Author: William A. Stein
1%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2%                                                                          %
3% shacomp_v5.tex                                                           %
4%                                                                          %
5% Visibility of Shafarevich-Tate groups of modular abelian varieties       %
6%                                                                          %
7% William A. Stein and Amod Agashe.                                        %
8%                                                                          %
9%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
10
11\documentclass{article}
12\hoffset=-.06\textwidth
13\voffset=-.06\textheight
14\textwidth=1.12\textwidth
15\textheight=1.12\textheight
16\bibliographystyle{amsplain}
17\include{macros}
18\usepackage[all]{xy}
19
20
21\DeclareMathOperator{\Res}{Res}
22\font\cyrbig=wncyr10 scaled \magstep 1
23\newcommand{\BigSha}{\mbox{\cyrbig X}}
24
25\newcommand{\marginalfootnote}{\footnote{#1}\marginpar{\hfill {\sf\thefootnote}}}
26\newcommand{\edit}{\marginalfootnote{#1}}
27
28\title{\sc Visibility of Shafarevich-Tate groups of modular abelian varieties
29and the Birch and Swinnerton-Dyer conjecture}
30\author{Amod Agashe\footnote{Warning: Agashe has not yet
31read this draft of our preprint.} \and William A.~Stein}
32
33\begin{document}
34\UseTips
35\maketitle
36
37\begin{abstract}
38We give examples of abelian subvarieties of $J_0(N)$ that have
39nontrivial visible Shafarevich-Tate groups.  These examples provide
40evidence for the Birch and Swinnerton-Dyer conjecture, and the methods
41used to compute them may lead to new connections between this
42conjecture and the theory of congruences between modular forms.
43\end{abstract}
44
45\section{Introduction}\label{sec:intro}
46Let~$N$ be a positive integer and let $J_0(N)$ be the Jacobian of the
47modular curve $X_0(N)$ (see, e.g., \cite{diamond-im}).  Let~$A$ be an
48abelian subvariety of $J_0(N)$.  Then the inclusion $A\hookrightarrow 49J_0(N)$ induces a map on Galois cohomology $H^1(\Q,A) \ra 50H^1(\Q,J_0(N))$, which restricts to a map $\Sha(A) \ra \Sha(J_0(N))$.
51Following \cite{cremona-mazur}, we call an element $c\in\Sha(A)$
52\defn{visible} in $J_0(N)$ if it lies in the kernel of this map.
53When~$c$ is visible the torsor attached to~$c$ is realized as a
54subvariety of $J_0(N)$; it is a fiber of the quotient map $J_0(N)\ra 55J_0(N)/A.$
56
57In~\cite{cremona-mazur}, Cremona and Mazur ask the following question:
58If $E\subset J_0(N)$ is a elliptic curve, how much of $\Sha(E)$ is
59visible in an abelian surface~$B$ that is itself a subvariety
60of~$J_0(N)$?  Among the $52$ examples of odd nontrivial $\Sha$ that
61they consider, $\Sha$ is invisible in at least~$8$ cases.  However,
62they warn that the elliptic curves they consider have small conductor,
63and the general situation is probably much different.  Standard
64conjectures imply that when $\ell$ is sufficiently large, elements of
65order~$\ell$ in $\Sha(E)$ can not be visible in an abelian surface; it
66is thus essential to consider visibility in higher dimensional abelian
67varieties.
68
69We generalize the experiment of Cremona and
70Mazur by removing the constraint that only elliptic curves be
71considered.  The numbers become large at much smaller conductor, and
72the proportion of~$\Sha$ that is visible declines accordingly.
73
74In Section~\ref{sec:vis} we give a general definition of visibility,
75which is motivated by a restriction-of-scalars construction of
76torsors.  Then we prove a theorem that gives a criterion for
77the existence of nontrivial visible elements of Shafarevich-Tate
78groups.  Section~\ref{sec:computing} describes the algorithms we use to
79enumerate modular abelian varieties, compute conjectural
80orders of their Shafarevich-Tate groups, and verify the visibility
81criterion of Section~\ref{sec:vis} when possible.  Section~\ref{sec:tables}
82presents the results of extensive numerical investigations.  It provides
83the impetus for the conjectures we make and questions we ask in
84Section~\ref{sec:conj}.
85
86{\bf Acknowledgment.} It is a pleasure to thank B.~Mazur for his
87lectures and conversations about visibility, K.~Ribet for explaining
88congruences to us, and R.~Coleman for helping us to understand
89component groups.  The authors would also like to thank B.~Conrad,
90J.~Ellenberg, R.~Greenberg, D.~Gross, L.~Merel, B.~Poonen, and R.~Taylor
92
93\section{Visible cohomology classes}\label{sec:vis}
94In Section~\ref{sec:torsors} we define the notion of visibility and
95observe that every cohomology class
96arises from a torsor that can be constructed geometrically inside of
97an appropriate restriction of scalars, then we define visible
98cohomology classes.  Section~\ref{sec:visthm}, which should perhaps be
99omitted from a first reading, gives a criterion for the existence of
100visible elements of Shafarevich-Tate groups.
101
102\subsection{Geometric realization of torsors}\label{sec:torsors}
103Let~$A$ be an abelian variety over a field~$K$.  The following
104definition is due to Mazur.
105
106\begin{definition}[Visible]
107Let $\iota:A\hookrightarrow J$
108be an embedding of~$A$ into an abelian variety~$J$.  Then the \defn{visible part
109of $H^1(K,A)$ with respect to the embedding~$\iota$} is
110        $$\Vis_J(H^1(K,A)) = \Ker(H^1(K,A)\ra{}H^1(K,J)).$$
111\end{definition}
112
113The Galois cohomology group $H^1(K,A)$ has a geometric interpretation
114as the group of classes of torsors~$X$ for~$A$ (see~\cite{lang-tate}).
115To a cohomology class $c\in H^1(K,A)$, there is a corresponding
116variety~$X$ over~$K$ and a map $A\cross X \ra X$ that satisfies axioms
117similar to those for a simply transitive group action.
118Suppose $\iota: A\ra J$ is an embedding and $c\in \Vis_J(H^1(K,A))$.
119We have an exact sequence of abelian varieties
120$0\ra A\ra J\ra B\ra 0$.  A piece of the
121associated long exact sequence of Galois cohomology is
122$$\cdots \ra J(K)\ra B(K) \ra H^1(K,A) \ra H^1(K,J) \ra \cdots,$$
123so, because $\iota_*(c)=0$, there is a point $x\in B(K)$ that maps
124to $c\in H^1(K,A)$.  Then the fiber~$X$ over~$x$ is a subvariety of~$J$,
125which, when equipped with its natural action of~$A$, lies in the class
126of torsors corresponding to~$c$.
127
128\begin{proposition}
129Every element of $H^1(K,A)$ is visible in some abelian variety~$J$.
130\end{proposition}
131\begin{proof}
132Fix $c\in H^1(K,A)$.  There is a finite extension~$L$ of~$K$ such that
133$\res_L(c) = 0\in H^1(L,A)$.  Let $J=\Res_{L/K}(A_L)$ be the
134restriction of scalars down to~$K$ of the abelian variety~$A$, which
135we view as an abelian variety over~$L$ (see \cite[\S7.6]{neronmodels}).
136Thus~$J$ is an abelian variety over~$K$ of dimension $[L:K]\cdot \dim(A)$, and for any
137scheme~$S$ over~$K$, we have a natural bijection $J(S) \ncisom A_L(S_L)$.
138In particular, $J(A) = A_L(A_L)$, and there is an
139injection $\iota:A\hookrightarrow J$ attached to $\id_{A_L}\in A_L(A_L)$.
140Using the Shapiro lemma, one finds that there is a canonical isomorphism
141$H^1(L,A) \isom H^1(K,J)$ and that
142$\iota_*(c) = 0\in H^1(K,J)$.
143\end{proof}
144\begin{remark}
145In \cite{cremona-mazur}, J.~de Jong gives a sophisticated proof of
146the above proposition in the special case when~$A$ is an elliptic curve.
147His proof involves Azumaya algebras.
148\end{remark}
149
150\subsection{Visible elements of Shafarevich-Tate groups}\label{sec:visthm}
151The Shafarevich-Tate group of an abelian variety over a number
152field measures the failure of the local-to-global principle
153for its torsors.
154\begin{definition}[Shafarevich-Tate group]
155Let~$A$ be an abelian variety over~$\Q$.
156The \defn{Shafarevich-Tate group} of $A$ is
157  $$\Sha(A) = \Ker\left(H^1(\Q,A) \ra \prod_{v} H^1(\Q_v,A)\right),$$
158where the product is over all places of~$\Q$.
159\end{definition}
160
161If $\iota:A\hookrightarrow J$ is an embedding, then the
162visible part of $\Sha(A)$ is
163  $$\Vis_J(\Sha(A)) = \Sha(A) \intersect \Vis_J(H^1(\Q,A)) = \Ker(\Sha(A)\ra \Sha(J)).$$
164We use the following theorem to produce examples of visible
165elements of Shafarevich-Tate groups.
166\begin{theorem}\label{thm:shaexists}
167Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ such
168that $A\intersect B$ is finite and $A(\Q)$ is finite.
169Let~$N$ be an integer divisible by the primes of bad reduction for~$J$.
170Assume that~$B$ has purely toric reduction at each prime dividing~$N$.
171Suppose~$p$ is a prime such that
172$$p\nmid 2N\cdot \#(J/B)(\Q)_{\tor}\cdot\#B(\Q)_{\tor}\cdot 173 \prod_{\ell\mid N} \#\Phi_{A,\ell}(\Fbar_\ell)\cdot 174 \#\Phi_{B,\ell}(\F_\ell).$$
175Suppose furthermore that $B[p] \subset A\intersect B$.
176Then
177         $$B(\Q)/pB(\Q)\hookrightarrow \Vis_J(\Sha(A)).$$
178\end{theorem}
179The proof proceeds in four steps.  First the torsion and
180$p$-congruence hypothesis is used to produce an injection $B(\Q)/p 181B(\Q)\hookrightarrow \Vis_J(H^1(\Q,A))$.  Next we perform a local
182analysis at each place~$v$ of~$\Q$, which proceeds in three steps.  At
183places~$v$ of bad reduction, we use the Mumford-Tate uniformization;
184at odd primes of good reduction we apply an exactness theorem about
185N\'eron models; when~$2$ is a place of bad reduction, we modify the
186situation by a $2$-isogeny and apply another exactness theorem.
187
188
189\begin{proof}
190The quotient $J/A$ is an abelian variety~$C$.  The long exact sequence
191of Galois cohomology associated to the short exact sequence
192$$0 \ra A \ra J \ra C \ra 0$$
193begins
194$$0\ra A(\Q) \ra J(\Q) \ra C(\Q) \xrightarrow{\,\delta\,} 195 H^1(\Q,A) \ra \cdots.$$
196Let~$\phi$ be map $B\ra C$, which is obtained by composing
197the inclusion $B\hookrightarrow J$ with the quotient map $J\ra C$.
198Since $B[p]\subset A$, we see that~$\phi$ factors through multiplication by~$p$.
199We thus obtain the following commutative diagram:
200$$\xymatrix{ 201& B\ar[d] \ar[r]^{p}& B\ar[d]\\ 202A\ar[r]&J\ar[r]&C.}$$
203Using that $B(\Q)[p]=0$, we
204obtain the following diagram, all of whose rows and columns are exact:
205$$\xymatrix{ 206 & K_0\ar[d] & K_1\ar[d]& K_2\ar[d]\\ 2070 \ar[r] & B(\Q) \ar[r]^{p}\ar[d] & B(\Q)\ar[dr]^{\pi} \ar[r]\ar[d] 208 & B(\Q)/pB(\Q)\ar[r]\ar[d] & 0\\ 2090 \ar[r] & J(\Q)/A(\Q)\ar[r]\ar[d] & C(\Q) \ar[r] & \delta(C(\Q)) \ar[r] & 0\\ 210 & K_3, 211}$$
212where $K_0$, $K_1$ and $K_2$ are the indicated kernels and $K_3$ is the
213indicated cokernel.
214The snake lemma gives an exact sequence
215  $$K_0\ra K_1 \ra K_2 \ra K_3.$$
216Because $B\ra C$ is an isogeny, $K_1\subset B(\Q)_{\tor}$.
217Since $B(\Q)[p]=0$ and $K_2$ is a $p$-torsion group, the map
218$K_1\ra K_2$ is the~$0$ map.
219The quotient
220$J(\Q)/B(\Q)$ has no $p$-torsion because
221it is a subgroup  of $(J/B)(\Q)$; also, $A(\Q)$ is a finite group,
222so $K_3 = J(\Q)/(A(\Q)+B(\Q))$ has no $p$-torsion, and the map
223$K_2\ra K_3$ must be the~$0$ map.
224We conclude that $K_2=0$.
225
226The above argument shows that $B(\Q)/p B(\Q)$ is a subgroup of
227$H^1(\Q,A)$; however, the latter group contains infinitely many elements of
228order~$p$, whereas $\Sha(A)[p]$ is a finite group, so we must work
229harder in order to deduce that $B(\Q)/p B(\Q)$ actually lies in
230$\Sha(A)[p]$.   Let $x\in B(\Q)$; we must show
231that $\pi(x)\in \Sha(A)[p]$.  It suffices to show that
232$\res_v(\pi(x))=0$ for all places~$v$ of~$\Q$.
233
234At the archimedian place $v=\infty$, the restriction
235$\res_v(\pi(x))$ is killed by~$2$ and the odd prime~$p$,
236hence $\res_v(\pi(x))=0$.
237
238Suppose that~$v$ is a place at which~$J$ has bad reduction.
239By hypothesis, $B$ has purely toric reduction,
240so over $\Q_v^{\ur}$
241there is an isomorphism $B\isom\Gm^d/\Gamma$
242of $\Gal(\Qbar_v/\Q_v^{\ur})$-modules,
243for some lattice'' $\Gamma$.  For example, when
244$\dim B=1$, this is the Tate curve representation of~$B$.
245Let~$n$ be the order of the component group of~$B$ at~$v$; thus~$n$
246equals the order of the cokernel of the valuation
247map $\Gamma\ra \Z^d$.  Choose a representative
248$P=(x_1,\ldots,x_d)\in\Gm^d$ for the point~$x$.
249Let $n'=\#\Phi_{B,v}(\F_v)$.
250Then since~$P$ is rational over $\Q_v$,
251$n'P$ can be adjusted by elements of~$\Gamma$
252so that each of its components $x_i\in\Gm$ has valuation~$0$;
253since~$p$ does not equal the residue characteristic of~$v$,
254it follows that there is a point $Q\in\Gm^d(\Q_v^{\ur})$
255such that $pQ = n'P$.
256Thus the cohomology class $\res_v(\pi(n'x))$ is unramified at~$v$.
257By \cite[Prop.~3.8]{milne:duality},
258$$H^1(\Q_v^{\ur}/\Q_v,A(\Q_v^{\ur})) 259 =H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fpbar)),$$
260where $\Phi_{A,v}$ is the component group of~$A$ at~$v$.
261Since~$p$ does not divide $\#\Phi_{A,v}(\Fpbar)$,
262and $\pi(n'x)$ has order~$p$, it follows that
263  $$0=\res_v(\pi(n'x))=n'\res_v(\pi(x)).$$
264Since the order of $\res_v(\pi(x))$ is coprime to~$n'$,
265we conclude that $\res_v(\pi(x))=0$.
266
267Next suppose that~$J$ has good reduction at~$\ell$
268and that~$\ell$ is {\em odd}.
269Let $\cA$, $\cJ$, $\cC$, be the N\'eron models
270of~$A$,~$J$,~$C$, respectively.
271Since~$\ell$ is odd, $1=e<\ell-1$, so we may apply
272\cite[Thm.~7.5.4]{neronmodels} to conclude that
273the sequence of group schemes
274$$0\ra \cA \ra \cJ\ra \cC \ra 0$$
275is exact, in the sense that it
276is exact as a sequence of sheaves on the
277\'etale site (see the proof of~\cite[Thm.~7.5.4]{neronmodels}).
278Thus it is exact on the stalks, so by~\cite[2.9(d)]{milne:etale}
279the sequence
280$$0\ra \cA(\Z_v^{\ur})\ra \cB(\Z_v^{\ur}) \ra \cC(\Z_v^{\ur})\ra 0$$
281is exact. By the N\'eron mapping property, the sequence
282$$0\ra A(\Q_v^{\ur})\ra B(\Q_v^{\ur}) \ra C(\Q_v^{\ur})\ra 0$$
283is also exact, so
284 $\res_v(\pi(x))$ is unramified.
285By \cite[Prop.~I.3.8]{milne:duality},
286 $$H^1(\Q_v^{\ur}/\Q_v,A) \isom H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fbar_v))=0,$$
287since~$A$ has good reduction at~$v$.
288Hence $\res_v(\pi(x))=0$.
289
290If~$J$ has bad reduction at~$v=2$, then we already dealt with~$2$ above.
291Consider the case when~$J$ has good reduction at~$2$.   The
292absolute ramification index~$e$ of $\Z_2$ is~$1$, which is
293{\em not} less than $2-1=1$, so we can not apply \cite[Thm.~7.5.4]{neronmodels}.
294However, we can modify everything by an isogeny of degree a power
295of~$2$ and apply a different theorem, as follows.
296The $2$-primary subgroup~$\Psi$ of $A\intersect B$ is rational
297over~$\Q$.   The abelian varieties
298$\tilde{J}=J/\Psi$, $\tilde{A}=A/\Psi$, and
299$\tilde{B}=B/\Psi$ also satisfy the hypothesis of
300the theorem we wish to prove.
301By \cite[Prop.~7.5.3(a)]{neronmodels}, the corresponding sequence of
302N\'eron models
303$$0\ra\tilde{\cA}\ra\tilde{\cJ}\ra\tilde{\cC}\ra 0$$
304is exact, so the sequence
305$$0\ra \tilde{A}(\Q_v^{\ur})\ra\tilde{J}(\Q_v^{\ur}) 306\ra\tilde{C}(\Q_v^{\ur})\ra 0$$
307is exact. Thus the image of
308$\res_v(\pi(x))$  in $H^1(\Q_v,\tilde{A})$ is unramified.
309It equals~$0$, again by \cite[Prop.~3.8]{milne:duality},
310since the component group of $\tilde{A}$ at~$v$ has order a power
311of~$2$, whereas $\pi(x)$ has odd prime order~$p$.
312Thus $\res_v(\pi(x))=0$, since
313the kernel of $H^1(\Q_v,A)\ra H^1(\Q_v,\tilde{A})$ is a
314finite group of $2$-power order.
315\end{proof}
316
317\subsubsection{Visibility when $A$ also has positive rank}
318In Theorem~\ref{thm:shaexists}, if the condition that $A(\Q)$ has rank~$0$ is removed,
319then the proof can be easily modified to show that the kernel of
320    $B(\Q)/p B(\Q) \ra \Vis_J(\Sha(A))$
321has dimension at most the rank of $A(\Q)$.
322
323According to \cite{cremona:algs},
324the smallest conductor elliptic curve~$E$ of rank~$3$ is found in
325$J=J_0(5077)$.  The number $5077$ is prime, and~$J$ decomposes
326up to isogeny as
327             $A \cross B \cross E,$
328where each of~$A$, $B$, and~$E$ are abelian subvarieties of~$J$
329associated to newforms, which have
330dimensions $205$, $216$, and~$1$, respectively.
331The modular degree of~$E$ is $1984=2^6\cdot 31$, and
332the sign of the Atkin-Lehner involution on~$E$ is the same
333as its sign on~$A$, so $E\subset A$.
334The numerator of $(5077-1)/12$ is $3^2\cdot 47$, so
335$31$ is coprime to the orders of any relevant component groups or torsion.
336Thus $\Vis_J(\Sha(A))$ contains $(\Z/31\Z)^2$.
337
338
339
340\section{Guide to computing on $J_0(N)$}\label{sec:computing}
341The Jacobian $J_0(N)$ is equipped with an action of the Hecke algebra~$\T$.
342Let $f\in S_2(\Gamma_0(N))$ be a newform, and let $I_f\subset\T$
343be the annihilator of~$f$.  The abelian variety~$A_f$ attached
344to~$f$ is the quotient $J_0(N)/I_f J_0(N)$.  Thus $A_f$ is an
345abelian variety of dimension equal to the number of Galois
346conjugates of~$f$ and equipped with a faithful action of $\T/I_f$.
347For the remainder of this section, $A=A_f$ denotes the optimal quotient
348of $J_0(N)$ attached to the annihilator~$I=I_f$ of a newform~$f$.
349
350\subsection{The Birch and Swinnerton-Dyer conjecture}
351The Birch and Swinnerton-Dyer conjecture, as generalized by
352Tate in~\cite{tate:bsd}, furnishes a conjectural formula
353for the order of the Shafarevich-Tate group of any new
354optimal quotient~$A$.
355In general, it is difficult given~$A$ to compute
356the conjectural order of~$\Sha(A)$.
357However, the situation is more optimistic
358when~$A$ is a new modular abelian variety
359such that $L(A,1)\neq 0$.
360For these~$A$ we have devised an algorithm
361that we use to compute the odd part of the
362conjectural order of $\Sha(A)$ in many cases.
363The following is a special case of a much more general conjecture.
364\begin{conjecture}[Birch and Swinnerton-Dyer]\label{conj:bsd}
365Suppose $L(A,1)\neq 0$.   Then
366$$\frac{L(A,1)}{\Omega_A} = 367\frac{\#\Sha(A)\cdot\prod_{p\mid N} c_p} 368{\# A(\Q)\cdot\#\Adual(\Q)}.$$
369\end{conjecture}
370When $L(A,1)\neq 0$, work of Kolyvagin and Logachev
371\cite{kolyvagin-logachev:finiteness,kolyvagin-logachev:totallyreal}
372implies that $A(\Q)$, $\Adual(\Q)$, and $\Sha(A)$ are all finite,
373so the quantities appearing in the above formula make sense.
374Here $c_p=\#\Phi_{A,p}(\F_p)$, the positive real number~$\Omega_A$ is
375the measure of~$A(\R)$ with respect to a basis of differentials on
376the N\'eron model of~$A$, and~$\Adual$ is the abelian variety
377dual of~$A$.
378The algorithms described below enable us in my cases to compute the
379conjectural order of $\Sha(A)$.  However, for question of visibility,
380we instead need to compute the order of $\Sha(\Adual)$.  This is no
381different because the Cassels-Tate pairing implies that
382$\#\Sha(A) = \#\Sha(\Adual)$.
383
384\subsection{Modular symbols}\label{modsym}
385It is not possible to compute very much about $J_0(N)$ without
386modular symbols, which provide a finite presentation for the homology
387group $H_1(X_0(N),\Z)$ in terms of paths between elements of
388$\P^1(\Q) = \Q\union \{\infty\}$.
389
390The \defn{modular symbol} defined by a pair $\alpha,\beta\in\P^1(\Q)$
391is denoted $\{\alpha,\beta\}$.  This modular symbol should be viewed as
392the homology class, relative to the cusps,
393of a geodesic path from~$\alpha$ to~$\beta$ in $\h^*$.
394The homology group relative to the cusps is a slight enlargement
395of the usual homology group, in that
396we allow paths with endpoints in $\P^1(\Q)$ instead of restricting
397to closed loops.
398We declare that modular symbols satisfy
399the following homology relations:
400if $\alpha,\beta,\gamma \in \Q\union\{\infty\}$, then
401$$\{\alpha,\beta\} + \{\beta,\gamma\} + \{\gamma,\alpha\} = 0.$$
402Furthermore, the space of modular symbols is torsion free, so, e.g.,
403$\{\alpha,\alpha\} = 0$ and
404$\{\alpha,\beta\} = -\{\beta,\alpha\}$.
405
406Denote by~$\sM_2$ the free abelian group with basis the set of
407symbols $\{\alpha,\beta\}$ modulo the three-term homology relations
408above and modulo any torsion.
409There is a left action of $\GL_2(\Q)$ on $\sM_2$, whereby
410a matrix~$g$ acts by
411  $$g\{\alpha, \beta\} = \{g(\alpha), g(\beta)\},$$
412and~$g$ acts on~$\alpha$ and~$\beta$ by a linear fractional
413transformation.
414The space $\sM_2(N)$ of \defn{modular symbols for $\Gamma_0(N)$}
415is the quotient of $\sM_2$ by the submodule
416generated by the infinitely many elements
417of the form $x - g(x)$, for~$x$ in ~$\sM_2$
418and~$g$ in $\Gamma_0(N)$, and modulo any torsion.
419A \defn{modular symbol for $\Gamma_0(N)$} is an element of
420this space.   We frequently denote the equivalence
421class that defines a modular symbol by giving a
422representative element.
423
424In \cite{manin:parabolic}, Manin proved that there is
425a natural injection $H_1(X_0(N),\Z)\hookrightarrow \sM_2(N)$.
426The image of $H_1(X_0(N),\Z)$ in $\sM_2(N)$ can be identified as follows.
427Let $\sB_2(N)$ denote the free abelian group whose basis is the finite set
428$\Gamma_0(N)\backslash \P^1(\Q)$.
429The \defn{boundary map} $\delta: \sM_2(N)\ra \sB_2(N)$
430sends $\{\alpha,\beta\}$ to $[\beta]-[\alpha]$, where $[\beta]$
431denotes the basis element of $\sB_2(N)$ corresponding to $\beta\in\P^1(\Q)$.
432The kernel $\sS_2(N)$ of~$\delta$ is the subspace of
433\defn{cuspidal} modular symbols.
434An element of $\sS_2(N)$ can be thought of as a linear
435combination of paths
436in $\h^*$ whose endpoints are cusps, and whose images in $X_0(N)$
437are a linear combination of loops.
438We thus obtain a canonical isomorphism $\varphi:\sS_2(N)\ra H_1(X_0(N),\Z)$.
439
440Part of the utility of modular symbols comes from the classical Abel-Jacobi
441theorem, which allows us to view $J_0(N)(\C)$ as the quotient
442$\C^g/H_1(X_0(N),\Z)$,
443where $H_1(X_0(N),\Z)$ is embedded in
444$\C^d\ncisom\Hom(S_2(\Gamma_0(N)),\C)$ using the integration pairing.
445Thus modular symbols give an explicit description of $J_0(N)(\C)$
446and of its constituent parts as modules over the Hecke algebra.
447We can also compute Hecke operators using modular symbols.
448
449For further introductory remarks on modular symbols, see~\cite{stein:modsyms},
450and for detailed instructions as to how to compute the space of modular symbols
451and the action of Hecke operators on it, see~\cite{cremona:algs}.
452
453\subsection{Computing with quotients and subvarieties of $J_0(N)$}
454First, we describe how to enumerate the newforms of level~$N$.  Then
455we define the modular degree, whose square annihilates the visible part
456of~$\Sha$.  Finally, we describe how to intersect abelian subvarieties
457of $J_0(N)$.
458
459\subsubsection{Enumerating quotients}
460Let $H_1(X_0(N),\Z)^+$ denote the $+1$-eigenspace for the action
461of the involution induced by complex conjugation.
462We list all newforms of a given level~$N$ by decomposing the new
463subspace of $H_1(X_0(N),\Q)^+$ under the action of the the
464Hecke operators.  First we compute the characteristic polynomial of~$T_2$,
465and use it to break up the full space.  We apply this process
466recursively with $T_3, T_5, \ldots$ until either we have exceeded the
467bound coming from~\cite{sturm:cong}, or we have found a Hecke
468operator~$T_n$ whose characteristic polynomial is irreducible.  After
469computing the decomposition, we order the newforms in a way that
470extends the systematic ordering in~\cite{cremona:algs}: First sort by
471dimension, with smallest dimension first; within each dimension, sort
472in binary by the signs of the Atkin-Lehner involutions, e.g., $+++$,
473$++-$, $+-+$, $+--$, $-++$, etc.  When two forms have the same sign
474sequence, order by $|\Tr(a_p)|$ with ties broken by taking the
475positive trace first.
476
477We denote a Galois conjugacy class of newforms by a bold symbol such
478as $\mathbf{389E}$, which consists of the level and the isogeny class,
479where $\mathbf{A}$ denotes the first class, $\mathbf{B}$ the second,
480and so on.
481
482As discussed in \cite[pg.~5]{cremona:algs}, for certain small levels
483the above ordering when restricted to elliptic curves does not agree
484with the ordering used in Cremona's tables.  For example, in the
485present paper our $\mathbf{446B}$ is Cremona's $\mathbf{446D}$.
486
487\subsubsection{The modular degree\label{modpolar}}
488A \defn{polarization}~$\lambda$ of an abelian variety~$A$ over~$\Q$ is an isogeny
489$\lambda:A\ra \Adual$ such that $\lambda_{\Qbar}$
490arises from an ample invertible sheaf on $A_{\Qbar}$ (see, e.g.,
491\cite[\S13]{milne:abvars}).
492Since $J_0(N)$ is a Jacobian, it possesses a canonical
493polarization arising from the $\theta$-divisor, and this
494polarization induces the \defn{modular polarization}
495$\theta: \Adual\ra A$ of $\Adual$.
496$$\[email protected]=3pc{ 497{\Adual}\[email protected]{^(->}[r]^{\pi^{\vee}\qquad}\ar[dr]^{\theta} & 498 J_0(N)^{\vee} \isom J_0(N)\,\,\,\[email protected]{->>}[d]^{\pi}\\ 499 &A.}$$
500If we view $\Adual$ as an abelian subvariety of $J_0(N)$, then the
501kernel of~$\theta$ is the intersection of $\Adual$ with $I J_0(N)$; thus
502the kernel of $\theta$ measures intersections between $\Adual$ and other
503factors of $J_0(N)$.
504\begin{definition}[Modular degree]
505The \defn{modular degree} $m_A$ of $A$ is
506$\sqrt{\deg(\theta)}$.
507\end{definition}
508\noindent{}By \cite[Thm.~13.3]{milne:abvars}, $\deg(\theta)$ is a perfect
509square, so $m_A$ is an integer.
510For an algorithm to compute $m_A$, see~\cite{kohel-stein:ants4}.
511
512The modular degree is of interest because its square
513annihilates the visible cohomology classes.
514\begin{proposition}
515$\displaystyle \Vis_{J_0(N)}(H^1(\Q,\Adual))\subset H^1(\Q,\Adual)[m_A^2]$
516\end{proposition}
517\begin{proof}
518Let $\delta$ be the composite map
519$\Adual \ra J_0(N)\ra A$.  There is a map $\hat{\delta}:A\ra \Adual$
520such that $\hat{\delta}\circ\delta$ is multiplication by $\deg(\delta)=m_A^2$.
521Thus
522$\Ker(H^1(\Q,\Adual)\ra H^1(\Q,J_0(N)))$
523is contained in $H^1(\Q,\Adual)[m_A^2]$.
524\end{proof}
525\begin{remark}
526When~$A$ has dimension one, the visible part of $H^1(\Q,\Adual)$ is contained
527in $H^1(\Q,A)[m_A]$.  Is this true for~$A$ of all dimensions?
528\end{remark}
529
530\subsubsection{Intersecting complex tori}\label{sec:intersect}
531Consider a complex torus $J=V/\Lambda$, and let
532$A=V_A/\Lambda_A$ and $B=V_B/\Lambda_B$ be subtori whose
533intersection $A\intersect B$ is finite.
534Here $V_A$ and $V_B$ are subspaces of~$V$ and $\Lambda_A$ and $\Lambda_B$
535are submodules of~$\Lambda$.
536\begin{proposition}\label{prop:intersection}
537There is a natural isomorphism of groups
538$$A\intersect B \isom 539 \left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right)_{\tor.}$$
540\end{proposition}
541\begin{proof}
542There is an exact sequence
543$$0\ra A\intersect B \ra A \oplus B \ra J.$$
544Consider the diagram
545$$\xymatrix{ 546 & {\Lambda_A \oplus\Lambda_B}\ar[d] \ar[r] & {\Lambda} \ar[r]\ar[d]& 547 {\Lambda/(\Lambda_A + \Lambda_B)}\ar[d]\\ 548 & {V_A \oplus V_B}\ar[d] \ar[r] & V \ar[r]\ar[d] & {V/(V_A+V_B)}\ar[d]\\ 549 {A\intersect B}\ar[r] & A\oplus B\ar[r] & J \ar[r] & J/(A+ B).}$$
550The snake lemma\index{Snake lemma} gives an exact sequence
551$$0 \ra 552 A\intersect B \ra 553 \Lambda/(\Lambda_A + \Lambda_B) \ra 554 V/(V_A+V_B).$$
555Since $V/(V_A+V_B)$ is a $\C$-vector space, the torsion
556part of $\Lambda/(\Lambda_A + \Lambda_B)$ must map to~$0$.
557No non-torsion in $\Lambda/(\Lambda_A + \Lambda_B)$ could
558map to~$0$, because if it did then $A\intersect B$ would not
559be finite.  The lemma follows.
560\end{proof}
561
562The following formula for the intersection of~$n$
563subtori is obtained  in a similar way.
564\begin{proposition}
565For $i=1,\ldots,n$ let $A_i = V_i/\Lambda_i$ be a subtorus of
566$J=V/\Lambda$, and assume that each pairwise intersection
567$A_i \intersect A_j$ is finite.
568Then
569$$A_1\intersect \cdots \intersect A_n 570 \isom 571\left(\frac{\Lambda\oplus \cdots \oplus \Lambda} 572 {f(\Lambda_1\oplus\cdots\oplus \Lambda_n)}\right),$$
573where $f(x_1,\ldots,x_n)=(x_1-x_2,x_2-x_3,x_3-x_4,\ldots,x_{n-1}-x_n)$.
574\end{proposition}
575
576
577\subsection{Computing the conjectural order of $\BigSha(A)$}
578In this section, we describe how in many cases we can compute the
579conjectural order of $\Sha(A)$ when $L(A,1)\neq 0$, at least up to a
580power of~$2$.
581
582In Section~\ref{sec:torsion}, we bound $\#A(\Q)$ and $\#\Adual(\Q)$.
583We compute each $c_p$ in Section~\ref{sec:tamagawa},
584for each~$p$ with $p\mid\mid N$. When $p^2\mid{}N$, it is
585possible to bound $c_p$; see, e.g., \cite[Cor.~15.2.1]{silverman:aec}
586where one finds that when $\dim A=1$ and $p^2\mid N$, we have $c_p\leq 4$.
587In Section~\ref{sec:bsdratio}, we use modular symbols to compute
588the rational number $L(A,1)/\Omega_A$, up to a bounded Manin constant.
589
590\subsubsection{Torsion subgroup}\label{sec:torsion}
591We obtain an upper bound on $\#A(\Q)_{\tor}$ and $\#\Adual(\Q)_{\tor}$
592as follows.
593The characteristic polynomial $\chi_p(X)$ of the Hecke operator~$T_p$
594acting on~$A$ is a monic polynomial having integer coefficients
595and degree equal to the dimension of~$A$.
596\begin{proposition}
597Both $\#A(\Q)_{\tor}$ and $\#\Adual(\Q)_{\tor}$ divide
598$$\gcd \{ \chi_p(p+1) : (p,2N)=1,\, \text{\rm p prime} \}.$$
599\end{proposition}
600\begin{proof}
601Use the Eichler-Shimura relation and that for primes~$p$ for which $p\nmid 2N$ the maps
602$A(\Q)_{\tor} \ra \tilde{A}(\F_p)$ and
603$\Adual(\Q)_{\tor} \ra \tilde{A}^{\vee}(\F_p)$
604are both injective, and that
605$\#\Adual(\F_p)_{\tor}=\#\tilde{A}^{\vee}(\F_p)$.
606\end{proof}
607
608The difference of two cusps $\alp,\beta \in{} X_0(N)$ defines
609a point $(\alp)-(\beta) \in J_0(N)(\C)$.  Manin observed
610in \cite{manin:parabolic} that $(0)-(\infty)$ is rational.
611The order of the image of $(0)-(\infty)$ in  $A(\Q)$ can be computed as follows.
612Let
613   $$V=\Hom(S_2(\Gamma_0(N)),\C),$$
614and $V_I = \Hom(S_2(\Gamma_0(N))[I],\C)$.
615The integration pairing $\langle f, \gamma \rangle = 2\pi i \int_\gamma f(z)dz$
616between homology and cusp forms gives rise to a map $P: H_1(X_0(N),\Q)\ra V_I$.
617By the Abel-Jacobi theory (see, e.g., \cite[Thm~IV.2.2]{lang:modular}),
618$A(\C) \isom V_I/P(H_1(X_0(N),\Z))$.
619\begin{proposition}
620The order of the image of $(\alp)-(\beta)$ in $A(\C)$
621equals the order of the image of the modular symbol $\{\alp,\beta\}$
622in $P(H_1(X_0(N),\Q))/P(H_1(X_0(N),\Z)).$
623\end{proposition}
624The quotient appearing in the proposition can be computed algebraically
625by replacing~$P$ by a map with the same kernel as~$P$.  Such a map can
626be computed using the Hecke operators (see \cite[\S3.7]{stein:phd}).
627
628\subsubsection{Tamagawa numbers}\label{sec:tamagawa}
629Suppose~$p$ is a prime that exactly divides~$N$ and let $\Phi_{A,p}$
630denote the component group of~$A$ at~$p$.
631We have an exact sequence,
632  $$0\ra \cA_{\Fp}^0\ra \cA_{\Fp} \ra \Phi_{A,p}\ra 0,$$
633where $\cA_{\Fp}$ is the closed fiber of the N\'eron model of~$A$ over $\Z_p$ and $\cA_{\Fp}^0$
634is the component of $\cA_{\Fp}$ that contains the identity.
635A formula for $\#\Phi_{A,p}(\Fbar_p)$ and, up to a power of $2$,
636for $\#\Phi_{A,p}(\F_p)$,
637is given in \cite{kohel-stein:ants4} and \cite{stein:compgroup}.
638
639\subsubsection{Rational part of the special value}\label{sec:bsdratio}
640As in Section~\ref{sec:torsion},
641let $P : H_1(X_0(N),\Z) \ra \Hom(S_2(\Gamma_0(N))[I],\C)$
642be the map induced by integration.
643Let $P(H_1(X_0(N),\Z))^+$ denote the $+1$-eigenspace for the action
644of the involution induced by complex conjugation on the image of~$P$.
645
646\begin{theorem}\label{thm:ratpart}
647$$\frac{L(A,1)}{\Omega_{A}} 648 = [P(H_1(X_0(N),\Z))^+ : P(\T\{0,\infty\})]/(c_\infty\cdot c_A),$$
649where $c_\infty$ is the number of components of $A(\R)$ and $c_A$
650is the Manin constant of~$A$, as defined below.
651\end{theorem}
652In order to define the Manin constant of~$A$,
653let $\cA$ denote the N\'eron model of~$A$ over~$\Z$.
654\begin{definition}[Manin constant]
655The \defn{Manin constant}~$c_A$ of~$A$ is the index
656 $$c_A := [S_2(\Gamma_0(N);\Z)[I]:H^0(\cA,\Omega_{\cA/\Z})].$$
657\end{definition}
658In the definition, we have implicitly mapped $H^0(\cA,\Omega_{\cA/\Z})$ into
659$S_2(\Gamma_0(N);\Q)$ using the composition of the following maps:
660$$H^0(\cA,\Omega_{\cA/\Z}) \ra 661 H^0(\cJ,\Omega_{\cJ/\Z})[I] \ra 662 H^0(J,\Omega_{J/\Q})[I] \ra 663 S_2(\Gamma_0(N);\Q)[I].$$
664For a discussion of why $H^0(\cA,\Omega_{\cA/\Z})$ is in fact
665contained in $S_2(\Gamma_0(N);\Z)[I]$, see \cite{agashe-stein:manin}.
666\begin{theorem}
667If $\ell \mid c_A$ then $\ell^2 \mid 4N$.
668\end{theorem}
669\begin{proof}
670See~\cite[\S4]{mazur:rational} when~$A$ has dimension~$1$,
671and \cite{agashe-stein:manin} in general.
672\end{proof}
673
674We now give the proof of Theorem~\ref{thm:ratpart}.
675\begin{proof}[Proof of Theorem~\ref{thm:ratpart}]
676Let $H=H_1(X_0(N),\Z)$ and $S=S_2(\Gamma_0(N))$.
677There is a perfect pairing $\T \cross S \ra \Z$ given by
678$\langle T_n, f\rangle = a_n(f)$, which
679induces a canonical isomorphism of rings $\T\isom \Hom_\Z(S,\Z)$,
680where $\Hom_\Z(S,\Z)$ is a ring under multiplication of functions.
681The subring $W=\Hom_\Z(S[I],\Z)$ of $\Hom_\Z(S[I],\R)$ is
682isomorphic to $\T/I$, since $S[I]$ is saturated in~$S$.
683Thus
684\begin{eqnarray*}
685 [W : P(\{0,\infty\})W] &=& [W:P(\T \{0,\infty\})]\\
686   &=& [W:P(H)^+] \cdot [P(H)^+ : P(\T \{0,\infty\})].
687\end{eqnarray*}
688To complete the proof, observe that
689that $\Omega_A = [W:P(H)^+] \cdot c_\infty\cdot c_A$ and observe
690that multiplication by $P(\{0,\infty\})$ has determinant
691$\prod_{i=1}^d 2\pi i \int_{\{0,\infty\}} f^{(i)} = \pm L(A,1)$.
692\end{proof}
693
694\subsection{Emerton's work}
695When~$N$ is prime, M.~Emerton has proved in \cite{emerton:myconj}
696that $\#A_f(\Q)$ and $c_p(A_f)$ divide the numerator of $(N-1)/12$.
697
698\section{Visibility tables}\label{sec:tables}
699The tables in this section guide and motivate the conjectures
700and questions of Section~\ref{sec:conj}.
701
702In {\bf Table~\ref{table:invisible}}, we list each of the $8$ invisible odd
703Shafarevich-Tate groups found in \cite{cremona-mazur}, and
704prove\footnote{This computation is currently only partially complete.} that
705they are visible in some $J_0(Nq)$.
706
707{\bf Table~\ref{table:prime}} lists every quotient $A_f$ of $J_0(p)$
708with $p\leq 2593$ and $L(A_f,1)\neq 0$ such that the BSD conjecture
709predicts that $\#\Sha(\Adual_f)$ is divisible by an odd
710prime.  In addition, the table contains data that can frequently be
711used in conjuction with Theorem~\ref{thm:shaexists} to deduce that
712there are visible elements of $\Sha(\Adual_f)$.  When the {\bf B}
713column is labeled NONE then there is definitely nothing in
714$\Sha(\Adual_f)$ of the predicted order.  When the {\bf B} column
715contains an elliptic curve, its rank has been computed and is~$2$, so
716there are visible elements of $\Sha(\Adual_f)$.  When the {\bf B}
717column contains an abelian variety of dimension greater than~$1$, we
718have verified that $L(B,1)=0$, so the BSD conjecture predicts that
719$B(\Q)$ is infinite; however, we have not proved that $B(\Q)$ is
720infinite.  If we assume that $B(\Q)$ is infinite, it follows in these cases that
721$\Sha(A)$ is visible in $J_0(p)$.  Note that~$B$ has rank~$2$ over the
722Hecke algebra here, so the results of \cite{gross-zagier} say
723nothing about $B(\Q)$.
724
725{\bf Table~\ref{table:prime2}} continues the computations of Table~\ref{table:prime}
726up to level $5647$.  For each prime~$p$ between $2609$ and $5674$, we computed each
727factor $A$ such that $L(A,1)\neq 0$ and the odd part of
728$\#\Shaan(A)$ is nontrivial.  We then found all factors~$B$ such that
729$L(B,1)=0$ and there is a mod~$\ell$ congruence between~$A$ and~$B$,
730where $\ell\mid \#\Shaan(A)$.  The column labeled~$N$ gives the level,
731the column labeled $d(A)$ gives the dimension of~$A$, the column labeled
732$d(B)$ gives the dimension of~$B$, and the column labeled cong'' gives
733the odd part of $\gcd(\# A\intersect B, \#\Shaan(A))$.
734
735{\bf Table~\ref{table:mordell-weil}}
736lists every quotient $A_f$ of $J_0(N)$ with $N\leq 1642$ such
737that $L(A_f,1)=0$ but the sign in the functional equation for~$f$ is $+1$.
738For each such $A_f$, we looked for an abelian variety~$B$ such that~$B$
739has rank~$0$ and $A_f^{\vee}$ probably gives rise to odd visible
740elements of $\Sha(B)$.  This table contains initial data towards the idea of constructing
741points on high-rank abelian varieties by constructing visible elements
742of Shafarevich-Tate groups using, e.g., Euler system methods.
743For example, to prove that $A=\mathbf{1061B}$ really has positive rank,
744we consider the variety $B=\mathbf{1061D}$.
745To prove that $A(\Q)\neq 0$, it suffices to construct an appropriate element of $\Sha(B)$
746and show that this element is visible in $A+B\subset J_0(1061)$.
747
748{\bf Table~\ref{table:motive}} suggests a first tenuous step towards a
749computational theory of motives attached to modular forms of weight
750greater than two.  This table is organized like
751Table~\ref{table:prime}, except that the abelian varieties are
752replaced by motives attached to weight~$4$ modular forms.  For
753example, at prime level~$127$ there is a $17$-dimensional motive $\cM$
754such that $\Sha(\cM)(2)$ seems to contain elements of order~$43$.  The
755computations used to suggest this conclusion were carried out using
756algorithms for higher weight modular symbols as described in
757\cite{merel:1585}, \cite{stein:phd}, and \cite{stein-verrill:periods}.
758
759
760\subsection{Odd invisible $\BigSha$ in \cite{cremona-mazur}}
761\label{table:invisible}
762$$763\begin{array}{lccll} 764\mbox{\rm\bf E}&\#\Sha(E)& \text{mod deg}(E) & \mbox{\rm\bf F} 765 & \text{Where \Sha(E) is visible}\\ 766& & & & \vspace{-3ex} \\ 767\mbox{\rm\bf 2849A}& 3^2 &2^5\cdot 5\cdot 61&\mbox{\rm\bf NONE}& \text{visible using an elliptic curve at level 3\cdot 2849}\\ 768\mbox{\rm\bf 3364C}& 7^2 &2^6\cdot3^2\cdot5^2\cdot7 &\mbox{\rm\bf none}& \text{visible using a 3-dimensional F at level 3364}\\ 769\mbox{\rm\bf 4229A}& 3^2 &2^3\cdot3\cdot7\cdot13 &\mbox{\rm\bf none}& \text{not visible at level 4299,}\\ 770 &&&&\text{???}\\ 771\mbox{\rm\bf 4343B}& 3^2 &2^4\cdot1583 &\mbox{\rm\bf NONE}& ???\\ 772\mbox{\rm\bf 4914N}& 3^2 &2^4\cdot 3^5 &\mbox{\rm\bf none}& ??? \\ 773\mbox{\rm\bf 5054C}& 3^2 &2^3\cdot 3^3\cdot 11&\mbox{\rm\bf none}& ???\\ 774\mbox{\rm\bf 5073D}& 3^2 &2^5\cdot 3\cdot 5\cdot7\cdot23 775 &\mbox{\rm\bf none}& ???\\ 776\mbox{\rm\bf 5389A}& 3^2 &2^2\cdot 2333 &\mbox{\rm\bf NONE}& \text{visible using an elliptic curve at level 7\cdot 5389}\\ 777\end{array} 778$$
779
780\comment{\subsubsection*{Remarks}
781The elliptic curve~$E$ denoted {\bf 3364C} is  labeled none'' because there is no
782{\em elliptic curve} that satisfies an appropriate $7$-congruence with
783{\bf 3364C}.  However, the modular degree is divisible by~$7$, so there
784must be some abelian subvariety that satisfies a $7$-congruence with~$E$.
785Computing, we find a $3$-dimensional abelian variety~$A$ such that $T_2$, $T_3$,
786and $T_5$ satisfies the polynomials $x^3$, $x^3 + 5x^2 + 6x + 1$, and
787$x^3 + 5x^2 + 6x + 1$ on~$A$, respectively.  Furthermore, $L(A,1)=0$, so
788the BSD conjecture strongly suggests that there are elements of $\Sha(E)$
789of order~$7$ that are visible in $E+A \subset J_0(3364)$.
790}
791
792\newpage
793\subsection{Visibility of $\BigSha$ at prime level}\label{table:prime}
794The entries in the columns mod deg'' and $\Shaan$'' are only really
795the odd parts of mod deg'' and $\Shaan$''.  Theorem~\ref{thm:shaexists} does not
796apply to the two entries marked with a $*$.\vspace{-.25ex}
797{\small
798$$799\begin{array}{lccclcc} 800\mbox{\rm\bf A}& \mbox{\rm dim}& \Shaan(A) & \mbox{\rm mod deg}(A) & \mbox{\rm\bf B} & \mbox{\rm dim} & \mbox{\rm mod deg} (B)\\ 801& & & & & & \vspace{-3ex} \\ 802\mbox{\rm\bf 389E}& 20 &5^{2}&5&\mbox{\rm\bf 389A}& 1 &5\\ 803\mbox{\rm\bf 433D}& 16 &7^{2}&3\cdot7\cdot37&\mbox{\rm\bf 433A}& 1 &7\\ 804\mbox{\rm\bf 563E}& 31 &13^{2}&13&\mbox{\rm\bf 563A}& 1 &13\\ 805\mbox{\rm\bf 571D}& 2 &3^{2}&3^{2}\cdot127&\mbox{\rm\bf 571B}& 1 &3\\ 806\mbox{\rm\bf 709C}& 30 &11^{2}&11&\mbox{\rm\bf 709A}& 1 &11\\ 807\mbox{\rm\bf 997H}& 42 &3^{4}&3^{2}&\mbox{\rm\bf 997B}& 1 &3\\ 808\mbox{\rm\bf 1061D}& 46 &151^{2}&61\cdot151\cdot179&\mbox{\rm\bf 1061B}& 2 &151\\ 809\mbox{\rm\bf 1091C}& 62 &7^{2}&1&\mbox{\rm NONE} & & \\ 810\mbox{\rm\bf 1171D}& 53 &11^{2}&3^{4}\cdot11&\mbox{\rm\bf 1171A}& 1 &11\\ 811\mbox{\rm\bf 1283C}& 62 &5^{2}&5\cdot41\cdot59&\mbox{\rm NONE} & & \\ 812\mbox{\rm\bf 1429B}& 64 &5^{2}&1&\mbox{\rm NONE} & & \\ 813\mbox{\rm\bf 1481C}& 71 &13^{2}&5^{2}\cdot2833&\mbox{\rm NONE} & & \\ 814\mbox{\rm\bf 1483D}& 67 &3^{2}\cdot5^{2}&3\cdot5&\mbox{\rm\bf 1483A}& 1 &3\cdot5\\ 815\mbox{\rm\bf 1531D}*& 73 &3^{2}&3&\mbox{\rm\bf 1531A}& 1 &3\\ 816\mbox{\rm\bf 1559B}& 90 &11^{2}&1&\mbox{\rm NONE} & & \\ 817\mbox{\rm\bf 1567D}& 69 &7^{2}\cdot41^{2}&7\cdot41&\mbox{\rm\bf 1567B}& 3 &7\cdot41\\ 818\mbox{\rm\bf 1613D}& 75 &5^{2}&5\cdot19&\mbox{\rm\bf 1613A}& 1 &5\\ 819\mbox{\rm\bf 1621C}& 70 &17^{2}&17&\mbox{\rm\bf 1621A}& 1 &17\\ 820\mbox{\rm\bf 1627C}& 73 &3^{4}&3^{2}&\mbox{\rm\bf 1627A}& 1 &3^{2}\\ 821\mbox{\rm\bf 1693C}& 72 &1301^{2}&1301&\mbox{\rm\bf 1693A}& 3 &1301\\ 822\mbox{\rm\bf 1811D}& 98 &31^{2}&1&\mbox{\rm NONE} & & \\ 823\mbox{\rm\bf 1847B}& 98 &3^{6}&1&\mbox{\rm NONE} & & \\ 824\mbox{\rm\bf 1871C}& 98 &19^{2}&14699&\mbox{\rm NONE} & & \\ 825\mbox{\rm\bf 1877B}& 86 &7^{2}&1&\mbox{\rm NONE} & & \\ 826\mbox{\rm\bf 1907D}& 90 &7^{2}&3\cdot5\cdot7\cdot11&\mbox{\rm\bf 1907A}& 1 &7\\ 827\mbox{\rm\bf 1913B}& 1 &3^{2}&3\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\ 828\mbox{\rm\bf 1913E}& 84 &5^{4}\cdot61^{2}&5^{2}\cdot61\cdot103&\mbox{\rm\bf 1913A,C}& 1,2 &3\cdot5^{2}, 5^2\cdot 61\\ 829\mbox{\rm\bf 1933C}*& 83 &3^{2}\cdot7^{2}&3\cdot7&\mbox{\rm\bf 1933A}& 1 &3\cdot7\\ 830\mbox{\rm\bf 1997C}& 93 &17^{2}&1&\mbox{\rm NONE} & & \\ 831\mbox{\rm\bf 2027C}& 94 &29^{2}&29&\mbox{\rm\bf 2027A}& 1 &29\\ 832\mbox{\rm\bf 2029C}& 90 &5^{2}\cdot269^{2}&5\cdot269&\mbox{\rm\bf 2029A}& 2 &5\cdot269\\ 833\mbox{\rm\bf 2039F}& 99 &3^{4}\cdot5^{2}&19\cdot29\cdot7759\cdot3214201&\mbox{\rm NONE} & & \\ 834\mbox{\rm\bf 2063C}& 106 &13^{2}&61\cdot139&\mbox{\rm NONE} & & \\ 835\mbox{\rm\bf 2089J}& 91 &11^{2}&3\cdot5\cdot11\cdot19\cdot73\cdot139&\mbox{\rm\bf 2089B}& 1 &11\\ 836\mbox{\rm\bf 2099B}& 106 &3^{2}&1&\mbox{\rm NONE} & & \\ 837\mbox{\rm\bf 2111B}& 112 &211^{2}&1&\mbox{\rm NONE} & & \\ 838\mbox{\rm\bf 2113B}& 91 &7^{2}&1&\mbox{\rm NONE} & & \\ 839\mbox{\rm\bf 2161C}& 98 &23^{2}&1&\mbox{\rm NONE} & & \\ 840\mbox{\rm\bf 2213C}& 101 & 3^4 & ? & \mbox{\rm NONE} & & \\ 841\mbox{\rm\bf 2239B}& 110 & 11^4 & 1 & \mbox{\rm NONE} & & \\ 842\mbox{\rm\bf 2251E}& 99 & 37^2 & 37 & \mbox{\rm\bf 2251A} & 1 & 37\\ 843\mbox{\rm\bf 2273C}& 105 & 7^2 & ? & \mbox{\rm NONE}& & \\ 844\mbox{\rm\bf 2287B}& 109 & 71^2 & 1 & \mbox{\rm NONE}& & \\ 845\mbox{\rm\bf 2293C}& 96 & 479^2& 479 & \mbox{\rm\bf 2293A} & 2 & 479\\ 846\mbox{\rm\bf 2311B}& 110 & 5^2 & 1 & \mbox{\rm NONE}& & \\ 847\mbox{\rm\bf 2333C}& 101 &83341^{2}&83341&\mbox{\rm\bf 2333A}& 4 &83341\\ 848\mbox{\rm\bf 2339C}& 114 &3^{8}&6791&\mbox{\rm NONE} & & \\ 849\mbox{\rm\bf 2411B}& 123 &11^{2}&1&\mbox{\rm NONE} & & \\ 850\mbox{\rm\bf 2593C}& 109 &67^2\cdot 2213^2 & 67 \cdot 2213&\mbox {\bf 2593A}& 4 851 & 67 \cdot 2213\\ 852\end{array} 853$$
854}
855
856\subsection{More $\Sha$ at prime level}\label{table:prime2}
857Only odd parts of $\Shaan$ and congruences are given.
858Observe that $\Shaan$ is only visible roughly 10 percent of the time!
859As the level gets large, we find that there is almost always some
860nontrivial $\Sha$ in a large-dimensional factor of $J_0(p)$, and that
861this $\Sha$ is invisible.
862(Warning: In making this table, $53$ primes below $5647$ were not analyzed.)
863{\tiny
864$$865\hspace{-6em}\begin{array}{lcccc|} 866 N & d(A) & \Shaan & d(B) & cong\\ 867\mathbf{2609} & 127 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\ 868\mathbf{2617} & 114 & 11^{2}\cdot19^{2} & 2 & 11\cdot19 \\ 869\mathbf{2647} & 117 & 13^{2} & \text{NONE} & \\ 870\mathbf{2659} & 123 & 53^{2} & \text{NONE} & \\ 871\mathbf{2663} & 132 & 43^{2} & \text{NONE} & \\ 872\mathbf{2671} & 122 & 37^{2} & \text{NONE} & \\ 873\mathbf{2677} & 115 & 3^{2} & 1 & 3 \\ 874\mathbf{2693} & 122 & 3^{4} & \text{NONE} & \\ 875\mathbf{2699} & 125 & 19^{2} & \text{NONE} & \\ 876\mathbf{2707} & 119 & 5^{2} & \text{NONE} & \\ 877\mathbf{2713} & 118 & 19^{2} & \text{NONE} & \\ 878\mathbf{2731} & 124 & 53^{2} & \text{NONE} & \\ 879\mathbf{2749} & 124 & 7^{2} & \text{NONE} & \\ 880\mathbf{2767} & 125 & 5^{2} & \text{NONE} & \\ 881\mathbf{2789} & 136 & 83^{2} & \text{NONE} & \\ 882\mathbf{2791} & 135 & 29^{2} & \text{NONE} & \\ 883\mathbf{2797} & 119 & 11^{2} & 1 & 11 \\ 884\mathbf{2819} & 138 & 13^{2} & \text{NONE} & \\ 885\mathbf{2837} & 128 & 23^{2} & 1 & 23 \\ 886\mathbf{2843} & 129 & 3^{6}\cdot587^{2} & \text{NONE} & \\ 887\mathbf{2851} & 129 & 7^{2} & \text{NONE} & \\ 888\mathbf{2861} & 133 & 11^{4}\cdot61^{2} & 2 & 11\cdot61 \\ 889\mathbf{2879} & 148 & 97^{2} & \text{NONE} & \\ 890\mathbf{2903} & 150 & 643^{2} & \text{NONE} & \\ 891\mathbf{2939} & 150 & 17^{2}\cdot19^{2} & \text{NONE} & \\ 892\mathbf{2953} & 127 & 29^{2} & 1 & 29 \\ 893\mathbf{2963} & 134 & 5^{2}\cdot31^{2}\cdot61^{2} & 2 & 31\cdot61 894\\ 895\mathbf{2969} & 136 & 103^{2} & \text{NONE} & \\ 896\mathbf{2999} & 161 & 1459^{2} & \text{NONE} & \\ 897\mathbf{3001} & 132 & 3^{4} & \text{NONE} & \\ 898\mathbf{3011} & 146 & 5^{2}\cdot101^{2} & \text{NONE} & \\ 899\mathbf{3019} & 130 & 3259^{2} & 2 & 3259 \\ 900\mathbf{3041} & 147 & 103^{2} & \text{NONE} & \\ 901\mathbf{3067} & 134 & 5^{4} & \text{NONE} & \\ 902\mathbf{3079} & 148 & 131^{2} & \text{NONE} & \\ 903\mathbf{3083} & 141 & 179^{2} & \text{NONE} & \\ 904\mathbf{3089} & 135 & 5^{2}\cdot131^{2} & 2 & 5\cdot131 \\ 905\mathbf{3109} & 136 & 5^{2} & \text{NONE} & \\ 906\mathbf{3119} & 164 & 11^{2}\cdot59^{2} & \text{NONE} & \\ 907\mathbf{3181} & 144 & 43^{2} & \text{NONE} & \\ 908\mathbf{3187} & 139 & 3^{4} & \text{NONE} & \\ 909\mathbf{3191} & 167 & 53^{2} & \text{NONE} & \\ 910\mathbf{3203} & 143 & 13^{2} & \text{NONE} & \\ 911\mathbf{3221} & 149 & 7^{2}\cdot41^{2} & \text{NONE} & \\ 912\mathbf{3229} & 142 & 3^{2} & \text{NONE} & \\ 913\mathbf{3251} & 166 & 3^{4} & \text{NONE} & \\ 914\mathbf{3257} & 143 & 13^{2} & \text{NONE} & \\ 915\mathbf{3271} & 146 & 7^{4}\cdot43^{2}\cdot71^{2} & 3 & 9167\cdot43\cdot71 \\ 917\mathbf{3299} & 164 & 6131^{2} & \text{NONE} & \\ 918\mathbf{3301} & 145 & 5^{2} & \text{NONE} & \\ 919\mathbf{3319} & 158 & 5^{4} & \text{NONE} & \\ 920\mathbf{3323} & 155 & 179^{2} & \text{NONE} & \\ 921\mathbf{3329} & 157 & 83^{2} & \text{NONE} & \\ 922\mathbf{3331} & 152 & 937^{2} & \text{NONE} & \\ 923\mathbf{3343} & 148 & 7^{2}\cdot53^{2} & \text{NONE} & \\ 924\mathbf{3347} & 150 & 139^{2} & \text{NONE} & \\ 925\mathbf{3359} & 174 & 67^{4} & \text{NONE} & \\ 926\mathbf{3371} & 159 & 1259^{2} & \text{NONE} & \\ 927\mathbf{3391} & 159 & 29^{2} & \text{NONE} & \\ 928\mathbf{3407} & 170 & 499^{2} & \text{NONE} & \\ 929\mathbf{3433} & 148 & 5^{4}\cdot7^{2} & \text{NONE} & \\ 930\mathbf{3449} & 168 & 107^{2} & \text{NONE} & \\ 931\mathbf{3461} & 167 & 83^{2} & \text{NONE} & \\ 932\mathbf{3463} & 151 & 199^{2} & 2 & 199 \\ 933\mathbf{3467} & 162 & 5^{4} & \text{NONE} & \\ 934\mathbf{3469} & 151 & 47^{2} & \text{NONE} & \\ 935\mathbf{3491} & 168 & 67^{2} & \text{NONE} & \\ 936\mathbf{3511} & 166 & 37^{2} & \text{NONE} & \\ 937\mathbf{3527} & 179 & 659^{2} & \text{NONE} & \\ 938\mathbf{3529} & 153 & 79^{2} & \text{NONE} & \\ 939\mathbf{3533} & 164 & 3^{4} & \text{NONE} & \\ 940\mathbf{3539} & 170 & 1871^{2} & \text{NONE} & \\ 941\mathbf{3541} & 156 & 5^{4} & \text{NONE} & \\ 942\mathbf{3557} & 156 & 229^{2} & \text{NONE} & \\ 943\mathbf{3559} & 170 & 1109^{2} & \text{NONE} & \\ 944\end{array} 945\begin{array}{|lcccc|} 946 N & d(A) & \Shaan & d(B) & cong\\ 947\mathbf{3571} & 163 & 67^{2} & \text{NONE} & \\ 948\mathbf{3583} & 161 & 3319^{2} & 2 & 3319 \\ 949\mathbf{3607} & 159 & 7^{4}\cdot19^{2} & \text{NONE} & \\ 950\mathbf{3613} & 156 & 7^{2} & \text{NONE} & \\ 951\mathbf{3617} & 165 & 3^{2} & \text{NONE} & \\ 952\mathbf{3623} & 172 & 3^{6} & \text{NONE} & \\ 953\mathbf{3631} & 172 & 433^{2} & \text{NONE} & \\ 954\mathbf{3643} & 160 & 5^{2} & \text{NONE} & \\ 955\mathbf{3659} & 181 & 3^{2}\cdot11^{4} & \text{NONE} & \\ 956\mathbf{3671} & 193 & 509^{2} & \text{NONE} & \\ 957\mathbf{3691} & 166 & 353^{2} & \text{NONE} & \\ 958\mathbf{3701} & 174 & 3^{4}\cdot281^{2} & 2 & 3^{2}\cdot281 \\ 959\mathbf{3709} & 164 & 3^{12} & \text{NONE} & \\ 960\mathbf{3719} & 188 & 13^{2}\cdot977^{2} & \text{NONE} & \\ 961\mathbf{3739} & 166 & 83^{2} & \text{NONE} & \\ 962\mathbf{3761} & 176 & 677^{2} & \text{NONE} & \\ 963\mathbf{3769} & 168 & 13^{2} & \text{NONE} & \\ 964\mathbf{3779} & 187 & 73^{2}\cdot149^{2} & 1 & 73 \\ 965\mathbf{3797} & 172 & 19^{2} & \text{NONE} & \\ 966\mathbf{3803} & 171 & 2531^{2} & \text{NONE} & \\ 967\mathbf{3821} & 182 & 307^{2} & \text{NONE} & \\ 968\mathbf{3823} & 173 & 7^{2} & \text{NONE} & \\ 969\mathbf{3863} & 191 & 11^{2}\cdot23^{2}\cdot311^{2} & 970\text{NONE} & \\ 971\mathbf{3907} & 168 & 3^{4} & \text{NONE} & \\ 972\mathbf{3919} & 182 & 71^{2} & \text{NONE} & \\ 973\mathbf{3929} & 185 & 877^{2} & \text{NONE} & \\ 974\mathbf{3931} & 174 & 31^{2} & \text{NONE} & \\ 975\mathbf{3943} & 173 & 2479319^{2} & 4 & 2479319 \\ 976\mathbf{3967} & 180 & 3^{6}\cdot13^{2} & 1 & 3\cdot13 \\ 977\mathbf{4007} & 195 & 7321^{2} & \text{NONE} & \\ 978\mathbf{4013} & 176 & 61^{2} & \text{NONE} & \\ 979\mathbf{4019} & 186 & 3^{4}\cdot5^{2}\cdot7^{4} & \text{NONE} & \\ 980\mathbf{4021} & 182 & 5^{4}\cdot71^{2} & \text{NONE} & \\ 981\mathbf{4027} & 174 & 29^{2}\cdot79^{2} & 2 & 29\cdot79 \\ 982\mathbf{4049} & 186 & 5^{2}\cdot3491^{2} & \text{NONE} & \\ 983\mathbf{4057} & 173 & 103^{2} & \text{NONE} & \\ 984\mathbf{4079} & 212 & 5^{2}\cdot157^{2}\cdot179^{2} & 985\text{NONE} & \\ 986\mathbf{4091} & 203 & 7^{4} & \text{NONE} & \\ 987\mathbf{4093} & 174 & 3^{2}\cdot89^{4} & 2 & 89^{2} \\ 988\mathbf{4099} & 185 & 3^{4}\cdot19^{2} & \text{NONE} & \\ 989\mathbf{4111} & 190 & 229^{2} & \text{NONE} & \\ 990\mathbf{4139} & 188 & 29^{2}\cdot67^{2} & 1 & 67 \\ 991\mathbf{4153} & 177 & 7^{2} & \text{NONE} & \\ 992\mathbf{4157} & 193 & 373^{2} & \text{NONE} & \\ 993\mathbf{4159} & 188 & 997^{2} & \text{NONE} & \\ 994\mathbf{4177} & 183 & 3^{2}\cdot17^{2} & \text{NONE} & \\ 995\mathbf{4217} & 186 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\ 996\mathbf{4219} & 190 & 71^{2} & \text{NONE} & \\ 997\mathbf{4229} & 1 & 3^{2} & \text{NONE} & \\ 998\mathbf{4229} & 194 & 3^{4} & \text{NONE} & \\ 999\mathbf{4231} & 201 & 3^{6} & \text{NONE} & \\ 1000\mathbf{4253} & 184 & 3^{6}\cdot2843^{2} & 3 & 3^{3}\cdot2843 \\ 1001\mathbf{4261} & 185 & 5^{2} & \text{NONE} & \\ 1002\mathbf{4271} & 210 & 163^{2}\cdot853^{2} & \text{NONE} & \\ 1003\mathbf{4273} & 183 & 181^{2} & \text{NONE} & \\ 1004\mathbf{4283} & 198 & 683^{2} & \text{NONE} & \\ 1005\mathbf{4289} & 205 & 8807^{2} & \text{NONE} & \\ 1006\mathbf{4339} & 196 & 17^{2} & \text{NONE} & \\ 1007\mathbf{4349} & 191 & 127^{2} & \text{NONE} & \\ 1008\mathbf{4357} & 187 & 7^{2}\cdot13^{2}\cdot17^{2} & 1 & 7\cdot13 1009\\ 1010\mathbf{4373} & 199 & 3^{12}\cdot29^{2} & \text{NONE} & \\ 1011\mathbf{4391} & 222 & 5^{4}\cdot372037^{2} & \text{NONE} & \\ 1012\mathbf{4409} & 200 & 157^{2} & \text{NONE} & \\ 1013\mathbf{4421} & 206 & 1523^{2} & \text{NONE} & \\ 1014\mathbf{4423} & 200 & 3^{6}\cdot587^{2} & \text{NONE} & \\ 1015\mathbf{4441} & 198 & 59^{2}\cdot101^{2} & \text{NONE} & \\ 1016\mathbf{4451} & 213 & 809^{2} & \text{NONE} & \\ 1017\mathbf{4457} & 199 & 337^{2} & \text{NONE} & \\ 1018\mathbf{4463} & 213 & 8951^{2} & \text{NONE} & \\ 1019\mathbf{4483} & 193 & 19^{2}\cdot61^{2} & 2 & 19\cdot61 \\ 1020\mathbf{4517} & 201 & 181^{2} & \text{NONE} & \\ 1021\mathbf{4519} & 202 & 2503^{2} & \text{NONE} & \\ 1022\mathbf{4547} & 205 & 73^{2} & 1 & 73 \\ 1023\mathbf{4549} & 203 & 19^{2}\cdot53^{2} & \text{NONE} & \\ 1024\mathbf{4591} & 215 & 6317^{2} & \text{NONE} & \\ 1025\end{array} 1026\begin{array}{|lcccc} 1027 N & d(A) & \Shaan & d(B) & cong\\ 1028\mathbf{4597} & 200 & 7^{2}\cdot17^{2} & \text{NONE} & \\ 1029\mathbf{4603} & 198 & 829^{2} & \text{NONE} & \\ 1030\mathbf{4621} & 196 & 13^{2} & \text{NONE} & \\ 1031\mathbf{4639} & 218 & 89^{2} & \text{NONE} & \\ 1032\mathbf{4649} & 215 & 751^{2} & \text{NONE} & \\ 1033\mathbf{4651} & 210 & 13^{4} & \text{NONE} & \\ 1034\mathbf{4673} & 207 & 11^{2}\cdot197^{2} & \text{NONE} & \\ 1035\mathbf{4691} & 216 & 43^{2} & \text{NONE} & \\ 1036\mathbf{4729} & 204 & 673^{2} & \text{NONE} & \\ 1037\mathbf{4733} & 210 & 17^{2} & \text{NONE} & \\ 1038\mathbf{4783} & 210 & 797^{2} & \text{NONE} & \\ 1039\mathbf{4789} & 206 & 13^{4} & \text{NONE} & \\ 1040\mathbf{4799} & 230 & 3^{2}\cdot7^{2}\cdot12203^{2} & 1 & 3\cdot7 1041\\ 1042\mathbf{4801} & 213 & 60271^{2} & \text{NONE} & \\ 1043\mathbf{4813} & 207 & 3^{2}\cdot6883^{2} & \text{NONE} & \\ 1044\mathbf{4817} & 214 & 283^{2} & \text{NONE} & \\ 1045\mathbf{4831} & 217 & 1151^{2} & \text{NONE} & \\ 1046\mathbf{4861} & 216 & 204749^{2} & \text{NONE} & \\ 1047\mathbf{4877} & 219 & 3^{4}\cdot103^{2} & \text{NONE} & \\ 1048\mathbf{4931} & 240 & 17^{2}\cdot37^{2}\cdot43^{2} & 1049\text{NONE} & \\ 1050\mathbf{4933} & 211 & 239^{2} & \text{NONE} & \\ 1051\mathbf{4957} & 212 & 5^{2} & \text{NONE} & \\ 1052\mathbf{4967} & 236 & 7^{2}\cdot53881^{2} & \text{NONE} & \\ 1053\mathbf{4969} & 220 & 11^{4} & \text{NONE} & \\ 1054\mathbf{4973} & 223 & 5^{2}\cdot11^{2} & \text{NONE} & \\ 1055\mathbf{4993} & 215 & 4013^{2} & \text{NONE} & \\ 1056\mathbf{4999} & 224 & 985121^{2} & \text{NONE} & \\ 1057\mathbf{5003} & 220 & 97^{2}\cdot1861^{2} & 3 & 97\cdot1861 \\ 1058\mathbf{5009} & 223 & 23^{2}\cdot977^{2} & \text{NONE} & \\ 1059\mathbf{5011} & 229 & 11^{4} & \text{NONE} & \\ 1060\mathbf{5021} & 225 & 1609^{2} & \text{NONE} & \\ 1061\mathbf{5023} & 221 & 51431^{2} & \text{NONE} & \\ 1062\mathbf{5039} & 251 & 166363^{2} & \text{NONE} & \\ 1063\mathbf{5051} & 239 & 13^{2}\cdot2633^{2} & \text{NONE} & \\ 1064\mathbf{5059} & 229 & 5^{2}\cdot13^{2}\cdot31^{2} & \text{NONE} & \\ 1065\mathbf{5077} & 216 & 283^{2} & \text{NONE} & \\ 1066\mathbf{5081} & 240 & 19^{2}\cdot149^{2} & \text{NONE} & \\ 1067\mathbf{5099} & 251 & 7^{4}\cdot11^{2}\cdot461^{2} & 1068\text{NONE} & \\ 1069\mathbf{5113} & 223 & 19^{2}\cdot61^{2} & \text{NONE} & \\ 1070\mathbf{5119} & 232 & 53^{2}\cdot103^{2} & \text{NONE} & \\ 1071\mathbf{5153} & 223 & 3^{4}\cdot41^{2} & \text{NONE} & \\ 1072\mathbf{5167} & 231 & 367^{2} & \text{NONE} & \\ 1073\mathbf{5171} & 249 & 73^{2}\cdot773^{2} & 1 & 73 \\ 1074\mathbf{5179} & 226 & 7^{2}\cdot13^{2} & \text{NONE} & \\ 1075\mathbf{5189} & 240 & 83^{2} & \text{NONE} & \\ 1076\mathbf{5197} & 223 & 37^{2} & \text{NONE} & \\ 1077\mathbf{5209} & 227 & 181^{2}\cdot1471^{2} & \text{NONE} & \\ 1078\mathbf{5227} & 232 & 3^{2}\cdot7717^{2} & \text{NONE} & \\ 1079\mathbf{5231} & 255 & 4507^{2} & \text{NONE} & \\ 1080\mathbf{5233} & 223 & 163^{2} & \text{NONE} & \\ 1081\mathbf{5237} & 229 & 7^{2} & \text{NONE} & \\ 1082\mathbf{5261} & 239 & 24103^{2} & \text{NONE} & \\ 1083\mathbf{5273} & 227 & 17389^{2} & \text{NONE} & \\ 1084\mathbf{5279} & 263 & 120431^{2} & \text{NONE} & \\ 1085\mathbf{5281} & 232 & 67^{2} & \text{NONE} & \\ 1086\mathbf{5297} & 238 & 397^{2} & \text{NONE} & \\ 1087\mathbf{5303} & 247 & 13^{2}\cdot73^{2}\cdot15467^{2} & 1088\text{NONE} & \\ 1089\mathbf{5309} & 247 & 1822693^{2} & \text{NONE} & \\ 1090\mathbf{5323} & 233 & 3^{4}\cdot120563^{2} & 3 & 120563 \\ 1091\mathbf{5333} & 237 & 967^{2} & \text{NONE} & \\ 1092\mathbf{5347} & 231 & 3643^{2} & \text{NONE} & \\ 1093\mathbf{5501} & 250 & 163^{2} & \text{NONE} & \\ 1094\mathbf{5503} & 241 & 7^{2}\cdot17^{2} & \text{NONE} & \\ 1095\mathbf{5507} & 252 & 103^{2}\cdot233^{2} & \text{NONE} & \\ 1096\mathbf{5519} & 278 & 61^{2}\cdot211469^{2} & \text{NONE} & \\ 1097\mathbf{5521} & 244 & 5^{4} & \text{NONE} & \\ 1098\mathbf{5531} & 253 & 977^{2} & \text{NONE} & \\ 1099\mathbf{5563} & 246 & 3^{4}\cdot1213^{2} & \text{NONE} & \\ 1100\mathbf{5569} & 239 & 3^{4}\cdot5^{2}\cdot13^{2} & \text{NONE} & \\ 1101\mathbf{5573} & 247 & 9901^{2} & \text{NONE} & \\ 1102\mathbf{5581} & 242 & 28927^{2} & \text{NONE} & \\ 1103\mathbf{5591} & 282 & 3^{2}\cdot13^{4}\cdot1061^{2} & 1104\text{NONE} & \\ 1105\mathbf{5639} & 278 & 229717^{2} & \text{NONE} & \\ 1106\mathbf{5641} & 244 & 41^{2}\cdot431^{2} & \text{NONE} & \\ 1107\mathbf{5647} & 245 & 4463^{2} & \text{NONE} & \\ 1108\end{array} 1109$$
1110}
1111
1112\subsection{Mordell-Weil groups of positive even rank and the $\BigSha$ they probably induce}
1113\label{table:mordell-weil}
1114
1115\noindent
1116$$\begin{array}{lclccc} 1117 L(1)=0 & d & 1118 \hspace{-.5em}L(1)\neq 0 & 1119 \hspace{-.6em}d & 1120 \hspace{-.8em}\text{cong} & 1121 \hspace{-.7em} 1122L(1)/\Omega \cdot 2^*\\ 1123 & & & & & \vspace{-2ex}\\ 1124\mbox{\bf 389A} & 1 & \mbox{\bf 389E} & 20 & 5 & 25/97\\ 1125\mbox{\bf 433A} & 1 & \mbox{\bf 433D} & 16 & 7 & 49/9\\ 1126\mbox{\bf 446B} & 1 & \mbox{\bf 446F} & 8 & 11 & 121/3\\ 1127\mbox{\bf 563A} & 1 & \mbox{\bf 563E} & 31 & 13 & 169/281\\ 1128\mbox{\bf 571B} & 1 & \mbox{\bf 571D} & 2 & 3 & 9\\ 1129\mbox{\bf 643A} & 1 & \text{NONE} & & & \\ 1130\mbox{\bf 655A} & 1 & \mbox{\bf 655D} & 13 & 9 & 81\\ 1131\mbox{\bf 664A} & 1 & \mbox{\bf 664F} & 8 & 5 & 25\\ 1132\mbox{\bf 681C} & 1 & \mbox{\bf 681B} & 1 & 3 & 9\\ 1133\mbox{\bf 707A} & 1 & \mbox{\bf 707G} & 15 & 13 & 169\\ 1134\mbox{\bf 718B} & 1 & \mbox{\bf 718F} & 7 & 7 & 49\\ 1135\mbox{\bf 794A} & 1 & \mbox{\bf 794G} & 12 & 11 & 121/3\\ 1136\mbox{\bf 817A} & 1 & \mbox{\bf 817E} & 15 & 7 & 49/5\\ 1137\mbox{\bf 916C} & 1 & \mbox{\bf 916G} & 9 & 11 & 121\\ 1138\mbox{\bf 944E} & 1 & \mbox{\bf 944O} & 6 & 7 & 49\\ 1139\mbox{\bf 997B} & 1 & \mbox{\bf 997H} & 42 & 3 & 81/83\\ 1140\mbox{\bf 997C} & 1 & \mbox{\bf 997H} & 42 & 3 & 81/83\\ 1141\mbox{\bf 1001C} & 1 & \mbox{\bf 1001F} & 3 & 3 & 27\\ 1142\mbox{\bf 1001C} & 1 & \mbox{\bf 1001L} & 7 & 7 & 49\\ 1143\mbox{\bf 1028A} & 1 & \mbox{\bf 1028E} & 14 & 11 & 3267\\ 1144\mbox{\bf 1034A} & 1 & \text{NONE} & & & \\ 1145\mbox{\bf 1041B} & 2 & \mbox{\bf 1041E} & 4 & 5 & 25\\ 1146\mbox{\bf 1041B} & 2 & \mbox{\bf 1041J} & 13 & 25 & 625\\ 1147\mbox{\bf 1058C} & 1 & \mbox{\bf 1058D} & 1 & 5 & 25\\ 1148\mbox{\bf 1061B} & 2 & \mbox{\bf 1061D} & 46 & 151 & 22801/265\\ 1149\mbox{\bf 1070A} & 1 & \mbox{\bf 1070M} & 7 & 15 & 75\\ 1150\mbox{\bf 1073A} & 1 & \text{NONE} & & & \\ 1151\mbox{\bf 1077A} & 1 & \mbox{\bf 1077J} & 15 & 9 & 81\\ 1152\mbox{\bf 1088J} & 1 & \mbox{\bf 1088R} & 2 & 3 & 9\\ 1153\mbox{\bf 1094A} & 1 & \mbox{\bf 1094F} & 13 & 11 & 121/3\\ 1154\mbox{\bf 1102A} & 1 & \mbox{\bf 1102K} & 4 & 3 & 9\\ 1155\mbox{\bf 1126A} & 1 & \mbox{\bf 1126F} & 11 & 11 & 121\\ 1156\mbox{\bf 1132A} & 1 & \mbox{\bf 1132F} & 12 & 5 & 225\\ 1157\mbox{\bf 1137A} & 1 & \mbox{\bf 1137C} & 14 & 9 & 81\\ 1158\mbox{\bf 1141A} & 1 & \mbox{\bf 1141I} & 22 & 7 & 1524537/41\\ 1159\end{array} 1160\begin{array}{lclccc} 1161 & & & & & \\ 1162 & & & & & \vspace{-2ex}\\ 1163\mbox{\bf 1143C} & 1 & \mbox{\bf 1143J} & 9 & 11 & 121\\ 1164\mbox{\bf 1147A} & 1 & \mbox{\bf 1147H} & 23 & 5 & 225/19\\ 1165\mbox{\bf 1171A} & 1 & \mbox{\bf 1171D} & 53 & 11 & 121/195\\ 1166\mbox{\bf 1246C} & 1 & \mbox{\bf 1246B} & 1 & 5 & 25\\ 1167\mbox{\bf 1309B} & 1 & \text{NONE} & & & \\ 1168\mbox{\bf 1324A} & 1 & \mbox{\bf 1324E} & 14 & 9 & 6561\\ 1169\mbox{\bf 1325E} & 1 & \mbox{\bf 1325T} & 11 & 9 & 2187\\ 1170\mbox{\bf 1363B} & 2 & \mbox{\bf 1363F} & 25 & 31 & 961/5\\ 1171\mbox{\bf 1431A} & 1 & \mbox{\bf 1431L} & 14 & 3 & 9\\ 1172\mbox{\bf 1436A} & 1 & \text{NONE} & & & \\ 1173\mbox{\bf 1443C} & 1 & \mbox{\bf 1443G} & 5 & 7 & 49\\ 1174\mbox{\bf 1446A} & 1 & \mbox{\bf 1446N} & 7 & 3 & 9\\ 1175\mbox{\bf 1466B} & 1 & \mbox{\bf 1466H} & 23 & 13 & \hspace{-2em}\mbox{{\tiny 4331806939187/367}}\\ 1176\mbox{\bf 1477A} & 1 & \mbox{\bf 1477C} & 24 & 13 & 169\\ 1177\mbox{\bf 1480A} & 1 & \mbox{\bf 1480G} & 5 & 7 & 49\\ 1178\mbox{\bf 1483A} & 1 & \mbox{\bf 1483D} & 67 & 15 & 225/247\\ 1179\mbox{\bf 1525C} & 1 & \mbox{\bf 1525O} & 16 & 7 & 49\\ 1180\mbox{\bf 1531A} & 1 & \mbox{\bf 1531D} & 73 & 3 & 3/85\\ 1181\mbox{\bf 1534B} & 1 & \mbox{\bf 1534J} & 6 & 3 & 3\\ 1182\mbox{\bf 1567B} & 3 & \mbox{\bf 1567D} & 69 & 287 & 82369/261\\ 1183\mbox{\bf 1570B} & 1 & \mbox{\bf 1570J} & 6 & 11 & 121\\ 1184\mbox{\bf 1576A} & 1 & \mbox{\bf 1576E} & 14 & 11 & 121\\ 1185\mbox{\bf 1591A} & 1 & \mbox{\bf 1591F} & 35 & 31 & 6727/19\\ 1186\mbox{\bf 1594A} & 1 & \mbox{\bf 1594J} & 17 & 3 & 3370648239/19\\ 1187\mbox{\bf 1608A} & 1 & \mbox{\bf 1608J} & 6 & 13 & 169\\ 1188\mbox{\bf 1611D} & 1 & \mbox{\bf 1611O} & 11 & 9 & 81\\ 1189\mbox{\bf 1613A} & 1 & \mbox{\bf 1613D} & 75 & 5 & 25/403\\ 1190\mbox{\bf 1615A} & 1 & \mbox{\bf 1615J} & 13 & 9 & 102141\\ 1191\mbox{\bf 1621A} & 1 & \mbox{\bf 1621C} & 70 & 17 & 289/135\\ 1192\mbox{\bf 1627A} & 1 & \mbox{\bf 1627C} & 73 & 9 & 81/271\\ 1193\mbox{\bf 1633A} & 3 & \mbox{\bf 1633D} & 27 & 189 & 35721\\ 1194\mbox{\bf 1639B} & 1 & \mbox{\bf 1639G} & 34 & 17 & 680017/25\\ 1195\mbox{\bf 1641B} & 1 & \mbox{\bf 1641J} & 24 & 23 & 529\\ 1196\mbox{\bf 1642A} & 1 & \mbox{\bf 1642D} & 14 & 7 & 49\\ 1197 & & & & & \\ 1198\end{array}$$
1199
1200\mbox{}\par\noindent
1201{{\bf 643A}, {\bf 1034A}, {\bf 1073A}, {\bf 1309B} all have modular degree
1202a power of~$2$; {\bf 1436A} has modular degree divisible by~$3$.
1203
1204\newpage
1205\subsection{Conjecturally visible $\BigSha$ of modular motives of weight~$4$}
1206\label{table:motive}
1207Suppose $f$ and $g$ are elements of $S_4(\Gamma_0(N))$ such that $p^2 \mid L(\sM_f,2)/\Omega$
1208and $L(\sM_g,2)=0$.   If~$f$ and~$g$ satisfy a $p$-congruence'',
1209does~$p$ then divide the visible part'' of $\Sha(\sM_f(2))$?
1210
1211$$1212\begin{array}{lcclcl} 1213%\hspace{2em}\chi\hspace{2em}\mbox{} & 1214 \sM_f \hspace{1em}\mbox{}& 1215 \text{dim} 1216 & 1217 \hspace{2em}p^2\hspace{2em}\mbox{} 1218 & \sM_g & \text{dim}\\ 1219 & & & & & \vspace{-2ex} \\ 1220 1221%(1,1) & \mbox{\bf 99C} & 8 & 19^2 & \mbox{\bf 99B} & 2 \\ 1222 \mbox{\bf 127k4C} & 17 & 43^2 & \mbox{\bf 127k4A} & 1\\ 1223 \mbox{\bf 159k4E} & 8 & 23^2 & \mbox{\bf 159k4B} & 1\\ 1224%(0,0,1) & \mbox{\bf 200E} & 8 & 7^2 & \mbox{\bf 200G} & 2\\ 1225 \mbox{\bf 365k4E} & 18 & 29^2 & \mbox{\bf 365k4A} & 1\\ 1226 \mbox{\bf 369k4I} & 9 & 13^2 & \mbox{\bf 369k4A} & 1\\ 1227 \mbox{\bf 453k4E} & 23 & 17^2 & \mbox{\bf 453k4A} & 1\\ 1228 \mbox{\bf 465k4H} & 7 & 11^2 & \mbox{\bf 465k4A} & 1\\ 1229 \mbox{\bf 477k4L} & 12 & 73^2 & \mbox{\bf 477k4A} & 1\\ 1230 \mbox{\bf 567k4G} & 8 & 13^2, 23^2 & \mbox{\bf 567k4A} & 1\\ 1231 \mbox{\bf 581k4E} & 34 & 19^2 & \mbox{\bf 581k4A} & 1 1232\end{array} 1233$$
1234%Here~$\chi$ is the common nebentypus character of~$f$ and~$g$.
1235
1236\section{Questions and conjectures}\label{sec:conj}
1237The following questions and conjectures were motivated by the
1238tables above and the computations that went into creating them.
1239The first conjecture suggests a generalization of a result of
1240Ribet on level raising.  The second conjecture asserts that
1241$\Sha$ is always visible in an appropriate modular Jacobian.
1242The third question suggests a new approach to the long-standing
1243open problem of constructing points on abelian varieties of
1244analytic rank greater than~$1$ over the Hecke algebra.
1245
1246\subsection{Level raising nonvanishing conjecture}
1247Let $f\in S_2(\Gamma_0(N))$ be a newform such that the sign
1248of the functional equation of $L(A_f,s)$ is equal to $+1$,
1249and fix a prime~$\lambda$ such
1250that the associated Galois representation $\rho_{f,\lambda}=A_f[\lambda]$
1251is irreducible.
1252For each prime~$q$ not dividing~$N$, let $\delta: J_0(N)\ra J_0(Nq)$ be
1253the injection obtained from the sum of the two degeneracy maps.
1254Ribet's construction in \cite{ribet:raising}
1255produces infinitely many primes~$q$ and newforms
1256$g\in S_2(\Gamma_0(qN))$ such that
1257 $$\delta(\Adual_f[\lambda]) \subset \delta(\Adual_f)\intersect \Adual_g$$
1258and the Tamagawa number $c_q$ of $\Adual_g$ is a power of~$2$.
1259\begin{conjecture}
1260Fix~$f$ and~$\lambda$.
1261\begin{enumerate}
1262\item Then there is a~$g$ among those constructed by Ribet
1263such that $L(A_g,1)\neq 0$.
1264\item If~$\lambda$ is in the support of the $\T$-module
1265$[P(H_1(X_0(N),\Z))^+: P(\T\{0,\infty\})]$
1266(see Theorem~\ref{thm:ratpart}),
1267then there is a~$g$ as above such that $L(A_g,1)=0$.
1268\end{enumerate}
1269\end{conjecture}
1270
1271\subsection{Eventual visibility conjecture}
1272Let~$S$ be the set of all square-free positive integers.
1273If $M, N\in S$ with $M\mid N$ then there is
1274a natural injection $J_0(M)\hookrightarrow J_0(N)$, and hence a map
1275$\Sha(J_0(M))\ra \Sha(J_0(N))$.
1276These maps are compatible, so the collection of groups $\Sha(J_0(N))$, with
1277$N\in S$, forms a directed system.  Let
1278$\lim_{N\in S} \Sha(J_0(N))$
1279be the direct limit of the $\Sha(J_0(N))$.
1280\begin{conjecture}
1281$\lim_{N\in S} \Sha(J_0(N))=0$
1282\end{conjecture}
1283If true, this would imply that if $A\subset J_0(N)$, then
1284each element of $\Sha(A)$ is visible in some $J_0(N')$,
1285for some multiple~$N'$ of~$N$.
1286
1287\subsection{Euler systems}
1288\label{sec:constructing}
1289In \cite{kolyvagin:structureofsha} and \cite{mccallum:kolyvagin}
1290one finds a construction using the Heegner point Euler system of
1291Kolyvagin of the Shafarevich-Tate groups of certain abelian varieties.
1292Under an unverified hypothesis on Heegner points, the construction
1293gives much of $\Sha(A_f/K)$, where~$K$ is a suitable imaginary
1294quadratic field.  Is it possible to verify the unverified hypothesis,
1295construct $\Vis_J(\Sha(A_f/K))$, and thus prove that $\Sha(\Adual_f/K)$
1296contains visible elements, when the BSD conjecture suggests that it should?  If so,
1297it would follow that there is a congruent $B_f^{\vee}$ having positive
1298algebraic rank, as predicted by the BSD conjecture.  Thus a construction of
1299visible elements of $\Sha(\Adual_f)$ also leads to a construction of points
1300on abelian varieties of positive analytic rank.
1301
1302\bibliography{biblio}
1303
1304\end{document}
1305
1306