Journal of the
Korean Mathematical Society JKMS

Let $k$ be a global field of characteristic unequal to two. Let $C$$:$ $y^2=f(x)$ be a nonsingular projective curve over $k$, where $f(x)$ is a quartic polynomial over $k$ with nonzero discriminant, and $K=k(C)$ be the function field of $C$. For each prime spot $\frp$ on $k$, let $\kp$ denote the corresponding completion of $k$ and $\kp(C)$ the function field of $C\times_k \kp$. Consider the map $$h : \, \br(K) \, \ra \, \Prod{\frp} \br\bigl(\kp(C)\bigr),$$ where $\frp$ ranges over all the prime spots of $k$. In this paper, we explicitly describe all the constant classes (coming from $\br(k)$) lying in the kernel of the map $h$, which is an obstruction to the Hasse principle for the Brauer groups of the curve. The kernel of $h$ can be expressed in terms of quaternion algebras with their prime spots. We also provide specific examples over $\qq$, the rationals, for this kernel.

Keywords: Brauer groups, Hasse principle, function fields of genus 1

MSC numbers: 16K20, 16K50, 14H05, 14H45