Finite vector bundle over an algebraic curve

Overview & Synopsis

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, generalizing 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,0} inside 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} inside M_{G}^{\ss,0} is not dense.