\documentclass[11pt]{article} \hoffset=-0.1\textwidth \voffset=-0.1\textheight \textheight=1.2\textheight \textwidth=1.2\textwidth \include{macros} \usepackage{amsmath} \usepackage[all]{xy} \DeclareMathOperator{\moddeg}{moddeg} \DeclareMathOperator{\op}{odd} \newcommand{\first}[1]{\hline#1} \newcommand{\add}[1]{\\} \newcommand{\sha}[3]{{\bf #1}, dim #2, $\ell=#3$} \newcommand{\higher}[3]{$\mathbf{p=#1}$: dim #2, rank #3} %opening \title{Visibility of Shafarevich-Tate Groups at Higher Level} \author{William Stein\\ {\tt http://modular.fas.harvard.edu}} \date{April 19, 2004} \begin{document} \maketitle \begin{abstract} I will begin by introducing the Birch and Swinnerton-Dyer conjecture in the context of abelian varieties attached to modular forms, and discuss some of the main results about it. I will then introduce Mazur's notion of visibility of Shafarevich-Tate groups and explain some of the basic facts and theorems. Cremona, Mazur, Agashe, and myself carried out large computations about visibility for modular abelian varieties of level $N$ in $J_0(N)$. These computations addressed the following question: If $A$ is a modular abelian variety of level $N$, how much of the Shafarevich-Tate group $\Sha(A)$ is modular of level $N$, i.e., visible in $J_0(N)$. The results of these computations suggest that often much of the Shafarevich-Tate group is not modular of level $N$. It is then natural to ask if every element of $\Sha(A)$ is modular of level~$M$, for some multiple $M=NR$, and if so, what can one say about the set of such $M$? I will finish the talk with some new data and a conjecture about this last question, which is still very much open. \end{abstract} \section{Modular Abelian Varieties} Let $N$ be a positive integer and consider the congruence subgroup $$ \Gamma_0(N) = \left\lbrace \mtwo{a}{b}{c}{d} \in \SL_2(\Z) \text{ such that } N \mid c \right\rbrace. $$ (Almost everything in this talk also makes sense with $\Gamma_0(N)$ replaced by $\Gamma_1(N)$.) The \textit{modular curve} $$ X_0(N) = \Gamma_0(N) \setminus \left(\left\lbrace z\in \C : \Im(z) > 0\right\rbrace \cup \Q \cup \left\lbrace \infty\right\rbrace \right) $$ is a Riemann surface that is the set of complex points of an algebraic curve over $\Q$. We will not use that $$ X_0(N)(\C) = \left\lbrace \text{ isomorphism classes of }(E,C)\,\, \right\rbrace \cup \left\lbrace \text{ cusps } \right\rbrace. $$ Our primary interest is the Jacobian $$ J_0(N) = \Jac(X_0(N)) $$ which is an abelian variety over $\Q$ of dimension equal to the genus of $X_0(N)$. The points on the Jacobian parametrize, in a natural way, the divisor classes of degree $0$ on $X_0(N)$. Let $ S_2(\Gamma_0(N)) $ be the cusp forms of weight $2$ for $\Gamma_0(N)$. This is the finite-dimensional complex vector space of holomorphic functions on the upper half plane such that $$f(z)dz = f(\gamma(z))d(\gamma(z))$$ for all $\gamma\in\Gamma_0(N)$, and which ``vanish at the cusps''. The map $f(z) \mapsto f(z)dz$ induces $$ S_2(\Gamma_0(N))\isom \H^0(X_0(N)_\C,\Omega^1) $$ so $S_2(\Gamma_0(N))$ has dimension the genus of $X_0(N)$. The \textit{Hecke algebra} is a commutative ring $$ \T = \Z[T_1,T_2,T_3, \ldots] $$ which acts on $S_2(\Gamma_0(N))$ and $J_0(N)$. A \textit{newform} $$ f = \sum_{n=1}^{\infty} a_n q^n \in S_2(\Gamma_0(N)) $$ is an eigenvector for every element of $\T$ normalized so $a_1 = 1$, which does not ``come from'' any lower level. Attached to $f$ there is an ideal $$ I_f = \Ann_{\T}(f) = \Ker(\T \to \Z[a_1,a_2,\ldots]), $$ and (following Shimura) to this ideal we attach an abelian variety $A_f$ and an $L$-function $L(A_f,s)$. Let $$ A_f = J_0(N)[I_f]^0 = \left( \bigcap_{\vphi \in I_f} \Ker(\vphi) \right)^0 $$ be the connected component of the intersections of the kernels of elements of $I_f$. Then $A_f$ has dimension $[K_f:\Q] = [\Q(a_1,a_2,\ldots):\Q)]$, and is define over $\Q$. Let $$ L(A_f,s) = \prod_{i=1}^d L(f_i,s) $$ where $d=[K_f:\Q]$ and the $f_i$ are the Galois conjugates of $f$. Also, $$ L(f,s) = \sum_{n=1}^\infty \dfrac{a_n}{n^s}. $$ Hecke proved that $L(f,s)$ is entire and satisfies a functional equation. The abelian varieties $A_f$ are a rich class of abelian varieties. The elliptic curves over~$\Q$ are all isogenous to some $A_f$ (the Wiles-Breuil-Conrad-Diamond-Taylor modularity theorem). \section{The Birch and Swinnerton-Dyer Conjecture} \subsection{Conjecture} \begin{conjecture}[Birch and Swinnerton-Dyer]\mbox{}\vspace{-4ex}\\ \begin{enumerate} \item $\rank A_f(\Q) = \ord_{s=1}L(A_f,s)$ \item $\ds \frac{L^{(r)}(A_f,1)}{r!} = \frac{\prod c_p \cdot \Omega_{A_f} \cdot \Reg_{A_f} \cdot \#\Sha(A_f)} {\#A_f(\Q)_{\tor}\cdot \#A_f^{\vee}(\Q)_{\tor}}. $ \end{enumerate} \end{conjecture} Remarks: Part of the conjecture is that $\Sha(A_f)$ is finite. There is also a conjecture for arbitrary abelian varieties over global fields. Clay Math Problem: \$1000000 prize for proof of (1) in case $\dim(A_f)=1$ Here: \begin{itemize} \item $c_p$ is the \textit{Tamagawa number} at the prime $p$, and the product is over the prime divisors of $N$. \item $\Omega_{A_f}$ is the canonical N\'eron measure of $A_f(\R)$. \item $\Reg_{A_f}$ is the regulator (absolute value of N\'eron-Tate canonical height pairing matrix). \item $A_f(\Q)_{\tor}$ is the torsion subgroup of $A_f(\Q)$. \item $\Sha(A_f)$ is the Shafarevich-Tate group. \end{itemize} \subsection{Evidence} \begin{itemize} \item Rubin: results in CM Case \item Kolyvagin, Logachev, Gross-Zagier, et al.: If $\ord_{s=1}L(f,s)=0$ or $1$, then (1) true and $\Sha(A_f)$ finite. \item Cremona: Compute $\Sha(A_f)_{?}$ (=conjectural order) for tens of thousands of $A_f$ of dimension $1$ and get approximate square order. (Theorem of Cassels: if $E$ an elliptic curve and $\Sha(E)$ finite then order a perfect square. Note that the analogue for abelian varieties is false; for exampe, I've constructed examples for each odd prime $p<25000$ of abelian varieties $A$ of dimension $p-1$ such that $\Sha(A) = p\cdot n^2$.) \end{itemize} In this talk I will focus on $A_f$ of possibly large dimension with $L(A_f,1)\neq 0$, since computation of $\Reg_{A_f}$ is difficult (impossible?) when one can't even reasonably hope to write down $A_f$ explicitly with equations. \section{Visibility of Shafarevich-Tate Groups} \subsection{Definitions} It 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. The idea of visibility of Shafarevich-Tate groups was introduced by Barry Mazur around 1998 to unify various constructions of elements of Shafarevich-Tate groups. \begin{definition}[Shafarevich-Tate Group]\label{defn:sha} $$\ds \Sha(A) = \Ker\left(\H^1(K,A) \to \bigoplus_v \H^1(K_v,A)\right). $$ \end{definition} Here $\H^1(K,A)$ is the first Galois cohomology, which can be interpreted geometrically as the Weil-Chatalet group $$ \text{WC}(A/K) = \{\text{ principal homogenous spaces }X\text{ for }A\,\}/\sim. $$ Then $\Sha(A)$ is the subgroup of locally trivial classes of homogenous spaces. For example $$3x^3+4y^3+5z^3=0 \in \Sha(x^3+y^3+60z^3=0)[3].$$ Fix an inclusion $i:A\hra B$ of abelian varieties and let $\pi: B\to C$ be the quotient of~$B$ by the image of $A$. \begin{definition}[Visible Subgroup] \begin{align*} \Vis_i(\H^1(K,A)) &= \Ker\left(\H^1(K,A) \to \H^1(K,B)\right)\\ &=\Coker(B(K) \to C(K)) \end{align*} and $$\Vis_i(\Sha(A)) &= \Ker(\Sha(A)\to \Sha(B)).$$ \end{definition} \begin{enumerate} \item The visible subgroup is finite because $B(K)$ is finitely generated and $\Vis_B(\H^1(K,A))$ is torsion. \item If $c\in\Vis_B(\H^1(K,A))$, then $c$ is also ``visible'' in the sense that if $c$ is the image of a point $x\in C(K)$, and if $X=\pi^{-1}(x)\subset B$, then $[X]\in\text{WC}(A)$ corresponds to $c$. \item The visibile subgroups depends on the choice of embedding $i:A\hra B$. \end{enumerate} \subsection{Theorems} ``Everything is visible somewhere.'' \begin{theorem}[Stein] If $c\in \H^1(K,A)$ then there exists $B=\Res_{L/K}(A_L)$ such that $A\hra B$ and $c\in\Vis_B(\H^1(K,A))$. (Here $L$ is such that $\res_{L/K}(c)=0$.) \end{theorem} \noindent``Visibility construction.'' \begin{theorem}[Agashe-Stein]\label{vis:const} Suppose $A,B\subset C$ over $\Q$, that $A+B=C$, that $A\cap B$ is finite. Suppose $N$ is divisible by all bad primes for $C$, and $p$ is a prime such that \begin{itemize} \item $B[p]\subset A$ \item $\ds p\nmid 2\cdot N\cdot \#B(\Q)_{\tor}\cdot \#(C/B)(\Q)_{\tor}\cdot \prod_{p\mid N} c_{A,p} \cdot c_{B,p}.$ \end{itemize} If $A$ has rank $0$, then there is a natural inclusion $$ B(\Q)/p B(\Q) \hra \Vis_C(\Sha(A)). $$ (And certain generalizations...) \end{theorem} \subsection{Example} \begin{example} For $N=389$, take $B$ the (first ever) rank $2$ elliptic curve, and $A$ the $20$-dimensional rank $0$ factor. $$ \xymatrix{ & E\ar[d] \\ A \ar[r] & {J_0(389)} } $$ Gives $$ (\Z/5\Z)^2 \isom E(\Q)/5 E(\Q) \hra \Sha(A). $$ Part 2 of the Birch and Swinnerton-Dyer conjecture predicts that $$ \Sha(A) = 5^2 \cdot 2^{?}, $$ so this gives evidence. \end{example} \section{Visibility in Modular Jacobians} Suppose now $A=A_f\subset J_0(N)$ is attached to a newform. \begin{definition}[Modular of level $M$] An element $c\in\Sha(A)[p]$ is \textit{modular of level}~$M$ if $c \in \Vis_{M}(\Sha(A))$, where $\Vis_{M}(\Sha(A))$ is the subgroup generated by all kernels of maps $\Sha(A)\to \Sha(J_0(M))$ induced by homomorphisms $A\to J_0(M)$ of degree coprime to $p$. \end{definition} Note that $M$ must be a multiple of $N$. \begin{question}[Mazur] Suppose $E\subset J_0(N)$ is an elliptic curve of conductor $N$. How much of $\Sha(E)$ is modular of level $N$? \end{question} Answer: In examples, surprisingly much. Expect not all, since $$ \Vis_{N}(\Sha(E)) \subset \Sha(E)[\text{modular degree}], $$ and modular degree annihilates symmetric square Selmer group (work of Flach). \subsection{Data and Experiments} \begin{itemize} \item {\bf Cremona-Mazur:} There are $52$ elliptic curves $E\subset J_0(N)$ with $N<5500$ such that $p\mid \#\Sha(E)_?$. Cremona-Mazur show that for $43$ of these that $\Sha(E)$ ``probably'' is modular of level $N$, and for $3$ that it is definitely not: $N=2849, 4343, 5389$. (``Probably'' was made ``provably'' in many cases in subsequent work.) \item {\bf Agashe-Stein:} Same question as Cremona-Mazur for $A_f\subset J_0(N)$ of any dimension. Using results of my Ph.D. thesis, MAGMA packages, etc. I computed a divisor and multiple of $\#\Sha(A_f)_?$ for the following: \begin{itemize} \item $10360$ abelian varieties $A_f\subset J_0(N)$ with $L(A_f,1)\neq 0$. \item Found $168$ with $\#\Sha(A_f)_?$ definitely divisible by an odd prime. \item For $39$ of these, prove that all $\#\Sha(A_f)_?^{\text{odd}}$ elements are modular of level $N$, and $106$ probably are. This gives strong evidence for the BSD conjecture, and a sense that maybe something further is going on. \item Of these $168$, at least $62$ have odd conjectural $\Sha$ that is definitely {\em not} modular of level $N$. Big mystery? Where is this $\Sha$ modular? Is it modular at all? Is it even there?? (Perhaps a good place to look for counterexample to BSD.) \end{itemize} \end{itemize} \section{Visibility at Higher Level} \begin{definition} Let $c\in\Sha(A_f)$. The {\em modularity levels} of $c$ are the set of integers $$\mathcal{N}(c) = \{M: c\in\Vis_{M}(\Sha(A_f))\}.$$ \end{definition} \begin{conjecture}[Stein] For any $c\in\Sha(A_f)$ we have $$ \mathcal{N}(c) \neq \emptyset, $$ i.e., every element of $\Sha(A_f)$ is modular. \end{conjecture} Motivation: This is a working hypothesis that makes \textit{computing} with modular abelian varieties easier. Also, if there were a common level at which all of $\Sha(A_f)$ were modular, then $\Sha(A_f)$ would be finite, and conversely (assuming the conjecture). \subsection{Ribet Level Raising} Suppose that $f=\sum a_n q^n\in S_2(\Gamma_0(N))$ is a newform and $\p$ is a nonzero prime ideal of $\Z[a_1,a_2,\ldots]$ such that $A_f[\p]$ is irreducible. If $$ a_\ell + \ell + 1 \con 0\pmod{\p} $$ then there exists an $\ell$-newform $g\in S_2(\Gamma_0(N\ell))$ such that $i(A_f[\p]) = A_g[\p]$ for an appropriate $i:J_0(N)\to J_0(N\ell)$ of degree coprime to $\text{char}(\p)$ and the sign of the functional equations for $L(f,s)$ and $L(g,s)$ are the same. If we instead require that $a_\ell - (\ell+1)\con 0\pmod{\p}$ then there is such a $g$, but the sign of the functional equation changes, and the new Tamagawa numbers of $A_g$ at $\ell$ will (or tends to be?) divisible by $\p$. \subsection{Evidence for Conjecture} I defined a precise notion of ``probably modular'' motivated by Theorem~\ref{vis:const} and what I can compute. In many cases I 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 to conjecture. Mazur proved that everything in $\Sha(E)[3]$, for $E$ an elliptic curve, is visible in an abelian surface, which, together with the modularity theorem, {\em might} imply modularity of $\Sha(E)[3]$ at higher level. Same for $2$, proved by me and by a different method by Thomas Klenke. \section{Some Tables} The first two pages of tables below give some of the data that I computed about visibility of Shafarevich-Tate groups at level $N$. The third table gives the new data about visibility at higher level. \newpage \begin{center} {\bf \Large Nontrivial Odd Parts of Shafarevich-Tate Groups} $\begin{array}{|l@{}ccc@{}c|l@{}c@{}|cc|}\hline \quad A& \dim& S_l & S_u & \moddeg(A)^{\op} & \quad B & \dim\, & \,\,A^{\vee}\intersect \tilde{B}^{\vee} & \Vis\\ \hline \mathbf{389E}*&20&5^{2}&=&5&\mathbf{389A}&1&[20^{2}]&5^{2} \\ \mathbf{433D}*&16&7^{2}&=&7\!\cdot\!\mbox{\tiny $111$}&\mathbf{433A}&1&[14^{2}]&7^{2} \\ \mathbf{446F}*&8&11^{2}&=&11\!\cdot\!\mbox{\tiny $359353$}&\mathbf{446B}&1&[11^{2}]&11^{2} \\ \mathbf{551H}&18&3^{2}&=&\mbox{\tiny $169$}&\text{NONE} & & & \\ \hline \mathbf{563E}*&31&13^{2}&=&13&\mathbf{563A}&1&[26^{2}]&13^{2} \\ \mathbf{571D}*&2&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $127$}&\mathbf{571B}&1&[3^{2}]&3^{2} \\ \mathbf{655D}*&13&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $9799079$}&\mathbf{655A}&1&[36^{2}]&3^{4} \\ \mathbf{681B}&1&3^{2}&=&3\!\cdot\!\mbox{\tiny $125$}&\mathbf{681C}&1&[3^{2}]&- \\ \hline \mathbf{707G}*&15&13^{2}&=&13\!\cdot\!\mbox{\tiny $800077$}&\mathbf{707A}&1&[13^{2}]&13^{2} \\ \mathbf{709C}*&30&11^{2}&=&11&\mathbf{709A}&1&[22^{2}]&11^{2} \\ \mathbf{718F}*&7&7^{2}&=&7\!\cdot\!\mbox{\tiny $5371523$}&\mathbf{718B}&1&[7^{2}]&7^{2} \\ \mathbf{767F}&23&3^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \hline \mathbf{794G}*&12&11^{2}&=&11\!\cdot\!\mbox{\tiny $34986189$}&\mathbf{794A}&1&[11^{2}]&- \\ \mathbf{817E}*&15&7^{2}&=&7\!\cdot\!\mbox{\tiny $79$}&\mathbf{817A}&1&[7^{2}]&- \\ \mathbf{959D}&24&3^{2}&=&\mbox{\tiny $583673$}&\text{NONE} & & & \\ \mathbf{997H}*&42&3^{4}&=&3^{2}&\mathbf{997B}&1&[12^{2}]&3^{2} \\ \hline &&& && \mathbf{997C}&1&[24^{2}]&3^{2} \\ \mathbf{1001F}&3&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $1269$}&\mathbf{1001C}&1&[3^{2}]&- \\ &&& && \mathbf{91A}&1&[3^{2}]&- \\ \mathbf{1001L}&7&7^{2}&=&7\!\cdot\!\mbox{\tiny $2029789$}&\mathbf{1001C}&1&[7^{2}]&- \\ \hline \mathbf{1041E}&4&5^{2}&=&5^{2}\!\cdot\!\mbox{\tiny $13589$}&\mathbf{1041B}&2&[5^{2}]&- \\ \mathbf{1041J}&13&5^{4}&=&5^{3}\!\cdot\!\mbox{\tiny $21120929983$}&\mathbf{1041B}&2&[5^{4}]&- \\ \mathbf{1058D}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $483$}&\mathbf{1058C}&1&[5^{2}]&- \\ \mathbf{1061D}&46&151^{2}&=&151\!\cdot\!\mbox{\tiny $10919$}&\mathbf{1061B}&2&[2^{2}302^{2}]&- \\ \hline \mathbf{1070M}&7&3 \!\cdot\! 5^{2}&3^{2} \!\cdot\! 5^{2}&3 \!\cdot\! 5\!\cdot\!\mbox{\tiny $1720261$}&\mathbf{1070A}&1&[15^{2}]&- \\ \mathbf{1077J}&15&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $1227767047943$}&\mathbf{1077A}&1&[9^{2}]&- \\ \mathbf{1091C}&62&7^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{1094F}*&13&11^{2}&=&11^{2}\!\cdot\!\mbox{\tiny $172446773$}&\mathbf{1094A}&1&[11^{2}]&11^{2} \\ \hline \mathbf{1102K}&4&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $31009$}&\mathbf{1102A}&1&[3^{2}]&- \\ \mathbf{1126F}*&11&11^{2}&=&11\!\cdot\!\mbox{\tiny $13990352759$}&\mathbf{1126A}&1&[11^{2}]&11^{2} \\ \mathbf{1137C}&14&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $64082807$}&\mathbf{1137A}&1&[9^{2}]&- \\ \mathbf{1141I}&22&7^{2}&=&7\!\cdot\!\mbox{\tiny $528921$}&\mathbf{1141A}&1&[14^{2}]&- \\ \hline \mathbf{1147H}&23&5^{2}&=&5\!\cdot\!\mbox{\tiny $729$}&\mathbf{1147A}&1&[10^{2}]&- \\ \mathbf{1171D}*&53&11^{2}&=&11\!\cdot\!\mbox{\tiny $81$}&\mathbf{1171A}&1&[44^{2}]&11^{2} \\ \mathbf{1246B}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $81$}&\mathbf{1246C}&1&[5^{2}]&- \\ \mathbf{1247D}&32&3^{2}&=&3^{2}\!\cdot\!\mbox{\tiny $2399$}&\mathbf{43A}&1&[36^{2}]&- \\ \hline \mathbf{1283C}&62&5^{2}&=&5\!\cdot\!\mbox{\tiny $2419$}&\text{NONE} & & & \\ \mathbf{1337E}&33&3^{2}&=&\mbox{\tiny $71$}&\text{NONE} & & & \\ \mathbf{1339G}&30&3^{2}&=&\mbox{\tiny $5776049$}&\text{NONE} & & & \\ \mathbf{1355E}&28&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $2224523985405$}&\text{NONE} & & & \\ \hline \mathbf{1363F}&25&31^{2}&=&31\!\cdot\!\mbox{\tiny $34889$}&\mathbf{1363B}&2&[2^{2}62^{2}]&- \\ \mathbf{1429B}&64&5^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{1443G}&5&7^{2}&=&7^{2}\!\cdot\!\mbox{\tiny $18525$}&\mathbf{1443C}&1&[7^{1}14^{1}]&- \\ \mathbf{1446N}&7&3^{2}&=&3\!\cdot\!\mbox{\tiny $17459029$}&\mathbf{1446A}&1&[12^{2}]&- \\ \hline \end{array}$ \end{center} \newpage \begin{center} {\bf \Large Nontrivial Odd Parts of Shafarevich-Tate Groups} $\begin{array}{|l@{}ccc@{}c|l@{}c@{}|cc|}\hline \quad A& \dim& S_l & S_u & \moddeg(A)^{\op} & \quad B & \dim\, & \,\,A^{\vee}\intersect \tilde{B}^{\vee} & \Vis\\ \hline \mathbf{1466H}*&23&13^{2}&=&13\!\cdot\!\mbox{\tiny $25631993723$}&\mathbf{1466B}&1&[26^{2}]&13^{2} \\ \mathbf{1477C}*&24&13^{2}&=&13\!\cdot\!\mbox{\tiny $57037637$}&\mathbf{1477A}&1&[13^{2}]&13^{2} \\ \mathbf{1481C}&71&13^{2}&=&\mbox{\tiny $70825$}&\text{NONE} & & & \\ \mathbf{1483D}*&67&3^{2} \!\cdot\! 5^{2}&=&3 \!\cdot\! 5&\mathbf{1483A}&1&[60^{2}]&3^{2} \!\cdot\! 5^{2} \\ \hline \mathbf{1513F}&31&3&3^{4}&3\!\cdot\!\mbox{\tiny $759709$}&\text{NONE} & & & \\ \mathbf{1529D}&36&5^{2}&=&\mbox{\tiny $535641763$}&\text{NONE} & & & \\ \mathbf{1531D}&73&3&3^{2}&3&\mathbf{1531A}&1&[48^{2}]&- \\ \mathbf{1534J}&6&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $635931$}&\mathbf{1534B}&1&[6^{2}]&- \\ \hline \mathbf{1551G}&13&3^{2}&=&3\!\cdot\!\mbox{\tiny $110659885$}&\mathbf{141A}&1&[15^{2}]&- \\ \mathbf{1559B}&90&11^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{1567D}&69&7^{2} \!\cdot\! 41^{2}&=&7 \!\cdot\! 41&\mathbf{1567B}&3&[4^{4}1148^{2}]&- \\ \mathbf{1570J}*&6&11^{2}&=&11\!\cdot\!\mbox{\tiny $228651397$}&\mathbf{1570B}&1&[11^{2}]&11^{2} \\ \hline \mathbf{1577E}&36&3&3^{2}&3^{2}\!\cdot\!\mbox{\tiny $15$}&\mathbf{83A}&1&[6^{2}]&- \\ \mathbf{1589D}&35&3^{2}&=&\mbox{\tiny $6005292627343$}&\text{NONE} & & & \\ \mathbf{1591F}*&35&31^{2}&=&31\!\cdot\!\mbox{\tiny $2401$}&\mathbf{1591A}&1&[31^{2}]&31^{2} \\ \mathbf{1594J}&17&3^{2}&=&3\!\cdot\!\mbox{\tiny $259338050025131$}&\mathbf{1594A}&1&[12^{2}]&- \\ \hline \mathbf{1613D}*&75&5^{2}&=&5\!\cdot\!\mbox{\tiny $19$}&\mathbf{1613A}&1&[20^{2}]&5^{2} \\ \mathbf{1615J}&13&3^{4}&=&3^{2}\!\cdot\!\mbox{\tiny $13317421$}&\mathbf{1615A}&1&[9^{1}18^{1}]&- \\ \mathbf{1621C}*&70&17^{2}&=&17&\mathbf{1621A}&1&[34^{2}]&17^{2} \\ \mathbf{1627C}*&73&3^{4}&=&3^{2}&\mathbf{1627A}&1&[36^{2}]&3^{4} \\ \hline \mathbf{1631C}&37&5^{2}&=&\mbox{\tiny $6354841131$}&\text{NONE} & & & \\ \mathbf{1633D}&27&3^{6} \!\cdot\! 7^{2}&=&3^{5} \!\cdot\! 7\!\cdot\!\mbox{\tiny $31375$}&\mathbf{1633A}&3&[6^{4}42^{2}]&- \\ \mathbf{1634K}&12&3^{2}&=&3\!\cdot\!\mbox{\tiny $3311565989$}&\mathbf{817A}&1&[3^{2}]&- \\ \mathbf{1639G}*&34&17^{2}&=&17\!\cdot\!\mbox{\tiny $82355$}&\mathbf{1639B}&1&[34^{2}]&17^{2} \\ \hline \mathbf{1641J}*&24&23^{2}&=&23\!\cdot\!\mbox{\tiny $1491344147471$}&\mathbf{1641B}&1&[23^{2}]&23^{2} \\ \mathbf{1642D}*&14&7^{2}&=&7\!\cdot\!\mbox{\tiny $123398360851$}&\mathbf{1642A}&1&[7^{2}]&7^{2} \\ \mathbf{1662K}&7&11^{2}&=&11\!\cdot\!\mbox{\tiny $16610917393$}&\mathbf{1662A}&1&[11^{2}]&- \\ \mathbf{1664K}&1&5^{2}&=&5\!\cdot\!\mbox{\tiny $7$}&\mathbf{1664N}&1&[5^{2}]&- \\ \hline \mathbf{1679C}&45&11^{2}&=&\mbox{\tiny $6489$}&\text{NONE} & & & \\ \mathbf{1689E}&28&3^{2}&=&3\!\cdot\!\mbox{\tiny $172707180029157365$}&\mathbf{563A}&1&[3^{2}]&- \\ \mathbf{1693C}&72&1301^{2}&=&1301&\mathbf{1693A}&3&[2^{4}2602^{2}]&- \\ \mathbf{1717H}*&34&13^{2}&=&13\!\cdot\!\mbox{\tiny $345$}&\mathbf{1717B}&1&[26^{2}]&13^{2} \\ \hline \mathbf{1727E}&39&3^{2}&=&\mbox{\tiny $118242943$}&\text{NONE} & & & \\ \mathbf{1739F}&43&659^{2}&=&659\!\cdot\!\mbox{\tiny $151291281$}&\mathbf{1739C}&2&[2^{2}1318^{2}]&- \\ \mathbf{1745K}&33&5^{2}&=&5\!\cdot\!\mbox{\tiny $1971380677489$}&\mathbf{1745D}&1&[20^{2}]&- \\ \mathbf{1751C}&45&5^{2}&=&5\!\cdot\!\mbox{\tiny $707$}&\mathbf{103A}&2&[505^{2}]&- \\ \hline \mathbf{1781D}&44&3^{2}&=&\mbox{\tiny $61541$}&\text{NONE} & & & \\ \mathbf{1793G}*&36&23^{2}&=&23\!\cdot\!\mbox{\tiny $8846589$}&\mathbf{1793B}&1&[23^{2}]&23^{2} \\ \mathbf{1799D}&44&5^{2}&=&\mbox{\tiny $201449$}&\text{NONE} & & & \\ \mathbf{1811D}&98&31^{2}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \hline \mathbf{1829E}&44&13^{2}&=&\mbox{\tiny $3595$}&\text{NONE} & & & \\ \mathbf{1843F}&40&3^{2}&=&\mbox{\tiny $8389$}&\text{NONE} & & & \\ \mathbf{1847B}&98&3^{6}&=&\mbox{\tiny $1$}&\text{NONE} & & & \\ \mathbf{1871C}&98&19^{2}&=&\mbox{\tiny $14699$}&\text{NONE} & & & \\ \hline \end{array}$ \end{center} \newpage \large {\bf\Large Visibility at Higher Level\vspace{3ex}} \begin{center} \begin{tabular}{|l|l|}\hline &\vspace{-2ex}\\ $A_f$ with odd invisible $\Sha_{\an}[\ell]$& All $\ell$-congruent\\ & $A_g\subset J_0(Np)_{\new}$\\ &with $Np\leq 5000$ and \\ & $\ord_{s=1}L(g,s)\geq 0$\\ & (and higher $Np$ if known)\\ &\vspace{-2ex}\\ % data is autogenerated by table.py \first{\sha{551}{18}{3}} \add{\higher{2}{1}{2}} \add{\higher{3}{1}{2}} \add{\higher{5}{25}{0}} \first{\sha{767}{23}{3}} \add{\higher{2}{1}{2}} \add{\higher{7}{1}{2}} \add{\higher{7}{52}{0}} \first{\sha{959}{24}{3}} \add{\higher{2}{1}{2}} \first{\sha{1091}{62}{7}} \add{\higher{7}{2}{2}} \first{\sha{1283}{62}{5}} \add{\higher{3}{2}{2}} \first{\sha{1337}{33}{3}} \add{\higher{2}{1}{2}} \first{\sha{1339}{30}{3}} \add{\higher{2}{1}{2}} \first{\sha{1355}{28}{3}} \add{\higher{2}{1}{2}} \first{\sha{1429}{64}{5}} \add{\higher{2}{2}{2}} \add{\higher{3}{66}{0}} \first{\sha{1481}{71}{13}} \add{Nothing in range} \first{\sha{1513}{31}{3}} \add{\higher{2}{1}{2}} \first{\sha{1529}{36}{5}} \add{\higher{7}{1}{2}} \first{\sha{1559}{90}{11}} \add{Nothing in range} \first{\sha{1589}{35}{3}} \add{Nothing in range} \first{\sha{1631}{37}{5}} \add{\higher{2}{1}{2}} \first{\sha{1679}{45}{11}} \add{\higher{2}{2}{2}} \first{\sha{1727}{39}{3}} \add{\higher{2}{1}{2}} \first{\sha{2849}{1}{3}} \add{\higher{3}{1}{2}} \first{\sha{4343}{1}{3}} \add{Nothing in range} \first{\sha{5389}{1}{3}} \add{\higher{7}{1}{2}} \hline\end{tabular} \end{center} \vspace{3ex} \noindent When the second column contains an $A_g$ of rank~$2$, then $\Sha(A_f)[\ell]$ is ``very likely'' to be visible of level $M=Np$. This is the case for most examples. The ``Nothing in range'' note means that the smallest~$p$ for which there exists~$g$ of even analytic rank congruent to~$f$ is beyond the range of my current tables. The examples of level 2849, 4343, and 5389 are the odd and definitely invisible examples in Cremona and Mazur's original paper on visibility. \end{document}