Atkin-Swinnerton-Dyer Congruences on Noncongruence Modular Forms

October 21, 2010
Winnie Li | Penn State

This presentation has two parts. The first half discusses the major factorization algorithms when ECM was discovered in 1985, stressing the similarities between ECM and P +- 1. The second half describes the recent discoveries of six large Mersenne factors using ECM on a network of PlayStations.

This is joint work with Joppe W. Bos, Thorsten Kleinjung, and Arjen K. Lenstra from EPFL.

Speaker Details

The understanding for the arithmetic of modular forms for noncongruence subgroups pales when compared to that for congruence subgroups. In large part, this is due to the lack of effective Hecke operators. The first pioneering work on noncongruence modular forms was done by Atkin and Swinnerton-Dyer in 1971. Based on a handful numerical data they gathered, Atkin and Swinnerton-Dyer proposed p-adic congruence relations, similar to the recursive relation satisfied by Hecke eigenforms, to be satisfied by a basis of a given space of noncongruence cusp forms. In this talk we shall survey subsequent developments and the current status of the ASD congruences.