Lens spaces isospectral on p-forms for every p

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

Seoul

Congreso; International Congress of Mathematicians; 2014

International Mathematical Union

To every lens space $L$ we associate a congruence lattice $\mathcal L$ in $\mathbb Z^m$, showing that two lens spaces $L$ and $L'$ are isospectral on functions if and only if the associated lattices $\mathcal L$ and $\mathcal L'$ are isospectral with respect to one-norm. We also prove that $L$ and $L'$ are isospectral on $p$-forms for every $p$ if and only if $L$ and $L'$ are one-norm isospectral and satisfy a stronger condition. By constructing such congruence lattices we give infinitely many pairs of 5-dimensional lens spaces that are $p$-isospectral for all $p$. Such pairs are the first example of compact connected Riemannian manifolds $p$-isospectral for all $p$ but not strongly isospectral; in particular, they cannot be constructed by Sunada's method.