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Failure of the local-global principle for isotropy of quadratic forms over function fields

Asher Auel, V. Suresh

Abstract

We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms over function fields of transcendence degree at least 2 over algebraically closed fields. Our construction involves generalized Kummer varieties as well as a new nontriviality result for the unramified cohomology of products of elliptic curves over discretely valued fields, which can be viewed as an arithmetic version of a theorem of Gabber.

Failure of the local-global principle for isotropy of quadratic forms over function fields

Abstract

We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms over function fields of transcendence degree at least 2 over algebraically closed fields. Our construction involves generalized Kummer varieties as well as a new nontriviality result for the unramified cohomology of products of elliptic curves over discretely valued fields, which can be viewed as an arithmetic version of a theorem of Gabber.

Paper Structure

This paper contains 6 sections, 16 theorems, 15 equations.

Table of Contents

  1. Bogomolov's trick
  2. Unramified cohomology of function fields
  3. Generalized Kummer varieties
  4. Hyperbolicity over a quadratic extension
  5. Failure of the local global principle
  6. A geometric presentation lemma

Key Result

Theorem 1

The local-global principle for isotropy of quadratic forms fails to hold in dimension $2^n$ over any function field $K$ of transcendence degree $n \geq 2$ over an algebraically closed field $k$ of characteristic $\neq 2$ other than possibly the algebraic closure of a finite field.

Theorems & Definitions (34)

  • Theorem 1
  • Lemma 1.1: Bogomolov's trick
  • proof
  • Corollary 1.2
  • Proposition 1.3
  • proof
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • ...and 24 more