Weak approximation on Châtelet surfaces

Abstract: We study weak approximation for Ch^{a}telet surfaces over number fields when all singular fibers are defined over rational points. We consider Ch^{a}telet surfaces which satisfy weak approximation over every finite extension of the ground field. We prove many of these results by showing that the Brauer-Manin obstruction vanishes, then apply results of Colliot-Th\'el`ene, Sansuc, and Swinnerton-Dyer.