Essentially finite G-torsors

2022/11/05 by Archia Ghiasabadi, Ghiasabadi, Archia, Stefan Reppen +1
Computer Science · #Digital Image Processing Techniques #Computability, Logic, AI Algorithms

paper · pdf · doi:10.48550/arxiv.2211.02909

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^ef,0, inside M_G^ss,0, the k-points of all semistable degree 0 G-torsors. We show that when g=1, M_G^ef⊂ M_G^ss,0 is dense. When g>1 and when char(k)=0, we show that for any reductive group of semisimple rank 1, M_G^ef,0⊂ M_G^ss,0 is not dense.

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