William A. Stein Project Summary

(617) 308-0144  was@math.harvard.edu  http://modular.fas.harvard.edu

The Birch and Swinnerton-Dyer conjecture and Mazur’s notion of visibility of Shafarevich-Tate groups motivate the computational and theoretical goals of this proposal. The computational goals are to develop new algorithms, tables, and software for studying modular forms and modular abelian varieties. The theoretical goals are to prove new theorems that relate Mordell-Weil and Shafarevich-Tate groups of elliptic curves and abelian varieties.

Intellectual Merit: The PI is one of the more sought-after people by the worldwide community of number theorists, for computational confirmation of conjectures, for modular forms algorithms, for data, and for ways of formulating problems so as to make them more accessible to algorithms. This project will provide 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 PI’s investigations into Mazur’s notion of visibility, and how it links Mordell-Weil and Shafarevich-Tate groups, may provide new insight into implications between cases of the Birch and Swinnerton-Dyer conjecture.

Broader Impact: The PI intends to complete the undergraduate textbook Elementary Number Theory and Elliptic Curves, which he is writing under contract with Springer-Verlag. He also intends to finish the graduate textbook Lectures on Modular Forms and Hecke Operators, which he is co-authoring with Ken Ribet, and which is likely to be published by Springer-Verlag. These textbooks distinguish themselves from similar titles by incorporating specific knowledge and intuition gathered by the PI from his past numerical investigations.

Continued development of his computational programs promises to have a broader impact on number theory, because his software and the modular forms database are standard tools for obtaining data about modular forms and associated objects.

Elliptic curves and abelian varieties over finite fields are used in everyday publickey cryptography. Though no specific applications to cryptography are given in this proposal, the PI hopes the tools and data that he develops will provide input to researchers who are analyzing and designing new cryptosystems.