Density results for automorphic forms on Hilbert modular groups II

TL;DR

This paper derives an asymptotic formula for weighted sums of cuspidal eigenvalues of automorphic forms over totally real fields, demonstrating the existence of infinitely many automorphic representations with eigenvalues in expanding regions.

Contribution

It provides a new asymptotic formula for eigenvalues of automorphic forms on Hilbert modular groups, extending previous results to more general regions and settings.

Findings

Existence of infinitely many automorphic representations with multi-eigenvalues in growing regions.

Asymptotic formula for weighted sums of cuspidal eigenvalues.

Application of a Kuznetsov-type sum formula to derive results.

Abstract

We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for $\SL_2$ over a totally real number field $F$, with discrete subgroup of Hecke type $\Gamma_0(I)$ for a non-zero ideal $I$ in the ring of integers of $F$. The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many cuspidal automorphic representations with multi-eigenvalues in various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips and products of prescribed small intervals for all but one of the infinite places of $F$. The main tool in the derivation is a sum formula of Kuznetsov type.