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

Abstract

To every $n$-dimensional lens space $L$, we associate a congruence lattice $\mathcal{L}$ in ${\mathbb{Z}}^m$, 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 $\Vert \cdot {\Vert }_1$-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 $\Vert \cdot {\Vert }_1$-isospectral (resp.\ $\Vert \cdot {\Vert }_1$-isospectral plus a geometric condition). Using this fact, we give, for every dimension $n\ge 5$, infinitely many examples of Riemannian manifolds that are isospectral on every level $p$ and are not strongly isospectral.

Accepted for publication in IMRN

References in corpus (1)

• Representation equivalence and p-Spectrum of constant curvature space forms

Cited by in corpus (7)

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• Weight multiplicity formulas for bivariate representations of classical Lie algebras
• Isospectral CR manifolds with respect to the Kohn Laplacian