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

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.