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 is 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.
