Rational Point on Conic Bundles

Abstract

In this paper, we focus on obstructions to the existence of rational points for a special class of algebraic varieties. In particular, we consider the case where $\pi \colon X \rightarrow \PPP_k^1$ is a smooth conic bundle and $k$ is a number field. We show that if $X/k$ has four geometric singular fibers with $X(\A_k)\neq \emptyset$ or $X$ has non-trivial Brauer group, then $X$ satisfies the Hasse principle over any even degree extension $L/k$. Furthermore for arbitrary conic bundles $X$ we show that, conditional on Schinzel’s hypothesis, $X$ satisfies the Hasse principle over all but finitely many quadratic extensions of $k$. We prove these results by showing the Brauer-Manin obstruction vanishes and then apply fibration method results of Colliot-Th\'el\`ene, following Colliot-Th\'el\`ene and Sansuc.}