Eisenstein series modulo prime powers

Paper

Abstract

is prime and

is an even integer with

we consider the Eisenstein series

modulo powers of

. It is classically known that for such

we have

. Here we obtain a generalization modulo prime powers

by giving an expression for

in terms of modular forms of weight at most

. As an application we extend a recent result of the first author with Hanson, Raum and Richter by showing that, modulo powers of

, every such Eisenstein series is congruent modulo

to a modular form of weight at most

. We prove a similar result for the normalized Eisenstein series

in the case that

and