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