One-norm isospectral lattices

LAURET, EMILIO AGUSTÍN

Gyeongju

Workshop; ICM 2014 Satellite Conference on Integral Quadratic Forms and Related Topics; 2014

Milnor in 1964 showed that two euclidean lattices representing the same numbers (with multiplicity) under the euclidean norm induce flat tori that are isospectral, that is, their Laplace operators have the same spectrum. Thus, the two 16-dimensional "isospectral" lattices constructed by Witt in 1941 give the first example of isospectral non-isometric Riemannian manifolds. In this talk we associate to each lens space (a manifold with constant curvature one and cyclic fundamental group) a congruence lattice in such a way that, two lens spaces are isospectral if and only if the one-norm represents the same numbers (with multiplicities) in the associated lattices. By adding to this condition an additional geometric condition we obtain that the associated lens spaces are isospectral on p-forms for every p.