On the trace formula for Hecke operators on congruence subgroups

2014/08/21 by Alexandru A. Popa, Popa, Alexandru A.
Mathematics · #11F11 #11F25 #11F67 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1408.4998

openalex publication_date 2014/08/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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