On the genera of semisimple groups defined over an integral domain of a global function field

Abstract

Let K = Fq(C) be the global field of rational functions on a smooth and projective curve C defined over a finite field Fq. Any finite but non-empty set S of closed points on C gives rise to a Hasse integral domain OS = Fq[C − S] of K. Given an almost-simple group scheme G defined over Spec OS with a smooth fundamental group F (G), we describe the finite set of (OS-classes of) twisted-forms of G in terms of geometric invariants of F (G) and the absolute type of the Dynkin diagram of G. This turns out in most cases to biject to a disjoint union of finite abelian groups. − −− → H 1 fl (O S , Aut(R)). Let A = D(B, σ) be the discriminant algebra. If R splits, namely, R ∼ = O S ×O S , then B ∼ = A×A op and σ is the exchange involution. Then U(B, σ) ∼ = GL 1 (A) and SU(B, σ) ∼ = SL 1 (A), so we are back in the previous situation. 20 Corollary 7.3. The map U(B, σ)(O S ) Nrd − − → R × is surjective if and only if the Hasse-principle holds for SU(B, σ). Acknowledgements: The authors thank P. Gille, B. Kunyavskiȋ and A. Quéguiner-Mathieu for valuable discussions concerning the topics of the present article.