Vanishing and non-vanishing of traces of Hecke operators

Rouse, Jeremy

doi:10.1090/S0002-9947-06-03896-7

Vanishing and non-vanishing of traces of Hecke operators
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Trans. Amer. Math. Soc. 358 (2006), 4637-4651 Request permission

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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