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The nonvanishing of the trace of $T_3$

Liubomir Chiriac, Daphne Kurzenhauser and Erin Williams

Vol. 17 (2024), No. 2, 263–272
Abstract

A generalized Lehmer conjecture predicts that, for every positive integer n, the trace of the Hecke operator Tn in level 1 does not vanish, unless the space of cusp forms acted upon is trivial. So far, this has only been established for n = 2. We use p-adic methods to prove the statement for n = 3.

Keywords
modular forms, Hecke operators, trace formula
Mathematical Subject Classification
Primary: 11F30
Secondary: 11B37, 11F85
Milestones
Received: 28 June 2022
Revised: 15 September 2022
Accepted: 7 February 2023
Published: 20 May 2024

Communicated by Kenneth S. Berenhaut
Authors
Liubomir Chiriac
Fariborz Maseeh Department of Mathematics and Statistics
Portland State University
Portland, OR
United States
Daphne Kurzenhauser
Fariborz Maseeh Department of Mathematics and Statistics
Portland State University
Portland, OR
United States
Erin Williams
Fariborz Maseeh Department of Mathematics and Statistics
Portland State University
Portland, OR
United States