Computation of étale cohomology on curves in single exponential time

In this paper, we describe an algorithm that, for a smooth connected curve X over a field k, a finite locally constant sheaf A on Xét of abelian groups of torsion invertible in k, computes the first étale cohomology H(Xksep,ét,A) and the first étale cohomology with proper support Hc(Xksep,ét,A) as sets of torsors. The complexity of this algorithm is exponential in nlog , pa(X), and pa(A), where pa(X) is the arithmetic genus of the normal completion of X, pa(A) is the arithmetic genus of the normal completion Y of the smooth curve representing A, and n is the degree of Y over X. The computation in this algorithm is done via the computation of a groupoid scheme classifying the A-torsors with some extra rigidifying data.