\chapter*{Conjectures}

The various theorems and examples that we presented illustrate that looking at \emph{visible elements} might 
be useful for a better understanding of the Shafarevich-Tate group, since rational points on abelian varieties 
are much easier to understand than cohomology classes. 

The visualization theorem, together with the various interesting examples might serve as a good motivation for 
the following question \\

\noindent {\bf Question 1.} If $A$ is an abelian variety over $K$, does there exists a variety $J$ and an embedding 
$i : A \hookrightarrow J$, such that the whole Shafarevich-Tate group becomes visible, i.e. 
$$
\textrm{Vis}_J^{(i)}(\Sha(A / K)) = \Sha(A / K)? 
$$ 

The above question is too general and it is likely that the answer might be negative. However, if we specialize the 
question to subvarieties of modular Jacobians $J_0(N)$, the level-raising example generates the following question \\

\noindent {\bf Question 2.} If $A$ is an abelian variety over $\Q$, whose dual is an optimal quotient of 
$J_0(N)$ (hence $A$ is a subvariety of $J_0(N)$), does there exists $M \in \mathbb{N}$ and a linear combination of 
degeneracy map $J_0(N) \ra J_0(MN)$ \footnote{Note that we allow this map to have a kernel - recall that the general definition of 
visibility did not require that $A$ is embedded in $J$.}, such that every element of $A$ becomes visible in $J_0(MN)$, i.e. 
$$
\textrm{Vis}_{J_0(MN)}(\Sha(A / \Q)) = \Sha(A / \Q)?
$$   

Why should one even bother to ask these questions? Indeed, there is a very subtle connection between 
visualizing elements of the Shafarevich-Tate group and what is called \emph{capitulation} of ideal classes. 

In fact, suppose that $L / K$ is an extension of number fields and consider the kernel of the natural map 
$C_K \ra C_L$ between the ideal class groups of $K$ and $L$. The elements of the kernel are those 
ideal classes of $\mathcal O_K$, which become trivial in $\mathcal O_L$ (we say that these ideal classes capitulate in 
$C_L$). In some sense, capitulation is the analogue of 
visibility for ideal classes. Class field theory tells us that there exists a Hilbert class field $H / K$, in which 
all ideal classes of $K$ become trivial, i.e. the whole ideal class group $C_K$ capitulates in $C_H$. It is natural to ask 
the following \\

\noindent {\bf Question 3.} Is there some analogue of the Hilbert class field $H / K$ for the case of abelian varieties?  

Since abelian varieties are in some sense much more difficult to work with than ideal classes, it might be impossible to answer 
the above question, or it might as well be that the general answer is negative. However, the connection between \emph{capitulation} 
and \emph{visibility} might be interesting to study and understand better.