Explicit Approaches to Modular Forms and Modular Abelian Varieties

点击翻译按钮获取中文摘要

ABSTRACT for the award DMS-0400386 of Stein The Birch and Swinnerton-Dyer conjecture and Mazur's notion ofvisibility of Shafarevich-Tate groups motivate the computational andtheoretical goals of this research project. The computational goalsare to develop new algorithms, tables, and software for studyingmodular forms and modular abelian varieties. The PI hopes to createnew computational tools, including a major new package for computingwith modular abelian varieties over number fields, and enhance themodular forms database, which is used by many mathematicians who studymodular forms. The theoretical goals are to prove new theorems thatrelate Mordell-Weil and Shafarevich-Tate groups of elliptic curves andabelian varieties. These investigations into Mazur's notion ofvisibility, and how it links Mordell-Weil and Shafarevich-Tate groups,may provide new insight into relationships between different cases ofthe Birch and Swinnerton-Dyer conjecture.Elliptic curves and modular forms play a central role in modern numbertheory and arithmetic geometry. For example, Andrew Wiles provedFermat's Last Theorem by showing that the elliptic curve attached byGerhard Frey to a counterexample to Fermat's claim would be attachedto a modular form, and that this modular form cannot exist. Ourunderstanding of the world of elliptic curves and modular forms isextensive, but many questions remain unresolved. The first goal ofthis project is to provide theoretical and computational tools to makemodular forms and objects attached to them very explicit, so thatmathematicians can compute with them, test their conjectures onthem, and gain a better feeling for them. The second goal is to usethese tools and other ideas to gain a deeper understanding of theconjecture of Bryan Birch and Peter Swinnerton-Dyer about the arithmetic of elliptic curves.