English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $X$ be a smooth projective curve of genus $g$, defined over analgebraically 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 toa finite group, generalising the notion of essentially finite vector bundles toarbitrary groups $G$. We give a Tannakian interpretation of such torsors, andwe prove that all essentially finite $G$-torsors have torsion degree, and thatthe degree is 0 if $X$ is an elliptic curve. We then study the density of theset 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 allsemistable 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.

Essentially finite $G$-torsors