Essentially finite $G$-torsors

TL;DR

This paper introduces the concept of essentially finite $G$-torsors on algebraic curves, explores their properties, and analyzes the density of their moduli points within the space of semistable torsors, revealing differences based on genus and characteristic.

Contribution

It generalizes the notion of essentially finite vector bundles to arbitrary reductive groups and provides a Tannakian interpretation, along with results on torsion degree and density properties.

Findings

All essentially finite $G$-torsors have torsion degree.

Degree is zero for elliptic curves.

Density of essentially finite torsors depends on genus and characteristic.

Abstract

Let $X$ be a smooth projective curve of genus $g$, defined over an algebraically closed field $k$, and let $G$ be a connected reductive group over $k$. We say that a $G$-torsor is essentially finite if it admits a reduction to a finite group, generalising the notion of essentially finite vector bundles to arbitrary groups $G$. We give a Tannakian interpretation of such torsors, and we prove that all essentially finite $G$-torsors have torsion degree, and that the degree is 0 if $X$ is an elliptic curve. We then study the density of the set of $k$-points of essentially finite $G$-torsors of degree $0$, denoted $M_{G}^{\text{ef},0}$, inside $M_{G}^{\text{ss},0}$, the $k$-points of all semistable degree 0 $G$-torsors. We show that when $g=1$, $M_{G}^{\text{ef}}\subset M_{G}^{\text{ss},0}$ is dense. When $g>1$ and when $\text{char}(k)=0$, we show that for any reductive group of semisimple rank 1, $M_{G}^{\text{ef},0}\subset M_{G}^{\text{ss},0}$ is not dense.