Hilbert Modular Forms Modulo pm: The Unramified Case

Abstract

This paper is about Hilbert modular forms on certain Hilbert modular varieties associated with a totally real field L. Let p be unramified in L. We reduce to the inert case and consider modular forms modulo pm. We study the ideal of modular forms with q-expansion equal to zero modulo pm, find canonical elements in it, and obtain as a corollary the congruences for the values of the zeta function of L at negative integers. Our methods are geometric and also have applications to lifting of Hilbert modular forms and compactification of certain modular varieties.