On Hilbert modular forms modulo $p$: explicit ring structure

Abstract

H.P.F. Swinnerton-Dyer determined the structure of the ring of modular forms modulo $p$ in the elliptic modular case. In this paper, the structure of the ring of Hilbert modular forms modulo $p$ is studied. In the case where the discriminant of corresponding quadratic field is 8 (or 5), the explicit structure is determined.