CoCalc Public Fileswww / nsf / abstract.txt
Author: William A. Stein
1TITLE:
2Explicit Approaches to Modular Forms and Modular Abelian Varieties
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4ABSTRACT:
5The Birch and Swinnerton-Dyer conjecture and Mazur's notion of
6visibility of Shafarevich-Tate groups motivate the computational and
7theoretical goals of this research project.  The computational goals
8are to develop new algorithms, tables, and software for studying
9modular forms and modular abelian varieties.  The PI hopes to create
10new computational tools, including a major new package for computing
11with modular abelian varieties over number fields, and enhance the
12modular forms database, which is used by many mathematicians who study
13modular forms.  The theoretical goals are to prove new theorems that
14relate Mordell-Weil and Shafarevich-Tate groups of elliptic curves and
15abelian varieties.  These investigations into Mazur's notion of
16visibility, and how it links Mordell-Weil and Shafarevich-Tate groups,
17may provide new insight into relationships between different cases of
18the Birch and Swinnerton-Dyer conjecture.
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20Elliptic curves and modular forms play a central role in modern number
21theory and arithmetic geometry.  For example, Andrew Wiles proved
22Fermat's Last Theorem by showing that the elliptic curve attached by
23Gerhard Frey to a counterexample to Fermat's claim would be attached
24to a modular form, and that this modular form cannot exist.  Our
25understanding of the world of elliptic curves and modular forms is
26extensive, but many questions remain unresolved. The first goal of
27this project is to provide theoretical and computational tools to make
28modular forms and objects attached to them very explicit, so that
29mathematicians can compute with them, test their conjectures on them,
30and gain a better feeling for them.  The second goal it to use these
31tools and other ideas to gain a deeper understanding of the conjecture
32of Bryan Birch and Peter Swinnerton-Dyer about the arithmetic of
33elliptic curves.
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