Spectra of lens spaces from 1-norm spectra of congruence lattices

Abstract

To every n-dimensional lens space L, we associate a congruence lattice L in ℤm, with n=2m - 1 and we prove a formula relating the multiplicities of Hodge-Laplace eigenvalues on L with the number of lattice elements of a given ||·||1-length in l. As a consequence, we show that two lens spaces are isospectral on functions (respectively, isospectral on p-forms for every p) if and only if the associated congruence lattices are ||·||1-isospectral (respectively, ||·||1-isospectral plus a geometric condition). Using this fact, we give, for every dimension n≥ 5, infinitely many examples of Riemannian manifolds that are isospectral on every level p and are not strongly isospectral.