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 elliptic curves and modular forms is extensive, but many questions remain unresolved.
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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