Explicit computation of the first étale 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 étale cohomology $H^1(X_{k^{sep},et},\mathcal{A})$ and the first étale cohomology with proper support $H_c^1(X_{k^{sep},et},\mathcal{A})$ 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).

arXiv comment29 pages, based in part on author's dissertation, comments are very welcome! Removed the condition that the sheaf of groups must be commutative wherever possible. End of introduction is expanded and is now Section 2, Section 5.5 is reworked a bit and is now Section 7; hopefully this improves exposition a bit. The numbering of the remaining sections changed accordingly