Sur le comptage des fibr\'es de Hitchin nilpotents

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.