p$, and the 618product is over the prime divisors of$N$. 619\item$\Omega_{A_f}$is the canonical N\'eron measure 620of$A_f(\R)$. 621\item$\Reg_{A_f}$is the regulator (absolute value 622of N\'eron-Tate canonical height pairing matrix). 623\item$A_f(\Q)_{\tor}$is the torsion subgroup of$A_f(\Q)$. 624\item$\Sha(A_f)$is the Shafarevich-Tate group. 625\end{itemize} 626\subsection{Evidence} 627\begin{itemize} 628\item Rubin: results in CM Case 629\item Kolyvagin, Logachev, 630Gross-Zagier, et al.: If$\ord_{s=1}L(f,s)=0$or$1$, 631then (1) true and$\Sha(A_f)$finite. 632\item Cremona: Compute$\Sha(A_f)_{?}$(=conjectural order) for tens 633of thousands of$A_f$of dimension$1$and get approximate square order. 634(Theorem of Cassels: if$E$an elliptic curve and 635$\Sha(E)$finite then order a perfect square. Note that the analogue for 636abelian varieties is false; for exampe, I've constructed examples for 637each odd prime$p<25000$of abelian varieties$A$of dimension$p-1$638such that$\Sha(A) = p\cdot n^2$.) 639\end{itemize} 640 641In this talk I will focus on$A_f$of possibly large dimension with$L(A_f,1)\neq 0$, since computation 642of$\Reg_{A_f}$is difficult (impossible?) when one can't even reasonably 643hope to write down$A_f$explicitly with equations. 644 645\section{Visibility of Shafarevich-Tate Groups} 646\subsection{Definitions} 647It is easy to write down a point on an elliptic curve$E$. You simply write down a pair of rational numbers, which are a solution to a Weierstrass equation. In contrast, imagine describing explicitly an element of$\Sha(E)$of order$2003$. The most direct way would be to give a genus one curve (with principal homogeneous space structure), embedded in$\P^3$of degree at least$2003$(!), hence very complicated. 648 649The idea of visibility of Shafarevich-Tate groups was introduced 650by Barry Mazur around 1998 to unify various constructions of 651elements of Shafarevich-Tate groups. 652\begin{definition}[Shafarevich-Tate Group]\label{defn:sha} 653$$654\Sha(A) = \Ker\left(\H^1(K,A) \to \bigoplus_v \H^1(K_v,A)\right). 655$$ 656\end{definition} 657Here$\H^1(K,A)$is the first Galois cohomology, which can 658be interpreted geometrically as the Weil-Chatalet group 659$$660\text{WC}(A/K) = \{\text{ principal homogenous spaces }X\text{ for }A\,\}/\sim. 661$$ 662 663Then$\Sha(A)$is the subgroup of locally trivial classes of 664homogenous spaces. For example 665$$3x^3+4y^3+5z^3=0 666\in \Sha(x^3+y^3+60z^3=0)[3].$$ 667 668Fix an inclusion$i:A\hra B$of abelian varieties and let$\pi: B\to C$669be the quotient of~$B$by the image of$A, so we have an exact sequence 670$$671 0 \to A \to B \to C \to 0 672$$ 673of abelian varieties. 674\begin{definition}[Visible Subgroup] 675\begin{align*} 676\Vis_i(\H^1(K,A)) &= \Ker\left(\H^1(K,A) \to \H^1(K,B)\right)\\ 677 &=\Coker(B(K) \to C(K)) 678\end{align*} 679and 680$$\Vis_i(\Sha(A)) &= \Ker(\Sha(A)\to \Sha(B)).$$ 681\end{definition} 682\begin{enumerate} 683\item The visible subgroup is finite because 684B(K)$is finitely generated and$\Vis_i(\H^1(K,A))$685is torsion. 686\item If$c\in\Vis_i(\H^1(K,A))$, then$c$is 687also visible'' in the sense that if$c$is the image 688of a point$x\in C(K)$, and if$X=\pi^{-1}(x)\subset B$, then 689$[X]\in\text{WC}(A)$corresponds to$c$. 690\item The visibile subgroups depends on the choice of embedding 691$i:A\hra B$. I've also considered defining 692$\Vis_B(\H^1(K,A))$to be the subgroup generated by all visible 693subgroups with respect to all embeddings$A\to B$, but I'm not 694sure what properties this definition has. 695\end{enumerate} 696 697 698\subsection{Theorems} 699 700Everything is visible somewhere.'' 701\begin{theorem}[Stein] 702If$c\in \H^1(K,A)$then there exists 703$B=\Res_{L/K}(A_L)$704such that$i:A\hra B$and$c\in\Vis_i(\H^1(K,A))$. 705(Here$L$is such that$\res_{L/K}(c)=0$.) 706\end{theorem} 707 708\noindentVisibility construction.'' 709\begin{theorem}[Agashe-Stein]\label{vis:const} 710Suppose$A,B\subset C$over$\Q$, that$A+B=C$, that 711$A\cap B$is finite. Suppose$N$is divisible by all 712bad primes for$C$, and$p$is a prime such that 713\begin{itemize} 714\item$B[p]\subset A$715\item$\ds p\nmid 2\cdot N\cdot \#B(\Q)_{\tor}\cdot \#(C/B)(\Q)_{\tor}\cdot
716\prod_{p\mid N} c_{A,p} \cdot c_{B,p}.$717\end{itemize} 718If$A$has rank$0$, then there is a natural inclusion 719$$720 B(\Q)/p B(\Q) \hra \Vis_C(\Sha(A)). 721$$ 722(And certain generalizations...) 723\end{theorem} 724 725 726 727\subsection{Example} 728 729\begin{example} 730For$N=389$, take$B$the (first ever) rank$2$elliptic curve, and 731$A$the$20$-dimensional rank$0$factor. 732$$733\xymatrix{ & B\ar[d] \\ 734 A \ar[r] & {J_0(389)} 735 } 736$$ 737Gives 738$$739 (\Z/5\Z)^2 \isom B(\Q)/5 B(\Q) \hra \Sha(A). 740$$ 741Part 2 of the Birch and Swinnerton-Dyer conjecture predicts that 742$$743 \Sha(A) = 5^2 \cdot 2^{?}, 744$$ 745so this gives evidence. 746\end{example} 747 748\section{Visibility in Modular Jacobians} 749Suppose now$A=A_f\subset J_0(N)$is attached to a newform. 750\begin{definition}[Modular of level$M$] 751An element$c\in\Sha(A)[p]$is \textit{modular of level}~$M$if 752$c \in \Vis_{M}^p(\Sha(A))$, 753where$\Vis_{M}^p(\Sha(A))$is 754the subgroup generated by all 755kernels of maps$\Sha(A)[p^\infty]\to \Sha(J_0(M))[p^\infty]$756induced by homomorphisms$A\to J_0(M)$757of degree coprime to$p$. 758\end{definition} 759Note that$M$must be a multiple of$N$. 760 761\begin{question}[Mazur] 762Suppose$E\subset J_0(N)$is an elliptic curve of conductor$N$. How much of$\Sha(E)$is modular of level$N$? 763\end{question} 764Answer: In examples, surprisingly much. Expect not all visible, since 765$$766 \Vis_{N}(\Sha(E)) \subset \Sha(E)[\text{modular degree}], 767$$ 768and modular degree annihilates symmetric square Selmer 769group (work of Flach). 770 771\subsection{Data and Experiments} 772\begin{itemize} 773\item {\bf Cremona-Mazur:} 774There are$52$elliptic curves$E\subset J_0(N)$with$N<5500$such that$p\mid \#\Sha(E)_?$. 775Cremona-Mazur show that for$43$of these that$\Sha(E)$776probably'' is modular of level$N$, and for$3$that it is definitely not: 777$N=2849, 4343, 5389$. (Probably'' was made provably'' in many cases 778in subsequent work.) 779 780\item {\bf Agashe-Stein:} 781Same question as Cremona-Mazur for$A_f\subset J_0(N)$of 782any dimension. Using results of my Ph.D. thesis, MAGMA packages, 783etc. I computed a divisor and multiple of$\#\Sha(A_f)_?$784for the following: 785\begin{itemize} 786\item$10360$abelian varieties$A_f\subset J_0(N)$with$L(A_f,1)\neq 0$. 787\item Found$168$with$\#\Sha(A_f)_?$definitely divisible by an odd prime. 788\item For$39$of these, prove that all$\#\Sha(A_f)_?^{\text{odd}}$789elements are modular of level$N$, and$106$probably are. This gives 790strong evidence for the BSD conjecture, and a sense that maybe 791something further is going on. 792\item Of these$168$, at least$62$have odd conjectural$\Sha$that 793is definitely {\em not} modular of level$N$. Big mystery? 794Where is this$\Sha$modular? Is it modular at all? Is it even there?? 795(Perhaps a good place to look for counterexample to BSD.) 796\end{itemize} 797 798\end{itemize} 799 800\section{Visibility at Higher Level} 801\begin{definition} 802Let$c\in\Sha(A_f)$. The {\em modularity levels} of$c$are the 803set of integers 804$$\mathcal{N}(c) = \{M: c\in\Vis_{M}(\Sha(A_f))\}.$$ 805\end{definition} 806 807\begin{conjecture}[Stein] 808For any$c\in\Sha(A_f)$we have 809$$810 \mathcal{N}(c) \neq \emptyset, 811$$ 812 i.e., every element of$\Sha(A_f)$is modular. 813\end{conjecture} 814Motivation: This is a working hypothesis that makes \textit{computing} 815with modular abelian varieties easier. 816Also, if there were a common level at which all of$\Sha(A_f)$were modular, 817then$\Sha(A_f)$would be finite, and conversely (assuming the conjecture). 818 819\subsection{Ribet Level Raising} 820Suppose that$f=\sum a_n q^n\in S_2(\Gamma_0(N))$821is a newform and$\p$is a nonzero prime ideal 822of$\Z[a_1,a_2,\ldots]$such that$A_f[\p]$823is irreducible. If 824$$825 a_\ell + \ell + 1 \con 0\pmod{\p} 826$$ 827then there exists an$\ell$-newform 828$g\in S_2(\Gamma_0(N\ell))$829such that$i(A_f[\p]) = A_g[\p]$for an appropriate 830$i:J_0(N)\to J_0(N\ell)$of degree coprime to$\text{char}(\p)$831and the sign 832of the functional equations for$L(f,s)$and$L(g,s)$833are the same. 834 835If we instead require that$a_\ell - (\ell+1)\con 0\pmod{\p}$836then there is such a$g$, but the sign of the functional 837equation changes, and the new Tamagawa numbers of 838$A_g$at$\ell$will (or tends to be?) divisible by$\p$. 839 840\subsection{Evidence for Conjecture} 841I defined a precise notion of probably modular'' motivated 842by Theorem~\ref{vis:const} and what I can compute. In many cases 843I could do extra work and actually prove modularity; however, at this stage it is more interesting to gather data to see what is going on, in order to have a sense for what 844to conjecture. 845 846Mazur proved that everything in$\Sha(E)[3]$, for$E$an elliptic curve, is visible 847in an abelian surface, which, together with the modularity theorem, {\em might} 848imply modularity of$\Sha(E)[3]$at higher level. Same for$2$, proved by me and by a 849different method by Thomas Klenke. 850 851 852\section{Some Tables} 853The first two pages of tables below give some of the data that 854I computed about visibility of Shafarevich-Tate groups 855at level$N$. The third table gives the new data about 856visibility at higher level. 857 858\newpage 859\begin{center} 860{\bf \Large Nontrivial Odd Parts of Shafarevich-Tate Groups} 861$\begin{array}{|l@{}ccc@{}c|l@{}c@{}|cc|}\hline
862\quad A& \dim& S_l & S_u & \moddeg(A)^{\op}
863    & \quad B  & \dim\, & \,\,A^{\vee}\intersect \tilde{B}^{\vee} & \Vis\\ \hline
864
865\mathbf{389E}*&20&5^{2}&=&5&\mathbf{389A}&1&[20^{2}]&5^{2} \\
866\mathbf{433D}*&16&7^{2}&=&7\!\cdot\!\mbox{\tiny $111$}&\mathbf{433A}&1&[14^{2}]&7^{2} \\
867\mathbf{446F}*&8&11^{2}&=&11\!\cdot\!\mbox{\tiny $359353$}&\mathbf{446B}&1&[11^{2}]&11^{2} \\
868\mathbf{551H}&18&3^{2}&=&\mbox{\tiny $169$}&\text{NONE} & & & \\
869\hline
870\mathbf{563E}*&31&13^{2}&=&13&\mathbf{563A}&1&[26^{2}]&13^{2} \\
871\mathbf{571D}*&2&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $127$}&\mathbf{571B}&1&[3^{2}]&3^{2} \\
872\mathbf{655D}*&13&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $9799079$}&\mathbf{655A}&1&[36^{2}]&3^{4} \\
873\mathbf{681B}&1&3^{2}&=&3\!\cdot\!\mbox{\tiny $125$}&\mathbf{681C}&1&[3^{2}]&- \\
874\hline
875\mathbf{707G}*&15&13^{2}&=&13\!\cdot\!\mbox{\tiny $800077$}&\mathbf{707A}&1&[13^{2}]&13^{2} \\
876\mathbf{709C}*&30&11^{2}&=&11&\mathbf{709A}&1&[22^{2}]&11^{2} \\
877\mathbf{718F}*&7&7^{2}&=&7\!\cdot\!\mbox{\tiny $5371523$}&\mathbf{718B}&1&[7^{2}]&7^{2} \\
878\mathbf{767F}&23&3^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
879\hline
880\mathbf{794G}*&12&11^{2}&=&11\!\cdot\!\mbox{\tiny $34986189$}&\mathbf{794A}&1&[11^{2}]&- \\
881\mathbf{817E}*&15&7^{2}&=&7\!\cdot\!\mbox{\tiny $79$}&\mathbf{817A}&1&[7^{2}]&- \\
882\mathbf{959D}&24&3^{2}&=&\mbox{\tiny $583673$}&\text{NONE} & & & \\
883\mathbf{997H}*&42&3^{4}&=&3^{2}&\mathbf{997B}&1&[12^{2}]&3^{2} \\
884\hline
885&&& && \mathbf{997C}&1&[24^{2}]&3^{2} \\
886\mathbf{1001F}&3&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $1269$}&\mathbf{1001C}&1&[3^{2}]&- \\
887&&& && \mathbf{91A}&1&[3^{2}]&- \\
888\mathbf{1001L}&7&7^{2}&=&7\!\cdot\!\mbox{\tiny $2029789$}&\mathbf{1001C}&1&[7^{2}]&- \\
889\hline
890\mathbf{1041E}&4&5^{2}&=&5^{2}\!\cdot\!\mbox{\tiny $13589$}&\mathbf{1041B}&2&[5^{2}]&- \\
891\mathbf{1041J}&13&5^{4}&=&5^{3}\!\cdot\!\mbox{\tiny $21120929983$}&\mathbf{1041B}&2&[5^{4}]&- \\
892\mathbf{1058D}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $483$}&\mathbf{1058C}&1&[5^{2}]&- \\
893\mathbf{1061D}&46&151^{2}&=&151\!\cdot\!\mbox{\tiny $10919$}&\mathbf{1061B}&2&[2^{2}302^{2}]&- \\
894\hline
895\mathbf{1070M}&7&3 \!\cdot\! 5^{2}&3^{2} \!\cdot\! 5^{2}&3 \!\cdot\! 5\!\cdot\!\mbox{\tiny $1720261$}&\mathbf{1070A}&1&[15^{2}]&- \\
896\mathbf{1077J}&15&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $1227767047943$}&\mathbf{1077A}&1&[9^{2}]&- \\
897\mathbf{1091C}&62&7^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
898\mathbf{1094F}*&13&11^{2}&=&11^{2}\!\cdot\!\mbox{\tiny $172446773$}&\mathbf{1094A}&1&[11^{2}]&11^{2} \\
899\hline
900\mathbf{1102K}&4&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $31009$}&\mathbf{1102A}&1&[3^{2}]&- \\
901\mathbf{1126F}*&11&11^{2}&=&11\!\cdot\!\mbox{\tiny $13990352759$}&\mathbf{1126A}&1&[11^{2}]&11^{2} \\
902\mathbf{1137C}&14&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $64082807$}&\mathbf{1137A}&1&[9^{2}]&- \\
903\mathbf{1141I}&22&7^{2}&=&7\!\cdot\!\mbox{\tiny $528921$}&\mathbf{1141A}&1&[14^{2}]&- \\
904\hline
905\mathbf{1147H}&23&5^{2}&=&5\!\cdot\!\mbox{\tiny $729$}&\mathbf{1147A}&1&[10^{2}]&- \\
906\mathbf{1171D}*&53&11^{2}&=&11\!\cdot\!\mbox{\tiny $81$}&\mathbf{1171A}&1&[44^{2}]&11^{2} \\
907\mathbf{1246B}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $81$}&\mathbf{1246C}&1&[5^{2}]&- \\
908\mathbf{1247D}&32&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $2399$}&\mathbf{43A}&1&[36^{2}]&- \\
909\hline
910\mathbf{1283C}&62&5^{2}&=&5\!\cdot\!\mbox{\tiny $2419$}&\text{NONE} & & & \\
911\mathbf{1337E}&33&3^{2}&=&\mbox{\tiny $71$}&\text{NONE} & & & \\
912\mathbf{1339G}&30&3^{2}&=&\mbox{\tiny $5776049$}&\text{NONE} & & & \\
913\mathbf{1355E}&28&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $2224523985405$}&\text{NONE} & & & \\
914\hline
915\mathbf{1363F}&25&31^{2}&=&31\!\cdot\!\mbox{\tiny $34889$}&\mathbf{1363B}&2&[2^{2}62^{2}]&- \\
916\mathbf{1429B}&64&5^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
917\mathbf{1443G}&5&7^{2}&=&7^{2}\!\cdot\!\mbox{\tiny $18525$}&\mathbf{1443C}&1&[7^{1}14^{1}]&- \\
918\mathbf{1446N}&7&3^{2}&=&3\!\cdot\!\mbox{\tiny $17459029$}&\mathbf{1446A}&1&[12^{2}]&- \\
919\hline
920
921\end{array}$922\end{center} 923 924\newpage 925\begin{center} 926{\bf \Large Nontrivial Odd Parts of Shafarevich-Tate Groups} 927$\begin{array}{|l@{}ccc@{}c|l@{}c@{}|cc|}\hline
928\quad A& \dim& S_l & S_u & \moddeg(A)^{\op}
929    & \quad B  & \dim\, & \,\,A^{\vee}\intersect \tilde{B}^{\vee} & \Vis\\ \hline
930
931\mathbf{1466H}*&23&13^{2}&=&13\!\cdot\!\mbox{\tiny $25631993723$}&\mathbf{1466B}&1&[26^{2}]&13^{2} \\
932\mathbf{1477C}*&24&13^{2}&=&13\!\cdot\!\mbox{\tiny $57037637$}&\mathbf{1477A}&1&[13^{2}]&13^{2} \\
933\mathbf{1481C}&71&13^{2}&=&\mbox{\tiny $70825$}&\text{NONE} & & & \\
934\mathbf{1483D}*&67&3^{2} \!\cdot\! 5^{2}&=&3 \!\cdot\! 5&\mathbf{1483A}&1&[60^{2}]&3^{2} \!\cdot\! 5^{2} \\
935\hline
936\mathbf{1513F}&31&3&3^{4}&3\!\cdot\!\mbox{\tiny $759709$}&\text{NONE} & & & \\
937\mathbf{1529D}&36&5^{2}&=&\mbox{\tiny $535641763$}&\text{NONE} & & & \\
938\mathbf{1531D}&73&3&3^{2}&3&\mathbf{1531A}&1&[48^{2}]&- \\
939\mathbf{1534J}&6&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $635931$}&\mathbf{1534B}&1&[6^{2}]&- \\
940\hline
941\mathbf{1551G}&13&3^{2}&=&3\!\cdot\!\mbox{\tiny $110659885$}&\mathbf{141A}&1&[15^{2}]&- \\
942\mathbf{1559B}&90&11^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
943\mathbf{1567D}&69&7^{2} \!\cdot\! 41^{2}&=&7 \!\cdot\! 41&\mathbf{1567B}&3&[4^{4}1148^{2}]&- \\
944\mathbf{1570J}*&6&11^{2}&=&11\!\cdot\!\mbox{\tiny $228651397$}&\mathbf{1570B}&1&[11^{2}]&11^{2} \\
945\hline
946\mathbf{1577E}&36&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $15$}&\mathbf{83A}&1&[6^{2}]&- \\
947\mathbf{1589D}&35&3^{2}&=&\mbox{\tiny $6005292627343$}&\text{NONE} & & & \\
948\mathbf{1591F}*&35&31^{2}&=&31\!\cdot\!\mbox{\tiny $2401$}&\mathbf{1591A}&1&[31^{2}]&31^{2} \\
949\mathbf{1594J}&17&3^{2}&=&3\!\cdot\!\mbox{\tiny $259338050025131$}&\mathbf{1594A}&1&[12^{2}]&- \\
950\hline
951\mathbf{1613D}*&75&5^{2}&=&5\!\cdot\!\mbox{\tiny $19$}&\mathbf{1613A}&1&[20^{2}]&5^{2} \\
952\mathbf{1615J}&13&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $13317421$}&\mathbf{1615A}&1&[9^{1}18^{1}]&- \\
953\mathbf{1621C}*&70&17^{2}&=&17&\mathbf{1621A}&1&[34^{2}]&17^{2} \\
954\mathbf{1627C}*&73&3^{4}&=&3^{2}&\mathbf{1627A}&1&[36^{2}]&3^{4} \\
955\hline
956\mathbf{1631C}&37&5^{2}&=&\mbox{\tiny $6354841131$}&\text{NONE} & & & \\
957\mathbf{1633D}&27&3^{6} \!\cdot\! 7^{2}&=&3^{5} \!\cdot\! 7\!\cdot\!\mbox{\tiny $31375$}&\mathbf{1633A}&3&[6^{4}42^{2}]&- \\
958\mathbf{1634K}&12&3^{2}&=&3\!\cdot\!\mbox{\tiny $3311565989$}&\mathbf{817A}&1&[3^{2}]&- \\
959\mathbf{1639G}*&34&17^{2}&=&17\!\cdot\!\mbox{\tiny $82355$}&\mathbf{1639B}&1&[34^{2}]&17^{2} \\
960\hline
961\mathbf{1641J}*&24&23^{2}&=&23\!\cdot\!\mbox{\tiny $1491344147471$}&\mathbf{1641B}&1&[23^{2}]&23^{2} \\
962\mathbf{1642D}*&14&7^{2}&=&7\!\cdot\!\mbox{\tiny $123398360851$}&\mathbf{1642A}&1&[7^{2}]&7^{2} \\
963\mathbf{1662K}&7&11^{2}&=&11\!\cdot\!\mbox{\tiny $16610917393$}&\mathbf{1662A}&1&[11^{2}]&- \\
964\mathbf{1664K}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $7$}&\mathbf{1664N}&1&[5^{2}]&- \\
965\hline
966\mathbf{1679C}&45&11^{2}&=&\mbox{\tiny $6489$}&\text{NONE} & & & \\
967\mathbf{1689E}&28&3^{2}&=&3\!\cdot\!\mbox{\tiny $172707180029157365$}&\mathbf{563A}&1&[3^{2}]&- \\
968\mathbf{1693C}&72&1301^{2}&=&1301&\mathbf{1693A}&3&[2^{4}2602^{2}]&- \\
969\mathbf{1717H}*&34&13^{2}&=&13\!\cdot\!\mbox{\tiny $345$}&\mathbf{1717B}&1&[26^{2}]&13^{2} \\
970\hline
971\mathbf{1727E}&39&3^{2}&=&\mbox{\tiny $118242943$}&\text{NONE} & & & \\
972\mathbf{1739F}&43&659^{2}&=&659\!\cdot\!\mbox{\tiny $151291281$}&\mathbf{1739C}&2&[2^{2}1318^{2}]&- \\
973\mathbf{1745K}&33&5^{2}&=&5\!\cdot\!\mbox{\tiny $1971380677489$}&\mathbf{1745D}&1&[20^{2}]&- \\
974\mathbf{1751C}&45&5^{2}&=&5\!\cdot\!\mbox{\tiny $707$}&\mathbf{103A}&2&[505^{2}]&- \\
975\hline
976\mathbf{1781D}&44&3^{2}&=&\mbox{\tiny $61541$}&\text{NONE} & & & \\
977\mathbf{1793G}*&36&23^{2}&=&23\!\cdot\!\mbox{\tiny $8846589$}&\mathbf{1793B}&1&[23^{2}]&23^{2} \\
978\mathbf{1799D}&44&5^{2}&=&\mbox{\tiny $201449$}&\text{NONE} & & & \\
979\mathbf{1811D}&98&31^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
980\hline
981\mathbf{1829E}&44&13^{2}&=&\mbox{\tiny $3595$}&\text{NONE} & & & \\
982\mathbf{1843F}&40&3^{2}&=&\mbox{\tiny $8389$}&\text{NONE} & & & \\
983\mathbf{1847B}&98&3^{6}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\
984\mathbf{1871C}&98&19^{2}&=&\mbox{\tiny $14699$}&\text{NONE} & & & \\
985\hline
986
987\end{array}$988\end{center} 989 990 991\newpage 992\large 993{\bf\Large Visibility at Higher Level\vspace{3ex}} 994\begin{center} 995\begin{tabular}{|l|l|}\hline 996&\vspace{-2ex}\\ 997$A_f$with odd invisible$\Sha_{\an}[\ell]$& All$\ell$-congruent\\ 998&$A_g\subset J_0(Np)_{\new}$\\ 999&with$Np\leq 5000$and \\ 1000&$\ord_{s=1}L(g,s)\geq 0$\\ 1001& (and higher$Np$if known)\\ 1002&\vspace{-2ex}\\ 1003% data is autogenerated by table.py 1004\first{\sha{551}{18}{3}} 1005 \add{\higher{2}{1}{2}} 1006 \add{\higher{3}{1}{2}} 1007 \add{\higher{5}{25}{0}} 1008\first{\sha{767}{23}{3}} 1009 \add{\higher{2}{1}{2}} 1010 \add{\higher{7}{1}{2}} 1011 \add{\higher{7}{52}{0}} 1012\first{\sha{959}{24}{3}} 1013 \add{\higher{2}{1}{2}} 1014\first{\sha{1091}{62}{7}} 1015 \add{\higher{7}{2}{2}} 1016\first{\sha{1283}{62}{5}} 1017 \add{\higher{3}{2}{2}} 1018\first{\sha{1337}{33}{3}} 1019 \add{\higher{2}{1}{2}} 1020\first{\sha{1339}{30}{3}} 1021 \add{\higher{2}{1}{2}} 1022\first{\sha{1355}{28}{3}} 1023 \add{\higher{2}{1}{2}} 1024\first{\sha{1429}{64}{5}} 1025 \add{\higher{2}{2}{2}} 1026 \add{\higher{3}{66}{0}} 1027\first{\sha{1481}{71}{13}} 1028 \add{Nothing in range} 1029\first{\sha{1513}{31}{3}} 1030 \add{\higher{2}{1}{2}} 1031\first{\sha{1529}{36}{5}} 1032 \add{\higher{7}{1}{2}} 1033\first{\sha{1559}{90}{11}} 1034 \add{Nothing in range} 1035\first{\sha{1589}{35}{3}} 1036 \add{Nothing in range} 1037\first{\sha{1631}{37}{5}} 1038 \add{\higher{2}{1}{2}} 1039\first{\sha{1679}{45}{11}} 1040 \add{\higher{2}{2}{2}} 1041\first{\sha{1727}{39}{3}} 1042 \add{\higher{2}{1}{2}} 1043\first{\sha{2849}{1}{3}} 1044 \add{\higher{3}{1}{2}} 1045\first{\sha{4343}{1}{3}} 1046 \add{Nothing in range} 1047\first{\sha{5389}{1}{3}} 1048 \add{\higher{7}{1}{2}} 1049\hline\end{tabular} 1050\end{center} 1051\vspace{3ex} 1052 1053\noindent When the second column contains an$A_g$of rank~$2$, 1054then$\Sha(A_f)[\ell]$is very likely'' to be visible of level$M=Np$. 1055This is the case for most examples. The Nothing in range'' note 1056means that the smallest~$p$for which there exists~$g$of even 1057analytic rank congruent to~$f\$ is beyond the range of my current
1058tables.  The examples of level 2849, 4343, and 5389 are the odd and
1059definitely invisible examples in Cremona and Mazur's original paper on
1060visibility.
1061
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1064\end{document}
1065