On the Hasse Principle for conic bundles over even degree extensions

TL;DR

This paper investigates the Hasse principle for conic bundles over number fields, demonstrating its validity over even degree extensions under certain conditions and conditional on Schinzel's hypothesis for quadratic extensions.

Contribution

It establishes the Hasse principle for conic bundles over even degree extensions with specific fiber conditions and extends results to quadratic extensions assuming Schinzel's hypothesis.

Findings

Hasse principle holds over even degree extensions with four geometric singular fibers.

Brauer-Manin obstruction vanishes in the studied cases.

Conditional results for quadratic extensions depend on Schinzel's hypothesis.

Abstract

Let $k$ be a number field and let $\pi \colon X \rightarrow\mathbb{P}_k^1$ be a smooth conic bundle. We show that if $X/k$ has four geometric singular fibers and either $X(\mathbb{A}_k)\neq \emptyset$ or $X/k$ has 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-Th\'el\`ene, following Colliot-Th\'el\`ene and Sansuc.