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\begin{quote}
The object of numerical computation is theoretical advance.\\
\hfill --{\em A.\thinspace{}O.\thinspace{}L. Atkin, see~\cite{birch:atkin}}
\end{quote}

The definition of the spaces of modular 
forms as functions on the upper half plane satisfying 
a certain equation is very abstract.  The definition 
of the Hecke operators\index{Hecke operators}  even more so.
Nevertheless, one wishes to carry out explicit 
investigations into these objects. 

We are fortunate that we now have methods available that allow us to
transform the vector space of cusp forms of given weight and level
into a concrete object, which can be explicitly computed.  We have the
work of Atkin-Lehner, Birch, Swinnerton-Dyer, Manin, Merel, and many
others to thank for this (see \cite{antwerpiv, cremona:algs,
merel:1585}).  For example, the Eichler-Selberg trace formula, as
extended in \cite{hijikata:trace}, can be used to compute
characteristic polynomials of Hecke operators.\index{Hecke operators}   One can compute Hecke operators\index{Hecke operators} 
 using Brandt matrices and quaternion algebras
\cite{kohel:hecke, pizer:alg}; another closely related method involves
the module of enhanced supersingular elliptic
curves~\cite{mestre:graphs}.  In the course of computing large tables
of invariants of elliptic curves in \cite{cremona:algs}, Cremona
demonstrated the power of systematic computation using modular
symbols.

Various methods often must be used in concert to obtain information
about the package of invariants attached to a modular form.  For
example, computing orders of component groups of optimal quotients of
$J_0(N)$ involves computations on the module of supersingular elliptic
curves combined with modular symbols techniques (see
Chapter~\ref{chap:compgroups}).

Chapter~\ref{chap:bsd} is an attempt to systematically prove the Birch
and Swinnerton-Dyer conjecture for a certain finite list of rank-$0$
quotients of $J_0(N)$ that have nontrivial Shafarevich-Tate groups.
The key idea is to use Barry Mazur's notion of visibility, coupled
with explicit computations, to produce lower bounds on the
Shafarevich-Tate group.  I have not finished the proof of the
conjecture in these examples; this would require computing explicit
upper bounds on the order of this group.  However, I obtain explicit
formulas and data that will be helpful in further investigations.

The following three chapters describe the algorithms used in
Chapter~\ref{chap:bsd}, along with generalizations to eigenforms on
$\Gamma_1(N)$ of integral weight greater than two.  I have used these
algorithms to investigate the Artin
Conjecture~\cite{buzzard-stein:artin}, Serre's conjecture, and many
other problems not described in this thesis.  I have implemented
most of the algorithms
that are described in Chapters~\ref{chap:modsym}--\ref{chap:compgroups} 
in both \magma{} and {\sf C++}; this implementation should be available
in the standard release of \magma{} in versions 2.7 and greater.
\vspace{1ex}

\par\noindent
William A. Stein\\
University of California, Berkeley







