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\noindent{}The Birch and Swinnerton-Dyer conjecture and Mazur's
notion of visibility of Shafarevich-Tate groups motivate the
computational and theoretical goals of this proposal.  The
computational goals are to develop new algorithms, tables, and
software for studying modular forms and modular abelian varieties.
The theoretical goals are to prove new theorems that relate
Mordell-Weil and Shafarevich-Tate groups of elliptic curves and
abelian varieties.

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\noindent{\bf Intellectual Merit:}

\noindent{}The PI is one of the more sought-after people by the
worldwide community of number theorists, for computational
confirmation of conjectures, for modular forms algorithms, for
data, and for ways of formulating problems so as to make them more
accessible to algorithms.   This project will provide new
computational tools, including a major new package for computing
with modular abelian varieties over number fields, and enhance the
modular forms database, which is used by many mathematicians who
study modular forms.

\mypar{}The PI's investigations into Mazur's notion of visibility,
and how it links Mordell-Weil and Shafarevich-Tate groups, may
provide new insight into implications between cases of the Birch
and Swinnerton-Dyer conjecture.

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\noindent{\bf Broader Impact:}

\noindent{}The PI intends to complete the undergraduate textbook
{\sl Elementary Number Theory and Elliptic Curves}, which he is
writing under contract with Springer-Verlag. He also intends to
finish the graduate textbook {\sl Lectures on Modular Forms and
Hecke Operators}, which he is co-authoring with Ken Ribet, and
which is likely to be published by Springer-Verlag. These
textbooks distinguish themselves from similar titles by
incorporating specific knowledge and intuition gathered by the PI
from his past numerical investigations.

\mypar{}Continued development of his computational programs
promises to have a broader impact on number theory, because his
software and the modular forms database are standard tools for
obtaining data about modular forms and associated objects.

\mypar{}Elliptic curves and abelian varieties over finite fields
are used in everyday public-key cryptography.  Though no specific
applications to cryptography are given in this proposal, the PI
hopes the tools and data that he develops will provide input to
researchers who are analyzing and designing new cryptosystems.
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