On the Hasse Principle for conic bundles over even degree extensions

Let $k$ be a number field and let $\pi \text{\hspace{0.17em}⁣}:X\to {\mathbb{P}}_k^1$ be a smooth conic bundle. We show that if $X\mathrm{/}k$ has four geometric singular fibers and either $X({\mathbb{A}}_k)\mathrm{\ne }\mathrm{\varnothing }$ or $X\mathrm{/}k$ has non-trivial Brauer group, then $X$ satisfies the Hasse principle over any even degree extension $L\mathrm{/}k$. 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.