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

In a previous work, we applied lattice point theorems on hyperbolic spaces to obtain 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. That formula involved an error term that depended 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 provide evidence of the existence of exceptional eigenvalues for some arithmetic subgroups of \(\mathrm {SU}(3,1)\) , \(\mathrm {SU}(4,1)\) , and \(\mathrm {SU}(5,1)\) , and thus they contradict the generalized Selberg (and Ramanujan) conjecture in these cases. Furthermore, for several arithmetic subgroups of \(\mathrm {SO}(4,1)\) , \(\mathrm {SO}(6,1)\) , \(\mathrm {SO}(8,1)\) , and \(\mathrm {SU}(2,1)\) , there is evidence of a lower bound on the first nonzero eigenvalue that is better than the already known lower bound for congruences subgroups.