Periodicity of traces of Hecke operators modulo prime powers

TL;DR

This paper investigates the periodicity of traces of Hecke operators on elliptic and Drinfeld cusp forms modulo prime powers, revealing a new regularity in their behavior across weights.

Contribution

It demonstrates that traces of Hecke operators are periodic in the weight modulo any prime power, a novel insight into their arithmetic properties.

Findings

Traces are periodic modulo prime powers.

Periodicity holds for both elliptic and Drinfeld cusp forms.

Provides new understanding of Hecke operator traces in modular forms.

Abstract

We study traces of Hecke operators on spaces of elliptic cusp forms and Drinfeld cusp forms and show that, modulo any prime power, these traces are periodic in the weight.