\documentclass[11pt]{article}%1\newcommand{\thisdocument}{Project Summary}%2\include{macros}3\newcommand{\mypar}{\vspace{1ex}\par\noindent{}}45\begin{document}6\noindent{}The Birch and Swinnerton-Dyer conjecture and Mazur's7notion of visibility of Shafarevich-Tate groups motivate the8computational and theoretical goals of this proposal. The9computational goals are to develop new algorithms, tables, and10software for studying modular forms and modular abelian varieties.11The theoretical goals are to prove new theorems that relate12Mordell-Weil and Shafarevich-Tate groups of elliptic curves and13abelian varieties.1415\vspace{2ex}1617\noindent{\bf Intellectual Merit:}1819\noindent{}The PI is one of the more sought-after people by the20worldwide community of number theorists, for computational21confirmation of conjectures, for modular forms algorithms, for22data, and for ways of formulating problems so as to make them more23accessible to algorithms. This project will provide new24computational tools, including a major new package for computing25with modular abelian varieties over number fields, and enhance the26modular forms database, which is used by many mathematicians who27study modular forms.2829\mypar{}The PI's investigations into Mazur's notion of visibility,30and how it links Mordell-Weil and Shafarevich-Tate groups, may31provide new insight into implications between cases of the Birch32and Swinnerton-Dyer conjecture.3334\vspace{2ex}3536\noindent{\bf Broader Impact:}3738\noindent{}The PI intends to complete the undergraduate textbook39{\sl Elementary Number Theory and Elliptic Curves}, which he is40writing under contract with Springer-Verlag. He also intends to41finish the graduate textbook {\sl Lectures on Modular Forms and42Hecke Operators}, which he is co-authoring with Ken Ribet, and43which is likely to be published by Springer-Verlag. These44textbooks distinguish themselves from similar titles by45incorporating specific knowledge and intuition gathered by the PI46from his past numerical investigations.4748\mypar{}Continued development of his computational programs49promises to have a broader impact on number theory, because his50software and the modular forms database are standard tools for51obtaining data about modular forms and associated objects.5253\mypar{}Elliptic curves and abelian varieties over finite fields54are used in everyday public-key cryptography. Though no specific55applications to cryptography are given in this proposal, the PI56hopes the tools and data that he develops will provide input to57researchers who are analyzing and designing new cryptosystems.58\end{document}5960