On the analogy between L-functions and Atiyah-Bott-Lefschetz trace formulas for foliated spaces

TL;DR

This paper establishes an Atiyah-Bott-Lefschetz trace formula for certain foliated spaces and explores their potential connection to arithmetic L-functions, inspired by Deninger's programme.

Contribution

It proves a trace formula for cohomology of ramified leafwise flat line bundles on Riemannian foliations and suggests a new approach to studying L-functions through foliated space cohomology.

Findings

Proved a trace formula for ramified leafwise flat line bundles.

Indicated potential use of foliated space cohomology in arithmetic L-functions.

Connected trace formulas with Deninger's programme.

Abstract

This paper is motivated by Deninger's programme. First we prove, using Alvarez Lopez-Kordyukov results, an Atiyah-Bott-Lefschetz trace formula for the cohomology groups associated to a ramified leafwise flat line bundle on a riemannian foliation. Then we argue, by precise computations, that cohomology groups associated to ramified leafwise flat vector bundles on a suitable foliated space (whose existence is still unknown) might be useful to study arithmetic L-functions via Atiyah-Bott-Lefschetz trace formulas.