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

ROSSETTI, JUAN PABLO; ROSSETTI, JUAN PABLO; LAURET, EMILIO; LAURET, EMILIO; MIATELLO, ROBERTO J.; MIATELLO, ROBERTO J.

INTERNATIONAL MATHEMATICS RESEARCH NOTICES

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2016 vol. 2016 p. 1054 - 1089

1073-7928

To every n-dimensional lens space L, we associate a congruence lattice $mathcal L$ in Z^m, 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 norm-one-length in $mathcal L$. 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 norm-one-isospectral (resp. norm-one-isospectral plus a geometric condition). Using this fact, we give, for every dimension n>4, infinitely many examples of Riemannian manifolds that are isospectral on every level p and are not strongly isospectral.