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4\chapter*{Preface}
5\markboth{}{}
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8
9\begin{quote}
10The object of numerical computation is theoretical advance.\\
11\hfill --{\em A.\thinspace{}O.\thinspace{}L. Atkin, see~\cite{birch:atkin}}
12\end{quote}
13
14The definition of the spaces of modular
15forms as functions on the upper half plane satisfying
16a certain equation is very abstract.  The definition
17of the Hecke operators\index{Hecke operators}  even more so.
18Nevertheless, one wishes to carry out explicit
19investigations into these objects.
20
21We are fortunate that we now have methods available that allow us to
22transform the vector space of cusp forms of given weight and level
23into a concrete object, which can be explicitly computed.  We have the
24work of Atkin-Lehner, Birch, Swinnerton-Dyer, Manin, Merel, and many
25others to thank for this (see \cite{antwerpiv, cremona:algs,
26merel:1585}).  For example, the Eichler-Selberg trace formula, as
27extended in \cite{hijikata:trace}, can be used to compute
28characteristic polynomials of Hecke operators.\index{Hecke operators}   One can compute Hecke operators\index{Hecke operators}
29 using Brandt matrices and quaternion algebras
30\cite{kohel:hecke, pizer:alg}; another closely related method involves
31the module of enhanced supersingular elliptic
32curves~\cite{mestre:graphs}.  In the course of computing large tables
33of invariants of elliptic curves in \cite{cremona:algs}, Cremona
34demonstrated the power of systematic computation using modular
35symbols.
36
37Various methods often must be used in concert to obtain information
38about the package of invariants attached to a modular form.  For
39example, computing orders of component groups of optimal quotients of
40$J_0(N)$ involves computations on the module of supersingular elliptic
41curves combined with modular symbols techniques (see
42Chapter~\ref{chap:compgroups}).
43
44Chapter~\ref{chap:bsd} is an attempt to systematically prove the Birch
45and Swinnerton-Dyer conjecture for a certain finite list of rank-$0$
46quotients of $J_0(N)$ that have nontrivial Shafarevich-Tate groups.
47The key idea is to use Barry Mazur's notion of visibility, coupled
48with explicit computations, to produce lower bounds on the
49Shafarevich-Tate group.  I have not finished the proof of the
50conjecture in these examples; this would require computing explicit
51upper bounds on the order of this group.  However, I obtain explicit
52formulas and data that will be helpful in further investigations.
53
54The following three chapters describe the algorithms used in
55Chapter~\ref{chap:bsd}, along with generalizations to eigenforms on
56$\Gamma_1(N)$ of integral weight greater than two.  I have used these
57algorithms to investigate the Artin
58Conjecture~\cite{buzzard-stein:artin}, Serre's conjecture, and many
59other problems not described in this thesis.  I have implemented
60most of the algorithms
61that are described in Chapters~\ref{chap:modsym}--\ref{chap:compgroups}
62in both \magma{} and {\sf C++}; this implementation should be available
63in the standard release of \magma{} in versions 2.7 and greater.
64\vspace{1ex}
65
66\par\noindent
67William A. Stein\\
68University of California, Berkeley
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