\documentclass{article} \title{Visibility of Shafarevich-Tate groups} \pagestyle{empty} \author{William A. Stein} \date{Fall 2000} \include{macros} \DeclareMathOperator{\Vis}{Vis} \DeclareMathOperator{\Res}{Res} \usepackage[all]{xy} \begin{document} \maketitle \begin{abstract} These are notes for my talks on visibility of Shafarevich-Tate groups of quotients of $J_0(N)$. I first introduce the notion of visibility in the context of trying to understand the Birch and Swinnerton-Dyer conjecture. I then give some experimental data, which suggests what one might expect. Next I sketch the theory behind proving that some of Sha is visible. Finally, I discuss visibility for quotients of $J_1(N)$, and for motives attached to modular forms. \end{abstract} \section{Modular abelian varieties} A central class of abelian varieties of interest in number theory are those which are {\defn modular}, in these sense that they are a quotient of some modular Jacobian $J_0(N)$, or, more generally, of some $J_1(N)$. For example, we now know that every elliptic curve over~$\Q$ is a member of this class (see \cite{breuil-conrad-diamond-taylor}). Such abelian varieties are constructed as follows. Let $f=\sum a_n q^n\in S_2(\Gamma_0(N),\C)$ be a newform. The Hecke algebra $\T=\Z[\ldots T_n \ldots]$ acts on~$f$, and there is a surjective homomorphism $\T\ra \O_f = \Z[\ldots a_n \ldots]$ sending $T_n$ to $a_n$. The kernel $I_f$ of this homomorphism is the annihilator of~$f$ in $\T$. Attach to $I_f$ the quotient $A_f = J_0(N)/I_f(J_0(N))$, where $J_0(N)$ is the Jacobian of the modular curve $X_0(N)$. The abelian variety $A_f$ has dimension $d=[\Q(\ldots a_n\ldots):\Q]$, and the dual $A_f^{\vee}$ is an abelian subvariety of $J_0(N)$. There are two groups attached to an abelian variety~$A$ over $\Q$. One is the Mordell-Weil group $A(\Q)$, which is a finitely generated abelian group, and the other is the Shafarevich-Tate group $\Sha(A) = \ker(H^1(\Q,A) \ra \prod_v H^1(\Q_v,A))$, which is ({\em conjecturally}!) a finite abelian group. This talk is about a link between these two groups, which is frequently signaled by the term ``visibility''. Before proceeding further, we state a fundamental conjecture that both guides and stimulates our work. \begin{conjecture}[Birch and Swinnerton-Dyer] Let $A$ be an abelian variety over~$\Q$. Then $\Sha(A)$ is finite, and one has the following formula: $$L(A,1) = \frac{\prod c_p \cdot \Omega_A}{\#A(\Q)\cdot \#A^{\vee}(\Q)} \cdot \#\Sha(A),$$ where the right hand side should be interpreted as~$0$ if $A(\Q)$ is infinite. (When $L(A,1)=0$, there is a refinement involving the regulator of $A$ and the leading coefficient of the power series expansion of $L(A,s)$ about $s=1$.) \end{conjecture} For a nice summary of this question, and some of its generalization in the special case when~$A$ has dimension~$1$, see Wiles's short article at the CMI web page \cite{wiles:cmi}. \section{Mazur: Visualize Sha!} Let~$A$ be an abelian variety over~$\Q$. To give an element of $\Sha(A)$ is equivalent to giving an algebraic variety~$X$ over~$\Q$ equipped with an action $A\cross X \ra X$ satisfying analogues of the usual axioms for a simply transitive group action. Where can we find these varieties~$X$? They can be found as translates of~$A$ inside a bigger abelian variety~$J$! Fix an embedding of $A$ into an abelian variety $J$, and let $C$ be the quotient: $$A \hookrightarrow J \onto C.$$ Taking cohomology, we reveal the following diagram: $$\xymatrix{ & & & {\Vis_J(\Sha(A))}\ar@{^(->}[r] & {\Sha(A)}\ar@{^(->}[d] \ar[r] & {\Sha(J)}\ar@{^(->}[d]\\ 0\ar[r] & A(\Q) \ar[r] & J(\Q) \ar[r] & C(\Q) \ar[r] & H^1(\Q,A) \ar[r] & H^1(\Q,J) }$$ \begin{definition}[Visible part] The \defn{visible part of $\Sha(A)$} with respect to the embedding $A\hookrightarrow J$ is the subgroup $$\Vis_J(\Sha(A)) = \ker(\Sha(A) \ra \Sha(J)).$$ Likewise, the visible part of $H^1(\Q,A)$ is $$\Vis_J(H^1(\Q,A)) = \ker(H^1(\Q,A) \ra H^1(\Q,J)).$$ \end{definition} \begin{question}[Mazur] Is every element $c\in H^1(\Q,A)$ visible somewhere? \end{question} Johan de Jong answered this question in the affirmative, when $\dim A = 1$; however, his proof, which exploits Azamuya algebras over $\Spec(\Z)$, is not elementary. Fortunately, Mazur's question has an elementary answer. \begin{proposition}[Folklore?] Given $c\in H^1(\Q,A)$ there is an abelian variety $J$ and an embedding $\iota:A\hookrightarrow J$ such that $\iota_*(c) = 0 \in H^1(\Q,J)$. \end{proposition} \begin{proof} Cochains are continuous, so there is a finite extension $K$ of $\Q$ such that $\res_K(c) = 0 \in H^1(K,A)$. The Shapiro lemma implies that $H^1(K,A)$ is canonically isomorphic to $H^1(\Q,\Res_{K/\Q} A_K)$, where $\Res_{K/\Q} A_K$ is the Weil restriction of scalars of $A_K$. Furthermore, there is a canonical embedding $\iota: A\hookrightarrow J=\Res_{K/\Q} A_K$, and this embedding sends~$c$ to $\res_K(c) = 0$. \end{proof} \begin{corollary} Every $c\in \Sha(E)[2]$ is visible in an abelian surface, and every $c\in \Sha(E)[3]$ is visible in a three-dimensional abelian variety. \end{corollary} Using more sophisticated techniques, Mazur proved that every $c\in \Sha(E)[3]$ is visible in an abelian surface. For the rest of this talk, we return to the case where~$A$ is attached to a modular form. \begin{question}[Mazur] Let~$f$ be a newform of level~$N$. Can one visualize the elements $c\in \Sha(A_f^{\vee})$ inside $J_0(N)$? Inside $J_0(M)$ for some multiple $M$ of $N$? \end{question} \section{Some data} Mazur, Cremona, and A.~Logan collected data on visibility of Sha using only elliptic curves and congruences between elliptic curves. The data is impressive, but probably only because $\Sha$ is very small in the range considered. As Frey emphasized in his talk, there are very few intersections between elliptic curves. With the help of A.~Agashe, the author made a table of rational numbers $L(A_f,1)/\Omega_{A_f}$ for all $f \in S_2(\Gamma_0(p))$, for $p\leq 2161$. To analyze the resulting data, let us momentarily assume the truth of the Birch and Swinnerton-Dyer conjecture. We find~$38$ of the $A_f$ have $L(A_f,1)\neq 0$ and the property that there is an odd prime~$\ell$ that divides $\#\Sha(A_f)$. Of these, the full odd part of $\Sha(A_f)$ is visible in $22$ cases, whereas it is invisible in each of the remaining $16$ cases. A small {\em selection} of data is recorded in the table below. \begin{center} \begin{tabular}{lcll} $p$ & $\dim A_f$ & $\sqrt{\#\Sha(A_f)^{\text{odd}}}$ & $\dim B$\\ 389 & 20 & 5 (VIS)& 1\\ 433 & 16 & 7 (VIS)& 1\\ 1061 & 46 & 151 (VIS)& 2 \\ 1091 & 62 & 7 (INV, so far) & ---\\ 1429 & 64 & 5 (VIS at level $2\cdot 1429$) & 2\\ 2111 & 112 & 211 (INV, so far) & ---\\ 2333 & 101 & 83341 (VIS) & 4\\ 2849 & 1 & 3 (VIS at level $3\cdot 2849$) & 1\\ 5389 & 1 & 3 (VIS at level $7\cdot 3849$) & 1\\ \end{tabular} \end{center} \section{Criterion for visibility} \begin{theorem}[Stein] Let $A,B \subset J$, and let $N$ be divisible exactly by the primes of bad reduction for~$J$. Suppose that \begin{itemize} \item $(A \intersect B)(\Qbar)$ is finite, \item $A(\Q)$ is finite, \item $B$ has purely toric reduction at each $\ell\mid N$. \end{itemize} Let $p$ be a prime such that \begin{enumerate} \item $$p\nmid 2N \cdot \# (\frac{J(\Q)}{A(\Q)})_{\tor} \cdot \#B(\Q)_{\tor} \cdot \prod_{\ell \mid N} \#\Phi_{A,\ell}(\overline{\F})\cdot \#\Phi_{B,\ell}(\F_\ell),$$ \item $B[p]\subset A$ (this condition can be relaxed). \end{enumerate} Then $$B(\Q)/p B(\Q) \hookrightarrow \Vis_J(\Sha(A)).$$ \end{theorem} \begin{proof}[Sketch] Consider the diagram $$\xymatrix{ & {A \intersect B} \ar[r]\ar[d] & B\ar[d] \ar[r]^p& B\ar[d]\\ 0 \ar[r] & A \ar[r] & J \ar[r] & C \ar[r]& 0. }$$ Taking cohomology, we arrive at the following key diagram: $$\xymatrix{ 0\ar[r] & B(\Q)\ar[d] \ar[r]^{p} & B(\Q)\ar[d] \ar[r] & B(\Q)/p B(\Q)\ar[d] \ar[r] & 0\\ 0\ar[r] & J(\Q)/A(\Q) \ar[r] & C(\Q) \ar[r] & \Vis_J(H^1(\Q,A)) \ar[r] & 0. }$$ Upon applying the snake lemma and using the above hypothesis, we deduce that $B(\Q)/p B(\Q) \hookrightarrow \Vis_J(H^1(\Q,A))$. The deduction that $B(\Q)/p B(\Q)\hookrightarrow \Vis_J(\Sha(A))$ involves a local analysis at each prime $\ell$ of $\Q$. \end{proof} \section{Visibility at higher level} There is a $53$-dimensional abelian subvariety~$A$ of $J_0(1429)$ such that~$5$ divides the conjectural order of $\Sha(A)$. However, this part of $\Sha$ can not be visible in $J_0(1429)$. Ribet's level raising theorem supplies us with an infinite collection of primes $p$ such that an image of $A$ in $J_0(pN)$ meets an abelian variety $B$ in some $\m$-torsion, where $\m$ is an appropriate maximal ideal of residue characteristic $5$ in $\T$. For each of these $B$, {\em probably} (I haven't proved this) $$L(B,1)\con L(A,1) = 0 \pmod{\m},$$ so either $5\mid \#\Sha(B)$ or $L(B,1)=0$. In the latter case, the Birch and Swinnerton-Dyer conjecture predicts that $B(\Q)$ is infinite; the theorem above then implies that $$5\mid \#\Vis_{J_0(pN)}(\Sha(A)).$$ There are many similar examples, some of which have $\dim B=1$, and hence do not rely on any conjectures. Based on this evidence, we make the following conjecture. \begin{conjecture}[Stein] Let $N$ be a prime and $A=A_f^{\vee}\subset J_0(N)$. If $c\in \Sha(A)$, then there is a prime $p$ such that $\delta_1^*(c)+ \delta_p^*(c) = 0 \in \Sha(J_0(pN))$. \end{conjecture} \section{Speculation: constructing points?} Constructing elements of $\Vis_{J_0(N)}(\Sha(A))$ is equivalent to constructing points on~$B$. This observation might lead to an approach to proving the implication $L(B,1)=0$ implies $B(\Q)$ is infinite. At present, mathematicians seem to have no good construction of points when $L(B,1)=L'(B,1)=0$. However, we do have Kolyvagin's Euler systems when $L(A,1)\neq 0$! \section*{The tables} \voffset=-.3in \textheight=1.15\textheight \newpage \subsection*{Shafarevich-Tate groups of abelian subvarieties of $J_0(p)$} (The entries in the columns ``mod deg'' and ``$\Shaan$'' are only really the odd parts of ``mod deg'' and ``$\Shaan$''.)\vspace{-.25ex} $$ \begin{array}{lccclcc} \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)\\ & & & & & & \vspace{-3ex} \\ \mbox{\rm\bf 389E}& 20 &5^{2}&5&\mbox{\rm\bf 389A}& 1 &5\\ \mbox{\rm\bf 433D}& 16 &7^{2}&3\cdot7\cdot37&\mbox{\rm\bf 433A}& 1 &7\\ \mbox{\rm\bf 563E}& 31 &13^{2}&13&\mbox{\rm\bf 563A}& 1 &13\\ \mbox{\rm\bf 571D}& 2 &3^{2}&3^{2}\cdot127&\mbox{\rm\bf 571B}& 1 &3\\ \mbox{\rm\bf 709C}& 30 &11^{2}&11&\mbox{\rm\bf 709A}& 1 &11\\ \mbox{\rm\bf 997H}& 42 &3^{4}&3^{2}&\mbox{\rm\bf 997B}& 1 &3\\ \mbox{\rm\bf 1061D}& 46 &151^{2}&61\cdot151\cdot179&\mbox{\rm\bf 1061B}& 2 &151\\ \mbox{\rm\bf 1091C}& 62 &7^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1171D}& 53 &11^{2}&3^{4}\cdot11&\mbox{\rm\bf 1171A}& 1 &11\\ \mbox{\rm\bf 1283C}& 62 &5^{2}&5\cdot41\cdot59&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1429B}& 64 &5^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1481C}& 71 &13^{2}&5^{2}\cdot2833&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1483D}& 67 &3^{2}\cdot5^{2}&3\cdot5&\mbox{\rm\bf 1483A}& 1 &3\cdot5\\ \mbox{\rm\bf 1531D}& 73 &3^{2}&3&\mbox{\rm\bf 1531A}& 1 &3\\ \mbox{\rm\bf 1559B}& 90 &11^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1567D}& 69 &7^{2}\cdot41^{2}&7\cdot41&\mbox{\rm\bf 1567B}& 3 &7\cdot41\\ \mbox{\rm\bf 1613D}& 75 &5^{2}&5\cdot19&\mbox{\rm\bf 1613A}& 1 &5\\ \mbox{\rm\bf 1621C}& 70 &17^{2}&17&\mbox{\rm\bf 1621A}& 1 &17\\ \mbox{\rm\bf 1627C}& 73 &3^{4}&3^{2}&\mbox{\rm\bf 1627A}& 1 &3^{2}\\ \mbox{\rm\bf 1693C}& 72 &1301^{2}&1301&\mbox{\rm\bf 1693A}& 3 &1301\\ \mbox{\rm\bf 1811D}& 98 &31^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1847B}& 98 &3^{6}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1871C}& 98 &19^{2}&14699&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1877B}& 86 &7^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 1907D}& 90 &7^{2}&3\cdot5\cdot7\cdot11&\mbox{\rm\bf 1907A}& 1 &7\\ \mbox{\rm\bf 1913B}& 1 &3^{2}&3\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\ \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\\ \mbox{\rm\bf 1933C}& 83 &3^{2}\cdot7^{2}&3\cdot7&\mbox{\rm\bf 1933A}& 1 &3\cdot7\\ \mbox{\rm\bf 1997C}& 93 &17^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2027C}& 94 &29^{2}&29&\mbox{\rm\bf 2027A}& 1 &29\\ \mbox{\rm\bf 2029C}& 90 &5^{2}\cdot269^{2}&5\cdot269&\mbox{\rm\bf 2029A}& 2 &5\cdot269\\ \mbox{\rm\bf 2039F}& 99 &3^{4}\cdot5^{2}&19\cdot29\cdot7759\cdot3214201&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2063C}& 106 &13^{2}&61\cdot139&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2089J}& 91 &11^{2}&3\cdot5\cdot11\cdot19\cdot73\cdot139&\mbox{\rm\bf 2089B}& 1 &11\\ \mbox{\rm\bf 2099B}& 106 &3^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2111B}& 112 &211^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2113B}& 91 &7^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2161C}& 98 &23^{2}&1&\mbox{\rm NONE} & & \\\hline \mbox{\rm\bf 2333C}& 101 &83341^{2}&83341&\mbox{\rm\bf 2333A}& 4 &83341\\ \mbox{\rm\bf 2339C}& 114 &3^{8}&6791&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2411B}& 123 &11^{2}&1&\mbox{\rm NONE} & & \\ \mbox{\rm\bf 2593B}& 109 &67^2\cdot 2213^2 & 67 \cdot 2213 &\mbox {\bf 2593A}& 4 & 67 \cdot 2213\\ \end{array} $$ \vspace{2em} \noindent\small email: {\sf was@math.harvard.edu} \subsection*{Odd ``invisible'' $\Sha$ in Cremona-Mazur} $$ \begin{array}{lccll} \mbox{\rm\bf E}&\#\Sha_E& \text{mod deg}(E) & \mbox{\rm\bf F} & \text{Remarks}\\ & & & & \vspace{-3ex} \\ \mbox{\rm\bf 2849A}& 3^2 &2^5\cdot 5\cdot 61&\mbox{\rm\bf NONE}& \text{visible at level $3\cdot 2849$}\\ \mbox{\rm\bf 3364C}& 7^2 &2^6\cdot3^2\cdot5^2\cdot7 &\mbox{\rm\bf none}& \text{visible in $3$-dim $F$}\\ \mbox{\rm\bf 4229A}& 3^2 &2^3\cdot3\cdot7\cdot13 &\mbox{\rm\bf none}& \\ \mbox{\rm\bf 4343B}& 3^2 &2^4\cdot1583 &\mbox{\rm\bf NONE}& \\ \mbox{\rm\bf 4914N}& 3^2 &2^4\cdot 3^5 &\mbox{\rm\bf none}& \text{{\bf E} has rational $3$-torsion}\\ \mbox{\rm\bf 5054C}& 3^2 &2^3\cdot 3^3\cdot 11&\mbox{\rm\bf none}& \\ \mbox{\rm\bf 5073D}& 3^2 &2^5\cdot 3\cdot 5\cdot7\cdot23 &\mbox{\rm\bf none}& \\ \mbox{\rm\bf 5389A}& 3^2 &2^2\cdot 2333 &\mbox{\rm\bf NONE}& \text{visible at level $7\cdot 5389$}\\ \end{array} $$ \comment{ \subsubsection*{2849A is visible at higher level} $\Sha(E)$ becomes visible in an abelian surface at level $3\cdot 2849 = 3\cdot 7\cdot 11\cdot 37$. \subsubsection*{3364C is visible at higher level} The elliptic curve~$E$ denoted {\bf 3364C} is labeled ``none'' because there is no {\em elliptic curve} that satisfies an appropriate $7$-congruence with {\bf 3364C}. However, the modular degree is divisible by~$7$, so there must be some abelian subvariety that satisfies a $7$-congruence with~$E$. Computing, we find a $3$-dimensional abelian variety~$A$ such that $T_2$, $T_3$, and $T_5$ satisfies the polynomials $x^3$, $x^3 + 5x^2 + 6x + 1$, and $x^3 + 5x^2 + 6x + 1$ on~$A$, respectively. Furthermore, $L(A,1)=0$, so the BSD conjecture strongly suggests that there are elements of $\Sha(E)$ of order~$7$ that are visible in $E+A \subset J_0(3364)$. \subsubsection*{5389A is visible at higher level} $\Sha(E)$ becomes visible in an abelian surface at level $7\cdot 5389 = 37723 = 7\cdot 17\cdot 317$. } \subsection*{Modular motives of weight~$4$: Conjecturally visible $\Sha$} Suppose $f,g \in S_4(\Gamma_1(N))$ are such that $L(f,2)/\Omega_f = p^2\cdot *$, $L(g,2)=0$, and~$f$ and~$g$ satisfy a ``$p$-congruence''. Does this force~$p$ to divide $\#\Sha(\sM_f(2))$? $$ \begin{array}{clcclcl} \hspace{2em}\chi\hspace{2em}\mbox{} & \sM_f \hspace{1em}\mbox{}& \text{dim} & \hspace{2em}p^2\hspace{2em}\mbox{} & \sM_g & \text{dim}\\ & & & & & & \vspace{-2ex} \\ (1,1) & \mbox{\bf 99C} & 8 & 19^2 & \mbox{\bf 99B} & 2 \\ 1 & \mbox{\bf 127C} & 17 & 43^2 & \mbox{\bf 127A} & 1\\ 1 & \mbox{\bf 159E} & 8 & 23^2 & \mbox{\bf 159B} & 1\\ (0,0,1) & \mbox{\bf 200E} & 8 & 7^2 & \mbox{\bf 200G} & 2\\ 1 & \mbox{\bf 365E} & 18 & 29^2 & \mbox{\bf 365A} & 1\\ 1 & \mbox{\bf 369I} & ? & 13^2 & \mbox{\bf 369A} & ?\\ 1 & \mbox{\bf 453E} & ? & 17^2 & \mbox{\bf 453A} & ?\\ 1 & \mbox{\bf 465H} & ? & 11^2 & \mbox{\bf 465A} & ?\\ 1 & \mbox{\bf 477L} & ? & 73^2 & \mbox{\bf 477A} & ?\\ 1 & \mbox{\bf 567G} & ? & 13^2, 23^2 & \mbox{\bf 567A} & ?\\ 1 & \mbox{\bf 581E} & ? & 19^2 & \mbox{\bf 581A} & ? \end{array} $$ Here~$\chi$ is the common ``nebentypus'' character of~$f$ and~$g$. \subsection*{Visibility of Mordell-Weil in $J_0(N)$ via an odd congruence}\vspace{-3ex} \noindent\mbox{}\hspace{-2em}\begin{minipage}{2.5in} \tiny $$\begin{array}{lclccc} \hspace{-2em}L(f,1)=0 & \hspace{-.5em}\text{dim} & \hspace{-1em}L(g,1)\neq 0 & \hspace{-.6em}\text{dim} & \hspace{-.8em}\text{cong} & \hspace{-.7em} L(A_g,1)/\Omega_{A_g} \cdot 2^*\\ & & & & & \vspace{-2ex}\\ \mbox{\bf 389A} & 1 & \mbox{\bf 389E} & 20 & 5 & 25/97\\ \mbox{\bf 433A} & 1 & \mbox{\bf 433D} & 16 & 7 & 49/9\\ \mbox{\bf 446B} & 1 & \mbox{\bf 446F} & 8 & 11 & 121/3\\ \mbox{\bf 563A} & 1 & \mbox{\bf 563E} & 31 & 13 & 169/281\\ \mbox{\bf 571B} & 1 & \mbox{\bf 571D} & 2 & 3 & 9\\ \mbox{\bf 643A} & 1 & \text{INV} & & & \\ \mbox{\bf 655A} & 1 & \mbox{\bf 655D} & 13 & 9 & 81\\ \mbox{\bf 664A} & 1 & \mbox{\bf 664F} & 8 & 5 & 25\\ \mbox{\bf 681C} & 1 & \mbox{\bf 681B} & 1 & 3 & 9\\ \mbox{\bf 707A} & 1 & \mbox{\bf 707G} & 15 & 13 & 169\\ \mbox{\bf 718B} & 1 & \mbox{\bf 718F} & 7 & 7 & 49\\ \mbox{\bf 794A} & 1 & \mbox{\bf 794G} & 12 & 11 & 121/3\\ \mbox{\bf 817A} & 1 & \mbox{\bf 817E} & 15 & 7 & 49/5\\ \mbox{\bf 916C} & 1 & \mbox{\bf 916G} & 9 & 11 & 121\\ \mbox{\bf 944E} & 1 & \mbox{\bf 944O} & 6 & 7 & 49\\ \mbox{\bf 997B} & 1 & \mbox{\bf 997H} & 42 & 3 & 81/83\\ \mbox{\bf 997C} & 1 & \mbox{\bf 997H} & 42 & 3 & 81/83\\ \mbox{\bf 1001C} & 1 & \mbox{\bf 1001F} & 3 & 3 & 27\\ \mbox{\bf 1001C} & 1 & \mbox{\bf 1001L} & 7 & 7 & 49\\ \mbox{\bf 1028A} & 1 & \mbox{\bf 1028E} & 14 & 11 & 3267\\ \mbox{\bf 1034A} & 1 & \text{INV} & & & \\ \mbox{\bf 1041B} & 2 & \mbox{\bf 1041E} & 4 & 5 & 25\\ \mbox{\bf 1041B} & 2 & \mbox{\bf 1041J} & 13 & 25 & 625\\ \mbox{\bf 1058C} & 1 & \mbox{\bf 1058D} & 1 & 5 & 25\\ \mbox{\bf 1061B} & 2 & \mbox{\bf 1061D} & 46 & 151 & 22801/265\\ \mbox{\bf 1070A} & 1 & \mbox{\bf 1070M} & 7 & 15 & 75\\ \mbox{\bf 1073A} & 1 & \text{INV} & & & \\ \mbox{\bf 1077A} & 1 & \mbox{\bf 1077J} & 15 & 9 & 81\\ \mbox{\bf 1088J} & 1 & \mbox{\bf 1088R} & 2 & 3 & 9\\ \mbox{\bf 1094A} & 1 & \mbox{\bf 1094F} & 13 & 11 & 121/3\\ \mbox{\bf 1102A} & 1 & \mbox{\bf 1102K} & 4 & 3 & 9\\ \mbox{\bf 1126A} & 1 & \mbox{\bf 1126F} & 11 & 11 & 121\\ \mbox{\bf 1132A} & 1 & \mbox{\bf 1132F} & 12 & 5 & 225\\ \mbox{\bf 1137A} & 1 & \mbox{\bf 1137C} & 14 & 9 & 81\\ \mbox{\bf 1141A} & 1 & \mbox{\bf 1141I} & 22 & 7 & 1524537/41\\ \end{array}$$ \end{minipage} \begin{minipage}{3in} \tiny$$\begin{array}{lclccc} & & & & & \\ & & & & & \vspace{-2ex}\\ \mbox{\bf 1143C} & 1 & \mbox{\bf 1143J} & 9 & 11 & 121\\ \mbox{\bf 1147A} & 1 & \mbox{\bf 1147H} & 23 & 5 & 225/19\\ \mbox{\bf 1171A} & 1 & \mbox{\bf 1171D} & 53 & 11 & 121/195\\ \mbox{\bf 1246C} & 1 & \mbox{\bf 1246B} & 1 & 5 & 25\\ \mbox{\bf 1309B} & 1 & \text{INV} & & & \\ \mbox{\bf 1324A} & 1 & \mbox{\bf 1324E} & 14 & 9 & 6561\\ \mbox{\bf 1325E} & 1 & \mbox{\bf 1325T} & 11 & 9 & 2187\\ \mbox{\bf 1363B} & 2 & \mbox{\bf 1363F} & 25 & 31 & 961/5\\ \mbox{\bf 1431A} & 1 & \mbox{\bf 1431L} & 14 & 3 & 9\\ \mbox{\bf 1436A} & 1 & \text{INV} & & & \\ \mbox{\bf 1443C} & 1 & \mbox{\bf 1443G} & 5 & 7 & 49\\ \mbox{\bf 1446A} & 1 & \mbox{\bf 1446N} & 7 & 3 & 9\\ \mbox{\bf 1466B} & 1 & \mbox{\bf 1466H} & 23 & 13 & 4331806939187/367\\ \mbox{\bf 1477A} & 1 & \mbox{\bf 1477C} & 24 & 13 & 169\\ \mbox{\bf 1480A} & 1 & \mbox{\bf 1480G} & 5 & 7 & 49\\ \mbox{\bf 1483A} & 1 & \mbox{\bf 1483D} & 67 & 15 & 225/247\\ \mbox{\bf 1525C} & 1 & \mbox{\bf 1525O} & 16 & 7 & 49\\ \mbox{\bf 1531A} & 1 & \mbox{\bf 1531D} & 73 & 3 & 3/85\\ \mbox{\bf 1534B} & 1 & \mbox{\bf 1534J} & 6 & 3 & 3\\ \mbox{\bf 1567B} & 3 & \mbox{\bf 1567D} & 69 & 287 & 82369/261\\ \mbox{\bf 1570B} & 1 & \mbox{\bf 1570J} & 6 & 11 & 121\\ \mbox{\bf 1576A} & 1 & \mbox{\bf 1576E} & 14 & 11 & 121\\ \mbox{\bf 1591A} & 1 & \mbox{\bf 1591F} & 35 & 31 & 6727/19\\ \mbox{\bf 1594A} & 1 & \mbox{\bf 1594J} & 17 & 3 & 3370648239/19\\ \mbox{\bf 1608A} & 1 & \mbox{\bf 1608J} & 6 & 13 & 169\\ \mbox{\bf 1611D} & 1 & \mbox{\bf 1611O} & 11 & 9 & 81\\ \mbox{\bf 1613A} & 1 & \mbox{\bf 1613D} & 75 & 5 & 25/403\\ \mbox{\bf 1615A} & 1 & \mbox{\bf 1615J} & 13 & 9 & 102141\\ \mbox{\bf 1621A} & 1 & \mbox{\bf 1621C} & 70 & 17 & 289/135\\ \mbox{\bf 1627A} & 1 & \mbox{\bf 1627C} & 73 & 9 & 81/271\\ \mbox{\bf 1633A} & 3 & \mbox{\bf 1633D} & 27 & 189 & 35721\\ \mbox{\bf 1639B} & 1 & \mbox{\bf 1639G} & 34 & 17 & 680017/25\\ \mbox{\bf 1641B} & 1 & \mbox{\bf 1641J} & 24 & 23 & 529\\ \mbox{\bf 1642A} & 1 & \mbox{\bf 1642D} & 14 & 7 & 49\\ \end{array}$$ \end{minipage} \mbox{}\par\noindent\vspace{5ex} {\tiny {\bf 643A}, {\bf 1034A}, {\bf 1073A}, {\bf 1309B} all have modular degree a power of~$2$; {\bf 1436A} has modular degree divisible by~$3$.} \comment {% Bibliography \newpage \mbox{} \newpage \bibliographystyle{amsplain} \bibliography{biblio} } \end{document}