Spectra of Lens Spaces from 1-Norm Spectra of Congruence Lattices

Abstract

Abstract. To every n-dimensional lens space L, we associate a congruence lattice 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 · 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.

The spectrum on p-forms of a lens space

“…For Γ a finite subgroup of T , we want to determine Spec p (Γ\S 2n−1 ) by using Theorem 2.1. If Γ ⊂ T is finite, we have (see [LMR16a,Lem. 3…”

“…Later in [Ik88], using the same family, he found pairs of lens spaces that are p-isospectral for all 0 ≤ p < p 0 but not p 0 -isospectral, for any p 0 > 0. More recently, Miatello, Rossetti and the author [LMR16a] found families of pairs, in any odd dimension n ≥ 5, of lens spaces that are p-isospectral for all p, but are not strongly isospectral. In particular such pair cannot be constructed by the generalized Sunada method due to DeTurck and Gordon [DG89], which uses representation equivalent discrete subgroups.…”

“…Furthermore, [LMR16a] also gives an explicit formula for the multiplicity of each eigenvalue in the 0-spectrum of a lens space and a geometric characterization of 0-isospectral lens spaces. The one-norm · 1 of an element in Z n is given by the sum of the absolute values of its entries, and two subsets L and L ′ of Z n are called · 1 -isospectral if for each positive integer k, there are the same number of elements in L and in L ′ with one norm equal to k. Each (2n − 1)-dimensional lens space L has associated a congruence lattice L ⊂ Z n (see (2.5)).…”

“…The aim of this paper, as a continuation of the study begun in [LMR16a], is to give an explicit description of the p-spectrum of a lens space for any value of p (Theorem 3.3). As a consequence, for any p 0 ≥ 0, we obtain a geometric characterization of lens spaces p-isospectral for all 0 ≤ p ≤ p 0 (Corollary 2.3).…”

Recent results on the spectra of lens spaces

“…This article surveys recent results on the spectral geometry of lens and related spaces. In [LMR16b], the authors discussed the seminal work of Ikeda ([IY79, Ik80, Ik88]), together with results in [LMR16a]. A number of new results have appeared since then, and a purpose of the present article is to give a rather complete update.…”