\chapter*{Introduction}

In the 1960s, the British mathematicians Bryan Birch and Peter Swinnerton-Dyer stated 
several interesting conjectures about the arithmetic of the elliptic curves over $\Q$, after 
doing some computations at Cambridge University. Later on, John Tate formulated more 
functorial versions of these conjectures and generalized them to abelian varieties over 
$\Q$. The most famous version of the Birch and Swinnerton-Dyer conjecture is a 
relation between analytic and arithmetic invariants of an elliptic curve (more generally, 
abelian variety). 

\begin{conj}[Birch and Swinnerton-Dyer Conjecture]
Let $E$ be an elliptic curve over $\Q$ of Mordell-Weil rank $r$
and let $L(E, s)$ be the $L$-series, attached to the elliptic curve, then 
$$
\textrm{ord}_{s = 1} L(E, s) = r. 
$$
\end{conj}   
  
There is another version of the conjecture, which was described by John Tate in 
1974 and is known as the full BSD conjecture. We will state the conjecture for 
abelian varieties over $\Q$, attached to newforms. The quantities 
that are included 
into the conjecture are the real volume $\Omega_A$ (or the canonical volume of 
$A(\R)$, the Tamagawa numbers $c_{A, p}$, the order of the Shafarevich-Tate group 
$\Sha(A/\Q)$, the order of the torsion subgroups $A(\Q)_{\textrm{tors}}$ and 
$A^\vee(\Q)_{\textrm{tors}}$ and the order of vanishing of the $L$-function of $A$. 
All of the quantities will be discussed in more details later. 

Let $A$ be an abelian variety over $\Q$, which is attached to a newform 
$\ds f = \sum_{n \geq 1}a_n q^n$ of level $N$ and let 
$\ds f^{(\sigma)} = \sum_{n \geq 1}a_n^\sigma q^n$ be the different 
Galois conjugates of $f$. We can define the $L$-function of the variety $A$ as 
$\ds L(A, s) := \prod_{\sigma : K_f \hookrightarrow \overline{\Q}} \left( \sum_{n \geq 1} 
\frac{a_n^\sigma}{n^s} \right)$.    

\begin{conj}[Full BSD Conjecture]
Assume that $L(A, s)$ does not vanish at $s = 1$. Then 
$$
\frac{L^{(r)}(A, s)}{r! \cdot \Omega_A} = \frac{|\Sha(A / K)| \cdot \prod_{p \mid N} c_{A, p} \cdot \Reg_A}
{|A(\Q)_{\tors}| \cdot |A^{\vee}(\Q)_{\tors}|}. 
$$
\end{conj}

The above conjecture assumes that $\Sha(A/\Q)$ is finite. In fact, there is a close connection 
between the Birch and Swinnerton-Dyer conjecture, and the Tate-Shafarevich conjecture 
(according to which $\Sha(A/K)$ is always 
finite for any abelian variety over a number field). 

The main goal of this thesis is to introduce and study a subgroup of the Shafarevich-Tate 
group, known as the \emph{visible subgroup} and to describe various techniques for producing  
visible elements of certain order, which in turn could provide evidence for Conjecture 2. 
We prove a theorem (the \emph{visibility theorem}), which is due to William Stein and 
Amod Agashe, which exhibits embeddings of certain weak Mordell-Weil group into 
$\Sha(A/K)$ for abelian varieties of rank zero under certain hypothesis. We prove a general 
statement, according to which every element of $\Sha(A/K)$ can be visualized somewhere. 
The proof of this statement is original and was discovered by William Stein and the author. 
Finally, we present a technique for visualizing elements by raising the level of the modular 
Jacobian. This technique is based on a theorem of K. Ribet and can be applied for abelian 
varieties, for which the visibility theorem fails. We give a computational example to 
explicitely illustrate the technique. 

This thesis project should in no case be considered to be a self-contained presentation, 
since such an exposition would have been beyond the volume of a senior thesis. As such, 
we assume basic familiarity with elliptic curves, Galois theory, Galois cohomology, 
theory of schemes. We also assume that the reader is familiar with the existence and the 
basic properties of N\'eron models of abelian varieties. We sometimes sketch the more 
technical proofs and constructions, omitting the details and refering the reader to more 
detailed references. We tried, however, to present all the important steps and ideas in the 
main results of the thesis (the visibility theorem, the visualization theorem, etc.).  
Some of the chapters (such as Chapter 1 and Chapter 3) require much less background than the 
others.