Joint distribution of eigenvalues of Hecke and Casimir operators for Hilbert Maass forms

Abstract

Let $F$ be a totally real number field, ${\mathcal{O}}_F$ the ring of integers, $\mathfrak{a}$ and $\mathfrak{I}$ integral ideals and let $\chi$ a character of ${\mathbb{A}}_F^{\times }/F^{\times }$. For each prime ideal $\mathfrak{p}$ in ${\mathcal{O}}_F$, $\mathfrak{p}\nmid \mathfrak{I}$ let $T_{\mathfrak{p}}$ be the Hecke operator acting on the space of Maass cusp forms on $L^2({\mathrm{GL}}_2(F)\backslash {\mathrm{GL}}_2({\mathbb{A}}_F))$. In this paper we investigate the distribution of joint eigenvalues of the Hecke operators $T_{\mathfrak{p}}$ and of the Casimir operators $C_j$ in each archimedean component of $F$, for $1\le j\le d$. Summarily, we prove that given a family of expanding compact subsets ${\Omega }_t$ of ${\mathbb{R}}^d$ as $t\to \infty$, and an interval $I_{\mathfrak{p}}\subseteq [-2,2]$, then, if $\mathfrak{p}\nmid \mathfrak{I}$ is a square in the narrow class group of $F$, there are infinitely many automorphic forms having eigenvalues of $T_{\mathfrak{p}}$ in $I_{\mathfrak{p}}$, distributed on $I_{\mathfrak{p}}$ according to a polynomial multiple of the Sato-Tate measure and having their Casimir eigenvalues in the region ${\Omega }_t$, distributed according to the Plancherel measure.

References in corpus (1)

• Twisted L-functions over number fields and Hilbert's eleventh problem