On p-adic properties of Siegel modular forms

TL;DR

This paper proves that Siegel modular forms of certain levels are p-adic and that their derivatives are also p-adic, extending Serre's methods to the Siegel case with some modifications.

Contribution

It demonstrates that Siegel modular forms of level (p^m) are p-adic and that derivatives preserve this property, extending p-adic modular form theory.

Findings

Siegel modular forms of level (p^m) are p-adic

Derivatives of these forms are also p-adic

Results partially apply to vector-valued modular forms

Abstract

We show that Siegel modular forms of level \Gamma_0(p^m) are p-adic modular forms. Moreover we show that derivatives of such Siegel modular forms are p-adic. Parts of our results are also valid for vector-valued modular forms. In our approach to p-adic Siegel modular forms we follow Serre closely; his proofs however do not generalize to the Siegel case or need some modifications.