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%opening
\title{\Huge\bf\dblue Modularity of Shafarevich-Tate Groups}
\author{\rd{\LARGE William Stein}\\
{\tt http://modular.fas.harvard.edu}}
\date{May 22, 2004\vspace{1ex}\\
\includegraphics[width=0.7\textwidth]{sha_fractal.eps}}

\begin{document}

\maketitle


\begin{slide}
\h{\rd{Goal}}

The goal of this 40-minute talk is to explain the meaning of
the following conjecture and give evidence for it.

{\Large{\bf Conjecture (--).} If $A$ is a modular abelian
variety, then the Shafarevich-Tate
group \rd{$\Sha(A)$} is \blue{modular}.}


\end{slide}

\begin{slide}
\h{Table of Contents}

\begin{enumerate}
\item Elliptic Curves and \rd{Modular Abelian Varieties}
\item The \rd{Birch and Swinnerton-Dyer} Conjecture
\item \mbox{}\rd{Visibility} of Shafarevich-Tate groups
\item \mbox{}\rd{Modularity} of Shafarevich-Tate groups
\item Some \rd{Data}
\end{enumerate}
\end{slide}

\begin{slide}
\h{1. Elliptic Curves and Abelian Varieties}
\begin{tabular}{ll}
\hspace{-.8ex}\begin{minipage}{6in}
\vspace{-2in}
\rd{Elliptic curve over $\Q$:} $y^2=x^3+ax+b$ 
with $a,b\in\Q$
and $\Delta=-16(4a^3+27b^2)\neq 0$
\end{minipage}&
\hspace{0.5in}
\includegraphics[width=3in]{elliptic6_floating_2.eps}
\end{tabular}\vspace{-.7in}

\rd{Abelian variety:} Any complete group variety.  

\rd{Examples:} Jacobians of curves.  Elliptic curves
are the abelian varieties of dimension one.  
\rd{Modular abelian varieties}.
\end{slide}

\begin{slide}
\h{Modular Curves, Modular Forms}

\rd{Congruence Subgroup:}
$$
\Gamma_0(N) = \left\lbrace 
\mtwo{a}{b}{c}{d} \in \SL_2(\Z) \text{ such that } N \mid c
\right\rbrace.
$$
\vspace{-1ex}

\rd{Modular Curve:} 
$X_0(N) = \Gamma_0(N) \setminus \text{(upper half plane)} \cup \text{(cusps)}$
\vspace{-1ex}

\rd{Example:}{\dblue  $X_0(39)$}
\begin{center}\vspace{-.4in}
\includegraphics[width=0.6\textwidth]{torus39.eps}
\end{center}
\vspace{-4ex}
\rd{Algebraic structure over $\Q$:}
$$
 y^2 = (x^4-7x^3+11x^2-7x+1)(x^4+x^3-x^2+x+1).
$$
\end{slide}

\begin{slide}
\h{Modular Abelian Varieties}
\rd{Modular Jacobian: } $J_0(N) = \Jac(X_0(N))$

\rd{Modular Abelian Variety:} Any abelian variety quotient of $J_0(N)$ (or
of $J_1(N)$, where $J_1(N)$ is defined using $\Gamma_1(N)$).

\rd{Theorem (Wiles et al.):} All elliptic curves over $\Q$ are modular.

\rd{Cusp Forms: }
$S_2(\Gamma_0(N))\isom \H^0(X_0(N)_\C,\Omega^1)$

\rd{Hecke Algebra:} $\T = \Z[T_1,T_2,T_3, \ldots] \subset \End(S_2(\Gamma_0(N)))$

\rd{Shimura:}
$\T$-eigenform $f \in S_2(\Gamma_0(N))$ 
gives $A_f = J_0(N)/I_f J_0(N)$, where $I_f=\Ann_\T(f)$.
We have $\dim A_f = [\Q(a_2(f),\ldots):\Q]$.

\end{slide}

\begin{slide}
\h{2. The Birch and Swinnerton-Dyer Conjecture}

Let $A$ be an abelian variety over $\Q$ (e.g., $A=A_f$ modular).
\vspace{-1in}
\begin{center}
\begin{minipage}{7in}
\rd{\LARGE Conjecture of \includegraphics[height=1.1in]{bsd.eps}:}\\
\begin{enumerate}
\item $r = \ds \rank A(\Q) \ce \ord_{s=1}L(A,s)$, and\\
\item  
$\ds
\frac{L^{(r)}(A,1)}{r!} = 
\frac{\prod c_p \cdot \Omega_A \cdot \Reg(A)
\cdot \#\Sha(A)}
{\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}}.
$
\end{enumerate}
\end{minipage}
\end{center}
Results: Kolyvagin, Kato, Rubin, etc.
\end{slide}

\begin{slide}
\h{Shafarevich-Tate Group \includegraphics[width=2.2in]{sha_fractal2.eps}}
\rd{Definition:}
$$
\Sha(A) = \Ker\left(\H^1(K,A) \to \bigoplus_{\text{all }v} \H^1(K_v,A)\right).
$$

$\H^1(K,A)$ is \rd{Galois cohomology}.  
Interpret geometrically as the \rd{Weil-Chatalet group}:
$$
\text{WC}(A/K) = \{\,\text{principal homogenous spaces }X\text{ for }A\,\}/\sim.
$$

$\Sha(A)$ is the subgroup of {\dblue locally trivial classes}. 
\rd{Example:}
$$3x^3+4y^3+5z^3=0
\in \Sha(x^3+y^3+60z^3=0)[3].$$
\end{slide}

\begin{slide}
\h{3. Visibility of Sha \includegraphics[width=1.4in,angle=-10]{mazur.eps}}
\vspace{-1in}

{\LARGE$$0 \to A\xrightarrow{i} B \to C \to 0$$}

1998 -- Barry Mazur introduced \rd{Visibility}:
\begin{align*}
\Vis_B(\H^1(K,A)) &= \Ker\left(\H^1(K,A) \to \H^1(K,B)\right)\\
   &\isom \Coker(B(K) \to C(K))\\
  & \\
\Vis_B(\Sha(A)) &= \Ker(\Sha(A)\to \Sha(B)).
\end{align*}

\end{slide}

\begin{slide}
%\h{\begin{tabular}{lcr}
%\begin{minipage}{2in}\mbox{}\\Visibility\end{minipage}
%&&\includegraphics[width=3in]{eye.eps}
%\end{tabular}}
% The eye graphic is from
%  http://www.littlebeast.com/ images/eye.jpg
% and was found using the google image search.
\begin{center}
\pspicture(-1,-1)(5,1)
\rput(-4,0){\h{Visibility \hspace{2in}\mbox{}}}
\rput(20,3){\includegraphics[width=3in]{eye.eps}}
\endpspicture
\end{center}


\vspace{-.7in}

$$
\xymatrix{
&&X=\pi^{-1}(P)\ar@{-}[d]\ar[r]&P\ar@{-}[d]\ar@{|->}[r]&c\ar@{-}[d]\\
0\ar[r]&A(K)\ar[r]&B(K)\ar[r]^{\pi}&C(K) \ar[r]&{\Vis_B(\H^1(K,A))}\ar[r]&0
}
$$

\rd{Why?} To write down $X$ using equations is terrifying;
to give $P$ is just to give a rational point.  Visibility concisely
encodes connections between Mordell-Weil and Shafarevich-Tate groups.

Give nonzero $c\in \Sha(A)[5]$ with $\dim(A)=20$ by giving
   $$(0,0)\in [y^2 + y = x^3 + x^2 - 2x]$$
\end{slide}

\begin{slide}
\h{Everything is Visible Somewhere}

\rd{Theorem (--):}
If $c\in \H^1(K,A)$ then there exists
$B$
such that $i:A\hra B$ and $c\in\Vis_B(\H^1(K,A))$.


\rd{Proof.}  Let $L$ be such that $\res_{L/K}(c)=0$. Then
$$
\xymatrix{
{c} \ar\ar@{|->}[rr] && {\res_{L/K}(c)=0}\\
{\H^1(K,A)}\ar[rr]\ar[ddrr] && {\H^1(K,\rd{\Res_{L/K}(A_L)})}\ar[dd]\\
 & \\
 && {\H^1(L,A)}
}$$

Note: If $A/\Q$ is modular and $L$ is abelian,
then $B$ is modular.
\end{slide}


\begin{slide}
\h{4. Modularity of Sha}
\rd{\bf Definition (Modular):} An element $c\in \Sha(A)$ is
{\em modular} if it is visible in a modular abelian variety.
I.e., if there is a factor $B$ of some $J_0(N)$ (or $J_1(N)$) 
and an inclusion $i:A\hra B$ such that $i_*(c)=0$.  
Torsor corresponding to $c$ is modular in ``usual''
sense.

\rd{Modularity Conjecture (--):}\\
\mbox{}\hspace{.6in}{\dblue If $A$ is modular, then every element of
  $\Sha(A)$ is modular.}

%{\red\bf Should you believe my conjecture?}

\rd{Theorem (Klenke, Mazur, Stein):} If $c\in \Sha(E)[p]$ with $p=2,3$ and $E$
an elliptic curve, then $c$ is modular.

\rd{Related questions:}\\
1. Levels $N$ such that such that $c$
is modular of level $N$?\\
2. Which genus one curves are modular?
%2. If an element
%$c\in \Sha(A)$ is modular, does that imply that
%$A$ is modular?
\end{slide}

\begin{slide}
\h{\dred Why Care about Modularity?}

$\bullet$ It could yield results about {\dblue 
structure of $\Sha(A)$}, e.g., finiteness. 

$\bullet$ It would give a nice ``explanation'' of 
{\dblue where all $\Sha(A)$ 
``comes from''} --- it all comes from Mordell-Weil groups.

$\bullet$ It {\dblue motivates proving new results} about the arithmetic
of modular abelian varieties.

$\bullet$ It provides {\dblue powerful computational tools}
for explicitly working with Shafarevich-Tate groups.

\end{slide}

\begin{slide}
\h{Visibility Construction}
\rd{Theorem (Agashe, --):} {\em Suppose $A, B\subset J_0(N)$, 
that $B[p]\subset A$, and other technical hypotheses.
Then 
  $$ B(\Q)/p B(\Q) \hra \Vis_{J_0(N)}(\Sha(A)). $$}

\rd{Proof.} Use the following diagram, chase some exact sequences, and
apply subtle properties of N\'eron models.  (Also more general 
Hecke-equivariant version, proved recently with Jetchev.)
$$
{\dgreen\xymatrix@=0.9in{
{B[p]}\ar[r]\ar[d]& B \ar[r]^p\ar[d] & B \ar[d]\\
{A}\ar[r] & {J_0(N)} \ar[r] & C.}
}$$
\end{slide}

\begin{slide}
%\h{5. Some Data}
\begin{center}
\pspicture(-1,-1)(5,1)
\rput(-1,0){\h{\hspace{-2in}5. Some Data}}
\rput(20,3){\includegraphics[width=4in]{tables.eps}}
\endpspicture
\end{center}\vspace{-1.5in}

{\LARGE $$\text{Suppose } A \subset J_0(N) $$}

\begin{itemize}
\item[(a)] Visibility of $\Sha(A)$ in $J_0(N)$, when $A$ is an elliptic
curve.
\item[(b)] Visibility of $\Sha(A)$ in $J_0(N)$, general $A$.
\item[(c)] Visibility of $\Sha(A)$ in $J_0(Np)$, general $A$.  (Modularity)
\end{itemize}
\end{slide}

\begin{slide}
%\voffset=-1in
\h{(a) Elliptic Curve tables from Cremona-Mazur}

\begin{center}\Large
Visibility of $\Sha(A)$ in $J_0(N)$, when $A=E$ is an \rd{elliptic
curve}.
\end{center}
\begin{center}
\includegraphics[width=2in]{cremona_table_1.eps}
$\qquad$\includegraphics[width=2in]{cremona_table_1b.eps}
$\qquad$\includegraphics[width=2in]{cremona_table_1c.eps}
\includegraphics[width=1in,angle=-10]{cremona5.eps}
\end{center}
\end{slide}

\begin{slide}
\voffset=-1in
\begin{center}
\includegraphics[height=1.3\textheight]{cm1.ps}
\end{center}

\begin{center}
\includegraphics[height=1.3\textheight]{cm2.ps}
\end{center}
\end{slide}

\begin{slide}
\voffset=-1in
\h{(b) Abelian Variety Tables from Agashe-Stein}

\begin{center}\Large
 Visibility of $\Sha(A)$ in $J_0(N)$, general $A$.
\end{center}

\begin{center}
\includegraphics[height=1.5\textheight]{as1.ps}
\end{center}

\begin{center}
\includegraphics[height=1.5\textheight]{as2.ps}
\end{center}

\begin{center}
\includegraphics[height=1.5\textheight]{as3.ps}
\end{center}

\begin{center}
\includegraphics[height=1.5\textheight]{as4.ps}
\end{center}

\end{slide}

\begin{slide}
\h{(c) Visibility at Higher Level -- Evidence for Modularity Conjecture}

\begin{center}\Large
 Visibility of $\Sha(A)$ in $J_0(Np)$, general $A$.
\end{center}
Recall Ribet's theorem...
\end{slide}

\begin{slide}
\vspace{-0.5in}
\h{Ribet Level Raising \includegraphics[width=1.5in,angle=-5]{ribet.eps}}

\vspace{-0.1in}Suppose\vspace{-0.4in}
\begin{itemize}\setlength{\itemsep}{0in}
\item $f=\sum a_n q^n\in S_2(\Gamma_0(N))$ a newform
\item $\lambda\subset \Z[a_1,a_2,\ldots]$ a nonzero prime ideal
s.t. $A_f[\lambda]$ irreducible.
\end{itemize}\vspace{-0.3in}

\rd{\bf Theorem:} {\dblue\large $
  a_p + p + 1 \con 0\pmod{\lambda} \implies
$}
there exists a $p$-newform $g\in S_2(\Gamma_0(Np))$ such that
\vspace{-0.4in}
\begin{itemize}\setlength{\itemsep}{0in}
\item $i(A_f[\lambda]) = A_g[\lambda]$ some
$i:J_0(N)\to J_0(N p)$, and
\item sign of functional equations for $L(f,s)$ and $L(g,s)$
same. 
\end{itemize}

%{\small Note: If instead $a_p - (p+1) \con 0\pmod{\lambda}$
%then there is such a $g$, but the sign of the functional
%equation changes, and the new Tamagawa number of 
%$A_g$ at $p$ are divisible by $\lambda$.}
\end{slide}

\begin{slide}
  \rd{\bf Big Computation:} For every level $N$ up to $5000$ (and
  more), use my modular forms package in MAGMA to provably compute:
\begin{enumerate}
\item Each newform $f=\sum a_n q^n \in S_2(\Gamma_0(N))$.
\item Whether or not $L(f,1)=0$.
\item Whether or not $\ord_{s=1} L(f,s)$ is even.
\item Characteristic polynomials of $a_2, a_3, a_5, \ldots, a_{19}$.
\end{enumerate}
(I hope to redo this computation 
using only open-source software that I'm currently
writing.)
\end{slide}

\begin{slide}
\h{Probable Modularity}
$\bullet$ Two forms $f=\sum a_n q^n$ and $g=\sum b_n q^n$ 
are {\em\dred probably congruent mod $\ell$} (away from level) if 
for $p<20$ with $p\nmid N_f N_g$ we have
$$
  \ell \mid \text{resultant}(\charpoly(a_p),\charpoly(b_p)).
$$
%(Congruence would imply this, but not conversely in general.)

$\bullet$ If $A=A_g\subset J_0(N)$, then there is 
{\em\dred probably a nonzero element in 
$\Sha(A)[\ell]$ modular of level $N p$} if
there is $f$ of level $N p$ such that:\vspace{-.4in}
\begin{enumerate}\setlength{\itemsep}{0in}
\item $f$ and $g$ are probably congruent modulo $\ell$, and
\item $\ord_{s=1} L(f,s)$ is positive and even.
\end{enumerate}
\end{slide}


\begin{slide}
\voffset=-1.3in

\begin{center}
\includegraphics[height=1.7\textheight]{higher.ps}
\end{center}
\end{slide}

\begin{slide}
\h{Questions}
\begin{center}
\includegraphics[width=3in]{questions.eps}
%\Huge {\dred ?}
\end{center}

\end{document}

\begin{slide}

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} 
$$
\Sha(A) = \Ker\left(\H^1(K,A) \to \bigoplus_v \H^1(K_v,A)\right).
$$
\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 $A$, so we have an exact sequence
$$
  0 \to A \to B \to C \to 0
$$
of abelian varieties.
\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 $\Vis_i(\H^1(K,A))$
is torsion.  
\item If $c\in\Vis_i(\H^1(K,A))$, 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
$i:A\hra B$.  I've also considered defining 
$\Vis_B(\H^1(K,A))$ to be the subgroup generated by all visible
subgroups with respect to all embeddings $A\to B$, but I'm not
sure what properties this definition has. 
\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 $i:A\hra B$ and $c\in\Vis_i(\H^1(K,A))$.
(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.
$$
\xymatrix{ & B\ar[d] \\
      A \ar[r] & {J_0(389)}
      }
$$
Gives
$$
 (\Z/5\Z)^2 \isom B(\Q)/5 B(\Q) \hra \Sha(A).
$$
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
$c \in \Vis_{M}^p(\Sha(A))$,
where $\Vis_{M}^p(\Sha(A))$ is
the subgroup generated by all
kernels of maps $\Sha(A)[p^\infty]\to \Sha(J_0(M))[p^\infty]$
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 visible, 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}