A numerical study on exceptional eigenvalues of certain congruence subgroups of SO(n,1) and SU(n,1)

TL;DR

This paper numerically investigates eigenvalues of certain hyperbolic space subgroups, providing evidence of exceptional eigenvalues that challenge existing conjectures and suggesting improved bounds for eigenvalues in some cases.

Contribution

It offers the first numerical comparison of error terms related to eigenvalues in hyperbolic manifolds, revealing potential exceptions to the generalized Selberg and Ramanujan conjectures.

Findings

Evidence of exceptional eigenvalues in some SU(n,1) subgroups.

Contradiction of the generalized Selberg and Ramanujan conjectures in specific cases.

Improved lower bounds on the first nonzero eigenvalues for certain SO(n,1) and SU(n,1) subgroups.

Abstract

In a previous work we apply lattice point theorems on hyperbolic spaces obtaining asymptotic formulas for the number of integral representations of negative integers by quadratic and hermitian forms of signature (n,1) lying in Euclidean balls of increasing radius. This formula involves an error term that depends on the first nonzero eigenvalue of the Laplace-Beltrami operator on the corresponding congruence hyperbolic manifolds. The aim of this paper is to compare the error term obtained by experimental computations, with the error term mentioned above, for several choices of quadratic and hermitian forms. Our numerical results give evidences of the existence of exceptional eigenvalues for some arithmetic subgroups of SU(3,1), SU(4,1) and SU(5,1), thus they contradicts the generalized Selberg (and Ramanujan) conjecture in these cases. Furthermore, for several arithmetic subgroups of SO(4,1), SO(6,1), SO(8,1) and SU(2,1), there are evidences of a lower bound on the first nonzero eigenvalue better than the already known lower bound for congruences subgroups.