Third Galois cohomology group of function fields of curves over number fields

Suresh, Venapally

doi:10.2140/ant.2020.14.721

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Third Galois cohomology group of function fields of curves over number fields

Venapally Suresh

Vol. 14 (2020), No. 3, 701–729

DOI: 10.2140/ant.2020.14.721

Abstract

Let $K$ be a number field or a $p$ -adic field and $F$ the function field of a curve over $K$ . Let $ℓ$ be a prime. Suppose that $K$ contains a primitive $ℓ$ -th root of unity. If $ℓ = 2$ and $K$ is a number field, then assume that $K$ is totally imaginary. In this article we show that every element in $H^{3} (F, μ_{ℓ}^{\otimes 3})$ is a symbol. This leads to the finite generation of the Chow group of zero-cycles on a quadric fibration of a curve over a totally imaginary number field.

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Keywords

Galois cohomology, functions fields, number fields, symbols

Mathematical Subject Classification 2010

Primary: 11R58

Milestones

Received: 8 December 2018

Revised: 6 October 2019

Accepted: 22 November 2019

Published: 1 June 2020

Authors

Venapally Suresh
Department of Mathematics Emory University Atlanta, GA United States

Vol. 14, No. 3, 2020