Exceptional Congruences for Powers of the Partition Function

In Journal of London Math. Soc.31 (1956), 350–359, Morris Newman studied vector spaces of functions arising from lifts to ₀(p) of certain eta-products on the group ₀(pQ), Q = pⁿ. In this paper, the author considers vector spaces of modular functions obtained as lifts of more general eta-products from ₀(pQ) to ₀(p), (Q, p) = 1. Specifically considered are functions arising as lifts of functions of the form,the arithmetic of which allows us to construct an infinite family of functions on ₀(p) with bounded valence. As a consequence, extensions of the exceptional congruences listed in Kiming and Olsson (Arch. Math.59 (1992), 348–360) are given. Furthermore, we obtain fairly natural criteria equivalent to the existence of an exceptional congruence. Certain other types of congruences are investigated also. Much of this paper is a revised version of chapter 3 of the author's dissertation (Stanger, Ph.D. thesis, UC Santa Barbara, June 2001).