## On the trace formula for Hecke operators on congruence subgroups

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## 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 one varies the weight or the level in the trace formula.## References

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## Additional Information

**Alexandru A. Popa**- Affiliation: Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
- MR Author ID: 792375
- Email: aapopa@gmail.com
- Received by editor(s): June 14, 2017
- Received by editor(s) in revised form: July 16, 2017
- Published electronically: March 14, 2018
- Additional Notes: This work was partly supported by the European Community grant PIRG05-GA-2009-248569 and by the CNCS grant PN-II-RU-TE-2011-3-0259. Part of this work was completed during several visits at MPIM in Bonn, whose support the author gratefully acknowledges.
- Communicated by: Ken Ono
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**146**(2018), 2749-2764 - MSC (2010): Primary 11F11, 11F25, 11F67
- DOI: https://doi.org/10.1090/proc/13896
- MathSciNet review: 3787340

Dedicated: Dedicated to Don Zagier on the occasion of his 65th birthday