Brauer groups of conic bundles over elliptic curves

Abstract We study the Brauer groups of regular conic bundles over elliptic curves defined over a number field k . We explicitly compute the Brauer group of the conic bundle when the singular fibres lie above k -points that are 2-torsion on the elliptic curve, and the corresponding splitting fields are isomorphic. We apply the result to compute the Brauer group of a class of surfaces analogous to that of Châtelet surfaces. We investigate Brauer–Manin obstructions to weak approximation coming from the real places on such surfaces.