Sur le comptage des fibr\'es de Hitchin nilpotents

TL;DR

This paper explores counting Hitchin bundles on a curve and deriving an explicit formula for the nilpotent part of the Arthur-Selberg trace formula, revealing a connection between geometric and automorphic aspects.

Contribution

It provides an explicit formula for the nilpotent part of the trace formula and links it to counting Hitchin bundles, expanding understanding of their relationship.

Findings

Explicit formula for nilpotent trace formula component

Connection between Hitchin bundle counting and trace formula

Use of zeta function of the curve in formulas

Abstract

This paper is concerned with two problems. One is to count Hitchin bundles on a projective curve and the other is to get an explicit formula for the nilpotent part of the Arthur-Selberg trace formula for a simple test function. The fact that the two problems are in fact related has been noticed in a previous paper. We expand the nilpotent part of the Arthur-Selberg trace formula in a sum of adelic integrals indexed by nilpotent orbits. For "regular by blocks" orbits, we get an explicit formula for these integrals in terms of the zeta function of the curve.