The unramified Brauer group and the Pythagoras number of hyperelliptic curves with good reduction defined over henselian valued fields

Abstract

Let F be a henselian valued field with real closed residue field, C a hyper-elliptic curve over F with good reduction. A set of independend generators for the two component of the Brauer group of a curve C with good reduction defined by an affine equation $y^2 = f(x)$, $deg f(x)$ is odd, is calculated. As an application it is shown that the Pythagoras number of the function field F(C) of such curves is 2 if F(C) is a real field and 3 if F(C) is a non-real field.