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\Large\bf
Explicit Approaches to Elliptic Curves and Modular
Abelian Varieties
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\noindent{}{\bf Intellectual Merit:}
The projects in this proposal would generalize the highly influential
tables of Cremona to the next (ordered by discriminant) totally real
number field and to higher dimensional modular abelian varieties.
This would improve on algorithms available for computing with elliptic
curves and abelian varieties, and provide useful data and tools for
number theory researchers. The proposed research would also advance
techniques for constructing points and cohomology classes on elliptic
curves, and for understanding the arithmetic of elliptic curves over
number fields. This project would have as a concrete deliverable new
publicly available tables and software that will be of use to many
number theorists.
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\noindent{}{\bf Broader Impact:} The PI is co-authoring a popular
expository book with Barry Mazur on the Riemann Hypothesis,
co-authoring a graduate level book with Kenneth Ribet on modular forms
and Hecke operators, and intends to release a new edition of his
modular forms book. The PI has tables of data that are freely
available online, and whose creation has been supported by NSF FRG
grant DMS-0757627, and the proposed research would expand these tables
further. He will also continue to organize the development of the
free open source NSF-funded Sage mathematical software project that he
started. The PI organizes dozens of ``Sage Days'' workshops that
involve many undergraduate and graduate students, and touch on number
theory, algebraic topology, combinatorics, special functions,
numerical computation, and other areas. The PI is also a co-PI on the
UTMOST NSF grant (DUE-1020378), whose goal is to make Sage more
accessible to high school and college teachers and students.
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