#### Vol. 14, No. 3, 2020

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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
##### Abstract

Let $K$ be a number field or a $p$-adic field and $F$ the function field of a curve over $K$. Let $\ell$ be a prime. Suppose that $K$ contains a primitive $\ell$-th root of unity. If $\ell =2$ and $K$ is a number field, then assume that $K$ is totally imaginary. In this article we show that every element in ${H}^{3}\left(F,{\mu }_{\ell }^{\otimes 3}\right)$ 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
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