P-spectra of lens spaces from norm_1 spectra of congruence lattices

EMILIO LAURET; MIATELLO, ROBERTO J.; J.P. ROSSETTI

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