## Algebra & Number Theory

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

Venapally Suresh

#### 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 $H3(F,μℓ⊗3)$ 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.

#### Article information

Source
Algebra Number Theory, Volume 14, Number 3 (2020), 701-729.

Dates
Received: 8 December 2018
Revised: 6 October 2019
Accepted: 22 November 2019
First available in Project Euclid: 2 July 2020

Permanent link to this document
https://projecteuclid.org/euclid.ant/1593655269

Digital Object Identifier
doi:10.2140/ant.2020.14.701

Mathematical Reviews number (MathSciNet)
MR4113778

#### Citation

Suresh, Venapally. Third Galois cohomology group of function fields of curves over number fields. Algebra Number Theory 14 (2020), no. 3, 701--729. doi:10.2140/ant.2020.14.701. https://projecteuclid.org/euclid.ant/1593655269

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