Abstract:
Ramanujan's classic congruence families were the first important arithmetic properties that were discovered for the partition function p(n).  Since then, we have discovered that similar properties are exhibited by the Fourier coefficients of various different modular forms.  Some of these are much more difficult to prove than others; the most difficult families are associated with modular curves with a high cusp count.  We will show an altogether new proof method that was recently applied to one of the more difficult congruence families, exhibited by certain generalized Frobenius partitions, and associated with the modular functions on the curve X_0(20). The idea is that one constructs an automorphism on a certain free Z[t]-module R of functions which permutes the generators of R while fixing the functions on the curve X_0(5). To our knowledge, this is an entirely new approach to the problem of proving p-adic convergence of modular function sequences, and the implications for future work are enormous.  This is joint work with Frank Garvan and James A. Sellers.