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

Let K = F _q (C) be the global function field of rational functions over a smooth and projective curve C defined over a finite field F _q . The ring of regular functions on C − S Where S ≠ ∅ is any finite set of closed points on C is a Dedekind domain O _S of K. For a semisimple O _S -group G With a smooth fundamental group F, We aim to describe both the set of genera of G and its principal genus (the latter if G ⊗ _OS K is isotropic at S) in terms of abelian groups depending on O _S and F only. This leads to a necessary and sufficient condition for the Hasse local-global principle to hold for certain G. We also use it to express the TamagaWa number τ(G) of a semisimple K-group G by the Euler–Poincaré invariant. This facilitates the computation of τ(G) for tWisted K-groups.