La formule des traces pour les rev\^etements de groupes r\'eductifs connexes. I. Le d\'eveloppement g\'eom\'etrique fin

We study the genuine part of the Arthur-Selberg trace formula for some nonlinear covers of connected reductive groups. As a first step towards the invariant trace formula, we express the geometric side in terms of weighted orbital integrals. In particular, our results apply to the covers constructed by Brylinksi and Deligne.