This is related to my question Adelic description of moduli of $G$-bundles on a curve.

Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and $G$ a smooth connected affine algebraic group over $\mathbb{F}_q$. Under what conditions on $G$ does any $G$-bundle on $X$ trivialize over a Zariski open subset of $X$? *A priori* this can only be done over an étale cover of $X$.

In the accepted answer to the aforementioned question it is claimed that this holds for simply connected and semisimple $G$ by a theorem of Harder. Does anyone have a reference for this result, preferably in English or French? What about e.g. $G = PGL_2$? Non-split tori? Unipotent groups? I would love to see either positive results or counterexamples.

vanishessince $H^1(K,GL_n)=1$, so $PGL_n$ is OK too. $\endgroup$ – user29283 Mar 30 '13 at 18:27