Eigenvalues of Hecke operators on Hilbert modular groups

TL;DR

This paper investigates the distribution of eigenvalues of Casimir operators and Hecke operators for cuspidal automorphic representations of Hilbert modular groups, revealing their joint distribution as a product of known measures.

Contribution

It provides a detailed analysis of the joint distribution of eigenvalues for automorphic forms on Hilbert modular groups, connecting them to Plancherel and Sato-Tate measures.

Findings

Eigenvalues follow the product of Plancherel and Sato-Tate measures.

Joint distribution of eigenvalues is characterized explicitly.

Results extend understanding of automorphic representation spectra.

Abstract

We consider cuspidal representations in spaces of automorphic forms for the congruence subgroup $\Gamma_0(I)$ of Hilbert modular groups for some number field $F$. To each such representation are associated the eigenvalue $\lambda_j$ of the Casimir operator at each real place $j$ of $F$, and the number $\ld_{\mathfrak p}$ parametrizing the eigenvalue of the Hecke operator $T_{\mathfrak p^2}$ at each finite place $\mathfrak p$ outside the ideal $I$. We study the joint distribution of the $\lambda_j$ for all real places $j$, and the $\ld_{\mathfrak p}$ for finitely many $\mathfrak p$ outside $I$, over the cuspidal representations. This distribution is given by the product of the Plancherel measure at each real place and the Sato-Tate measure at each finite place.