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R-triviality for adjoint classical groups of type C

M. Archita

Abstract

For a central simple algebra with a symplectic involution (A,s) over a field of characteristic different from 2, we show that its group of projective similitudes PSim(A,s) is R-trivial in two new cases.

R-triviality for adjoint classical groups of type C

Abstract

For a central simple algebra with a symplectic involution (A,s) over a field of characteristic different from 2, we show that its group of projective similitudes PSim(A,s) is R-trivial in two new cases.
Paper Structure (3 sections, 7 theorems, 5 equations)

This paper contains 3 sections, 7 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results

Key Result

Theorem 2.1

We have ${K}^{\times 2}\cdot\mathsf{Hyp}(A,\sigma)\subseteq \mathsf{G}(A,\sigma)$ and In particular, the group ${\bf PSim} (A,\sigma)$ is $R$-trivial if and only if, for every field extension $K'/K$, one has $\mathsf{G}(A_{K'},\sigma_{K'})={K'}^{\times 2}\cdot\mathsf{Hyp}(A_{K'},\sigma_{K'})$.

Theorems & Definitions (15)

  • Theorem 2.1: Merkurjev
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Example 3.3
  • Proposition 3.4
  • proof
  • Lemma 3.5
  • ...and 5 more