On a relation between spectral theory of lens spaces and Ehrhart theory

Abstract

In this article Ehrhart quasi-polynomials of simplices are employed to determine isospectral lens spaces in terms of a finite set of numbers. Using the natural lattice associated with a lens space the associated toric variety of a lens space is introduced. It is proved that if two lens spaces are isospectral then the dimension of global sections of powers of a natural line bundle on these two toric varieties are equal and they have the same general intersection number. Also, harmonic polynomial representation of the group SO(n) is used to provide a direct proof for a theorem of Lauret, Miatello and Rossetti on isospectrality of lens spaces.

Recent results on the spectra of lens spaces

“…In this section, we first give a description of the spectrum of a lens space by using Ehrhart series, as in [MH17] (see also [La16,Thm. 3.9]), and we end the section by giving a connection with toric varieties, developed by Mohades and Honari in [MH17,§5].…”

“…In one result, we give an elementary proof of the theorem in [LMR16a] concerning the spectrum of the Laplace-Beltrami operator acting on functions on lens spaces. In particular, this approach avoids the use of irreducible representations of compact Lie groups (see Section 3 and also [MH17]). An important tool is an integral lattice of rank n, defined by a congruence relation, naturally associated to each (2n − 1)-dimensional lens space.…”

“…We review the description of the spectrum of a lens space by using the Ehrhart theory for counting integer points in a naturally associated polytope. Furthermore, we include applications due to Mohades and Honari [MH17], on the toric varieties associated to pairs of isospectral lens spaces (see Section 6).…”

A Computational Study on Lens Spaces Isospectral on Forms

“…The articles [LMR16b], [BL17], [La16a] and [La16b] follow this approach and [DD14] study in detail the examples of all-p-isospectral pairs in [LMR16a]. The articles [MH16a], [MH16] are also related to this approach.…”

The spectrum on p-forms of a lens space

“…Hence, the above mentioned characterization for 0-isospectral lens spaces can be stated as follows: L and L ′ are 0-isospectral if and only if L and L ′ are · 1 -isospectral. (See [BL17], [LMR16b], [La16], [MH17], [MH16] for related results. )…”