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Author: William A. Stein
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% intro.tex
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\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}
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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 even more so.
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Nevertheless, one wishes to carry out explicit
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computational investigations on these objects.
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% survey of methods.
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We are fortunate that we now have methods available which
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allow us to transform the vector space of cusp forms of given weight
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and level into a concrete object, which can be
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explicitly computed. We have the work of Atkin-Lehner,
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Birch, Swinnerton-Dyer, Manin, Merel, and many others to thank for this
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(see \cite{antwerpiv, cremona:algs, merel:1585}).
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The Eichler-Selberg trace formula, as extended
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in \cite{hijikata:trace}, can be used to compute
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characteristic polynomials of
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Hecke operators and hence gain some information about
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spaces of modular forms. It is also possible to compute
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Hecke operators and $q$-expansions using Brandt matrices and
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quaternion algebras as in \cite{kohel:hecke, pizer:alg};
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%pizer:arithmetic, pizer:arithmetictwo, pizer:rep};
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another
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closely related method relies on the module
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of enhanced supersingular elliptic curves as exploited by
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Mestre-Oesterl\'{e}~\cite{mestre:graphs} and extended
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by Edixhoven in his Ph.D. thesis.
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I take the approach of Cremona's book
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\cite{cremona:algs} and view modular symbols as central.
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No single computational method provides complete information
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about the package of invariants attached to a modular form.
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The various methods must be organized into a symphony;
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for example, the Mestre-Oesterl\'e method combined with
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modular symbols can be used to compute component groups.
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In this thesis, I present methods for explicitly
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computing with modular forms, Hecke operators, and the abelian
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varieties attached to them. In Chapter~\ref{chap:bsd},
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I give an example of how to coordinate these methods to
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gain data about real world problems.
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Chapters~\ref{chap:modsym} and~\ref{chap:computing}
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owe much to the publications of Merel and Cremona.
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