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Serre conjecture II for pseudo-reductive groups

Mac Nam Trung Nguyen

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.

Serre conjecture II for pseudo-reductive 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.
Paper Structure (3 sections, 11 theorems, 17 equations)

This paper contains 3 sections, 11 theorems, 17 equations.

Table of Contents

  1. Serre conjecture II
  2. Recollections on pseudo-reductive groups
  3. Structure of pseudo-semisimple, simply connected groups

Key Result

Theorem 2

Let $k$ be a field of cohomological dimension at most $2$ and of degree of imperfection at most $1.$ The following are equivalent

Theorems & Definitions (17)

  • Conjecture 1: Serre conjecture II for pseudo-reductive groups
  • Theorem 2: Corollary \ref{['main-thm']}
  • Corollary 3
  • Lemma 7
  • proof
  • Lemma 9
  • Lemma 10
  • Lemma 12
  • Lemma 13
  • Lemma 14
  • ...and 7 more