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Vanishing and non-vanishing of traces of Hecke operators

Author: Jeremy Rouse
Journal: Trans. Amer. Math. Soc. 358 (2006), 4637-4651
MSC (2000): Primary 11F25; Secondary 11F72
Published electronically: May 9, 2006
MathSciNet review: 2231391
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Abstract: Using a reformulation of the Eichler-Selberg trace formula, due to Frechette, Ono and Papanikolas, we consider the problem of the vanishing (resp. non-vanishing) of traces of Hecke operators on spaces of even weight cusp forms with trivial Nebentypus character. For example, we show that for a fixed operator and weight, the set of levels for which the trace vanishes is effectively computable. Also, for a fixed operator the set of weights for which the trace vanishes (for any level) is finite. These results motivate the ``generalized Lehmer conjecture'', that the trace does not vanish for even weights $ 2k \geq 16$ or $ 2k = 12$.

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Additional Information

Jeremy Rouse
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706

Received by editor(s): July 15, 2004
Received by editor(s) in revised form: November 8, 2004
Published electronically: May 9, 2006
Additional Notes: This research was supported by the NDSEG Fellowship Program, which is sponsored by the Department of Defense and the Office of Naval Research.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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