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The goal of this proposal is to carry out numerous computational and
theoretical investigations with elliptic curves and abelian varieties
motivated by the Birch and Swinnerton-Dyer conjecture.  These
investigations will hopefully improve our practical computational
capabilities, extend the data and tools that researchers have
available for formulating conjectures, and deepen our understanding of
theorems about the arithmetic of elliptic curves, abelian varieties,
and modular forms.

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\noindent{\bf Intellectual Merit:}

\mypar{}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 may lead to
new conjectures and theorems, provide substantial
new data relevant to work on the Birch and Swinnerton-Dyer
conjecture, and lead to the creation of new computational tools. 


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\noindent{\bf Broader Impact:}


\mypar{}The proposed research will likely 
result in significant improvements
to the PI's software SAGE, which is intended to be an optimal 
open source software environment
for research in algebra, geometry, number theory,
and related areas. 

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

\mypar{}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.

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