CoCalc Public Fileswww / papers / thesis-old / preface.tex
Author: William A. Stein
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1% intro.tex
2\chapter*{Preface}
3
4
5\begin{quote}
6The object of numerical computation is theoretical advance.\\
7\hfill --{\em A.\thinspace{}O.\thinspace{}L. Atkin}
8\end{quote}
9
10The definition of the spaces of modular
11forms as functions on the upper half plane satisfying
12a certain equation is very abstract.  The definition
13of the Hecke operators even more so.
14Nevertheless, one wishes to carry out explicit
15computational investigations on these objects.
16
17% survey of methods.
18We are fortunate that we now have methods available which
19allow us to transform the vector space of cusp forms of given weight
20and level into a concrete object, which can be
21explicitly computed.  We have the work of Atkin-Lehner,
22Birch, Swinnerton-Dyer, Manin, Merel, and many others  to thank for this
23(see \cite{antwerpiv, cremona:algs, merel:1585}).
24The Eichler-Selberg trace formula, as extended
25in \cite{hijikata:trace},  can be used to compute
26characteristic polynomials of
27Hecke operators and hence gain some information about
28spaces of modular forms. It is also possible to compute
29Hecke operators and $q$-expansions using Brandt matrices and
30quaternion algebras as in \cite{kohel:hecke, pizer:alg};
31%pizer:arithmetic, pizer:arithmetictwo, pizer:rep};
32another
33closely related method relies on the module
34of enhanced supersingular elliptic curves as exploited by
35Mestre-Oesterl\'{e}~\cite{mestre:graphs} and extended
36by Edixhoven in his Ph.D. thesis.
37I take the approach of Cremona's book
38\cite{cremona:algs} and view modular symbols as central.
39
40No single computational method provides complete information
41about the package of invariants attached to a modular form.
42The various methods must be organized into a symphony;
43for example, the Mestre-Oesterl\'e method combined with
44modular symbols can be used to compute component groups.
45
46In this thesis, I present methods for explicitly
47computing with modular forms, Hecke operators, and the abelian
48varieties attached to them.  In Chapter~\ref{chap:bsd},
49I give an example of how to coordinate these methods to
50gain data about real world problems.
51
52Chapters~\ref{chap:modsym} and~\ref{chap:computing}
53owe much to the publications of Merel and Cremona.
54