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