Serre conjecture II for pseudo-reductive groups

TL;DR

This paper extends the Serre conjecture II to pseudo-reductive groups, proving that torsors under certain pseudo-semisimple, simply connected groups over specific fields always have rational points, broadening the conjecture's scope.

Contribution

The paper generalizes Serre's conjecture II to pseudo-reductive groups and establishes the equivalence, demonstrating the existence of rational points over key fields.

Findings

Torsors under pseudo-semisimple, simply connected groups over global function fields have rational points.

The conjecture is proven to be equivalent for pseudo-reductive groups.

Extension of Serre's conjecture II to a broader class of algebraic groups.

Abstract

The Serre conjecture II predicts that every torsor under a semisimple, simply connected, algebraic group over a field of cohomological dimension at most 2 and of degree of imperfection at most 1 has a rational point. We generalize this conjecture to pseudo-reductive groups and prove their equivalence. In particular, we show that every torsor under a pseudo-semisimple, simply connected group over a global function field or a non-archimedean local field always has a rational point.