Explicit computation of the first \'etale cohomology on curves

TL;DR

This paper presents an algorithm to explicitly compute the first étale cohomology groups of curves over fields, including torsors, with complexity exponential in key geometric parameters.

Contribution

It introduces a novel algorithm that computes first étale cohomology and torsors on curves, with complexity depending on the genus and degree, advancing computational algebraic geometry.

Findings

Computes $H^1$ and $H^1_c$ as sets of torsors.

Complexity is exponential in $n^{ ext{log} n}$, $p_a(X)$, and $p_a( ext{A})$.

Uses groupoid schemes to classify torsors.

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).