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On the trace formula for Hecke operators on congruence subgroups


Author: Alexandru A. Popa
Journal: Proc. Amer. Math. Soc. 146 (2018), 2749-2764
MSC (2010): Primary 11F11, 11F25, 11F67
DOI: https://doi.org/10.1090/proc/13896
Published electronically: March 14, 2018
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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.


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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
Email: aapopa@gmail.com

DOI: https://doi.org/10.1090/proc/13896
Keywords: Trace formula, Hecke operators, holomorphic modular forms, period polynomials
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.
Dedicated: Dedicated to Don Zagier on the occasion of his 65th birthday
Communicated by: Ken Ono
Article copyright: © Copyright 2018 American Mathematical Society

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