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Similitudes over fields with I^4=0

M. Archita, Karim Johannes Becher

Abstract

This article studies the set of R-equivalence classes of the group of proper projective similitudes of an algebra with involution of the first kind. The main results concern base fields of characteristic different from 2 over which every 9-dimensional quadratic form has a nontrivial zero. This includes function fields of p-adic curves and extensions of transcendence degree 3 of C. Main results of [28] and [29] are extended by relaxing the condition on the base field as well as on the Clifford invariant for orthogonal involutions.

Similitudes over fields with I^4=0

Abstract

This article studies the set of R-equivalence classes of the group of proper projective similitudes of an algebra with involution of the first kind. The main results concern base fields of characteristic different from 2 over which every 9-dimensional quadratic form has a nontrivial zero. This includes function fields of p-adic curves and extensions of transcendence degree 3 of C. Main results of [28] and [29] are extended by relaxing the condition on the base field as well as on the Clifford invariant for orthogonal involutions.
Paper Structure (7 sections, 30 theorems, 35 equations)

This paper contains 7 sections, 30 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Quadratic forms and cohomological invariants
  3. Algebras with involution, similitudes and norms
  4. Reduced norms
  5. Norms from splitting $2$-extensions
  6. Involutions on index-$2$ algebras
  7. Orthogonal involutions of trivial discriminant when $\mathsf{I}^4=0$

Key Result

Theorem 2.1

The restriction of $c$ to $\mathsf{I}^2 K$ gives a surjective group homomorphism $c:\mathsf{I}^2K\rightarrow \operatorname{\mathsf{Br}}_2(K)$ whose its kernel is equal to $\mathsf{I}^3K$.

Theorems & Definitions (63)

  • Theorem 2.1: Merkurjev
  • proof
  • Theorem 2.2: Orlov-Vishik-Voevodsky
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 3.1
  • proof
  • ...and 53 more