On full asymptotics of analytic torsions for compact locally symmetric orbifolds

TL;DR

This paper computes the asymptotic behavior of analytic torsions for flat vector bundles on compact locally symmetric orbifolds using heat trace analysis and Selberg's trace formula, with explicit geometric formulas for orbital integrals.

Contribution

It provides explicit asymptotic formulas for analytic torsions on orbifolds by evaluating orbital integrals through a geometric localization approach.

Findings

Explicit asymptotic formulas for analytic torsions.

Development of a geometric localization formula for orbital integrals.

Application of Bismut's formula for semisimple orbital integrals.

Abstract

We consider a certain sequence of flat vector bundles on a compact locally symmetric orbifold, and we evaluate explicitly the associated asymptotic Ray-Singer real analytic torsion. The basic idea is to computing the heat trace via Selberg's trace formula, so that a key point in this paper is to evaluate the orbital integrals associated with nontrivial elliptic elements. For that purpose, we deduce a geometric localization formula, so that we can rewrite an elliptic orbital integral as a sum of certain identity orbital integrals associated with the centralizer of that elliptic element. The explicit geometric formula of Bismut for semisimple orbital integrals plays an essential role in these computations.