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Explicit Approaches to Modular Forms and Modular Abelian Varieties

The Birch and Swinnerton-Dyer conjecture and Mazur's notion of
visibility of Shafarevich-Tate groups motivate the computational and
theoretical goals of this research project.  The computational goals
are to develop new algorithms, tables, and software for studying
modular forms and modular abelian varieties.  The PI hopes to create
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.  The theoretical goals are to prove new theorems that
relate Mordell-Weil and Shafarevich-Tate groups of elliptic curves and
abelian varieties.  These investigations into Mazur's notion of
visibility, and how it links Mordell-Weil and Shafarevich-Tate groups,
may provide new insight into relationships between different cases of
the Birch and Swinnerton-Dyer conjecture.

Elliptic curves and modular forms play a central role in modern number
theory and arithmetic geometry.  For example, Andrew Wiles proved
Fermat's Last Theorem by showing that the elliptic curve attached by
Gerhard Frey to a counterexample to Fermat's claim would be attached
to a modular form, and that this modular form cannot exist.  Our
understanding of the world of elliptic curves and modular forms is
extensive, but many questions remain unresolved. The first goal of
this project is to provide theoretical and computational tools to make
modular forms and objects attached to them very explicit, so that
mathematicians can compute with them, test their conjectures on them,
and gain a better feeling for them.  The second goal it to use these
tools and other ideas to gain a deeper understanding of the conjecture
of Bryan Birch and Peter Swinnerton-Dyer about the arithmetic of
elliptic curves.