Congruences of Swinnerton-Dyer Type (iii) for the Fourier coefficients of modular forms
by
Basil Gordon
UCLA

Let f(\tau) be a holomorphic modular form of weight k on \Gamma(N) with Fourier expansion f(\tau)=\sum_n=1^\infty a(n)e^{2\pii n\tau}, a(n) in Z. Swinnerton-Dyer determined five types of congruences for a(n) modulo a prime p. When N=1, only types (i), (ii), and (iii) can occur. He conjectured one of type (iii) with f(\tau)=\Delta(\tau)E_r(\tau) and p=59; this was later proved by Haberland and others. The present talk gives a way to prove such congruences without Galois cohomology and presents several new examples with other forms and other primes; they have N > 1.

Date received: April 25, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caew-73.