Spectra of lens spaces from 1-norm spectra of congruence lattices
Emilio A. Lauret, Roberto J. Miatello, Juan Pablo Rossetti · Repositorio Digital de la UNC (National University of Cordoba) · 2016
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 (resp. isospectral on p-forms for every p) if and only if the associated congruence lattices are ‖·‖1-isospectral (resp. ‖·‖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.