# Can we classify reductive group schemes over curves

Let $C$ be a smooth quasi-projective connected curve over the complex numbers.

Can one classify all reductive group schemes over $C$?

Certainly, you have the trivial ones (coming from pulling-back those living over the complex numbers).

I can give a cohomological interpretation of this set as in Is every reductive group scheme etale locally trivial?, but I'm interested in a concrete description.

I would already be very happy to see some "non-trivial" examples of reductive group schemes over the affine line.

• You get an $n$-torus for any conjugacy class of homomorphisms $\pi_1^{et}(C) \to \mathrm{GL}_n(\mathbb{Z})$. – S. Carnahan Sep 24 '14 at 22:46
• The most interesting case is that of semisimple groups. Over any scheme $S$, every rank-$n$ vector bundle $E$ on $S$ gives rise to the Zariski-form ${\rm{SL}}(E)$ of ${\rm{SL}}_n$, and every rank-$n$ fiberwise non-degenerate line-bundle valued quadratic space $q:V \rightarrow L$ over $S$ (with $n \ge 3$) gives rise to the $S$-group ${\rm{SO}}(q)$ (whose isomorphism class encodes the isomorphism class of $(V,L,q)$ up to ${\rm{Pic}}(S)$-twisting), so it's hard to know what you might mean by a "concrete description", especially over a non-complete curve. – user27920 Sep 25 '14 at 2:27
• There is an old paper by Harder (in German) which I believe does the most one can hope for over a general connected Dedekind base, but I don't remember the reference; maybe someone else does. – user27920 Sep 25 '14 at 2:28
• Is it Harder, G.: Halbeinfache Gruppenschemata über Dedekindringen. Inventiones math.4, 165–191 (1967) ? I will take a look at it. – User12345 Sep 25 '14 at 8:21
• There is also this paper: link.springer.com/article/10.1007%2FBF01425451 – User12345 Sep 25 '14 at 8:23