Explicit computation of the first \'etale cohomology on curves

Abstract

In this paper, we describe an algorithm that, for a smooth connected curve $X$ over a field $k$ with normal completion having arithmetic genus $p_a(X)$, a finite locally constant sheaf $\mathcal A$ on $X_{et}$ of abelian groups of torsion invertible in $k$, represented by a smooth curve with normal completion having arithmetic genus $p_a(\mathcal A)$ and degree $n$ over $X$, computes the first \'etale cohomology $H^1(X_{k^{sep},et},\mathcal A)$ and the first \'etale cohomology with proper support $H^1_c(X_{k^{sep},et},\mathcal A)$ as sets of torsors, in arithmetic complexity exponential in $n^{\log n}$, $p_a(X)$, and $p_a(\mathcal A)$. This is done via the computation of a groupoid scheme classifying the relevant torsors (with extra rigidifying data).