One-norm spectrum of a lattice

LAURET, EMILIO A.

Potsdam, Alemania

Conferencia; Analysis and geometry on graphs and manifolds; 2017

University of Potsdam

In 1964, John Milnor gave the first example of isospectral non-isometric compact Riemannian manifolds.To do this, he related the spectrum of the Laplace operator on a torus, and the (Euclidean) norm of the vectors of the (corresponding) dual lattice. Consequently, a pair of lattices with the same theta function induces a pair of isospectral tori. In this talk, we will introduce a new relation between the spectrum of a lens space (a sphere over a cyclic group), and the one-norm (sum of the absolute values of the entries) of the vectors in an associated lattice. We associate to each lattice, the one-norm generating function defined as follows: the power series whose $k$-th term is the number of vectors in the lattice with one-norm equal to $k$. We will show that two lens spaces are isospectral if and only if their corresponding lattices have the same one-norm generating function. Furthermore, we will prove that the generating function is a rational function, and consequently, a finite part of the spectrum determines the whole spectrum. This is a joint work with Roberto Miatello and Juan Pablo Rossetti.