On the occurrence of Hecke eigenvalues and a lacunarity question of Serre

Let $\pi$ be a unitary cuspidal automorphic representation for $\mathrm{GL}(n)$ over a number field. We establish upper bounds on the number of Hecke eigenvalues of $\pi$ equal to a fixed complex number. For $\mathrm{GL}(2)$, we also determine upper bounds on the number of Hecke eigenvalues with absolute value equal to a fixed number $\gamma$; in the case $\gamma = 0$, this answers a question of Serre. These bounds are then improved upon by restricting to non-dihedral representations. Finally, we obtain analogous bounds for a family of cuspidal automorphic representations for $\mathrm{GL}(3)$.

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Published 2 April 2015