On the trace formula for Hecke operators on congruence subgroups

TL;DR

This paper presents a simplified proof of the trace formula for Hecke operators on modular forms for congruence subgroups, utilizing algebraic properties of universal Hecke operators and extending previous approaches.

Contribution

It introduces a new, straightforward proof method for the trace formula that generalizes earlier work and simplifies calculations for various weights and levels.

Findings

Derived a simple trace formula for cusp forms and modular forms

Analyzed the effects of varying weight and level on the trace formula

Extended previous approaches to a broader class of subgroups

Abstract

We give a new, simple proof of the trace formula for Hecke operators on modular forms for finite index subgroups of the modular group. The proof uses algebraic properties of certain universal Hecke operators acting on period polynomials of modular forms, and it generalizes an approach developed by Don Zagier and the author for the modular group. This approach leads to a very simple formula for the trace on the space of cusp forms plus the trace on the space of modular forms. As applications, we investigate what happens when varying the weight or the level in the trace formula.