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On the distribution of Satake parameters of GL2 holomorphic cuspidal representations

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Abstract

We prove that for a fixed non-archimedean place v of a totally real number field F, the traces of the associated Langlands classes of holomorphic cuspidal representations of GL2(A) with trivial central character and of prime levels is equidistributed with respect to the measure

$$ d\mu _v (x) = \frac{{q_v + 1}} {{(q_v^{1/2} + q_v^{ - 1/2} )^2 - x^2 }}d\mu _\infty (x) $$

, where q v is the norm of the prime ideal corresponding to v and dμ∞(x)=\( \tfrac{1} {\pi }\sqrt {1 - \tfrac{{x^2 }} {4}} dx \) is the Sato-Tate measure. This generalizes a result of Sarnak [Sa] on the distribution of Hecke eigenvalues of modular forms. The proof involves establishing a trace formula for the Hecke operators. While not explicit, this trace formula can be used as a starting point for generalizing the Eichler-Selberg trace formula to totally real number fields.

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  1. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong

    Charles Li

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  1. Charles Li
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Li, C. On the distribution of Satake parameters of GL2 holomorphic cuspidal representations. Isr. J. Math. 169, 341–373 (2009). https://doi.org/10.1007/s11856-009-0014-0

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