Equidistribution of the eigenvalues of Hecke operators

Key Points

The equidistribution of Hecke eigenvalues is proven, showing a significant property of Maass forms over any number field.
At a fixed prime ideal, the observed proportion of Maass forms failing the Ramanujan-Petersson conjecture is exactly 0.
The method involves analyzing Hecke operators in the context of the Sato-Tate measure, showcasing their relevance in number theory.
This research highlights the interplay between eigenvalues and conjectural frameworks in the study of modular forms.

Abstract

Abstract In this paper, we prove the equidistribution of the Hecke eigenvalues of Maass forms over an arbitrary number field at a fixed prime ideal, with respect to the Sato–Tate measure. As an application, we obtain that the proportion of Maass forms that do not satisfy the Ramanujan–Petersson conjecture at a fixed prime ideal is 0.

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Equidistribution of the eigenvalues of Hecke operators

Key Points

Abstract

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