On the Hasse Principle for conic bundles over even degree extensions

Abstract

Let $k$ be a number field and let $\pi :X\to {\mathbb{P}}_k^1$ be a smooth conic bundle. We show that if $X/k$ has four geometric singular fibers with $X({\mathbb{A}}_k)\ne \varnothing$ or non-trivial Brauer group, then $X$ satisfies the Hasse principle over any even degree extension $L/k$. Furthermore for arbitrary $X$ 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-Thélène, following Colliot-Thélène and Sansuc.

arXiv commentReplacing this paper with arXiv 2508.07564