On Siegel modular forms of level p and their properties mod p

Abstract

Using theta series we construct Siegel modular forms of level p which behave well modulo $p$ in all cusps. This construction allows us to show (under a mild condition) that all Siegel modular forms of level p and weight 2 are congruent mod $p$ to level one modular forms of weight $p+1$; in particular, this is true for Yoshidal lifts of level $p$.