On p-adic Siegel–Eisenstein series from a point of view of the theory of mod $$p^m$$ singular forms

Abstract

We show that the p-adic Siegel–Eisenstein series of general degree attached to two kind of number sequences are both linear combinations of genus theta series of level dividing p, by applying the theory of mod p-power singular forms. As special cases of this result, we derive the results of Nagaoka and Katsurada–Nagaoka.