On the Hasse Principle for conic bundles over even degree extensions

TL;DR

This paper investigates the Hasse principle for conic bundle surfaces over number fields, demonstrating it holds over even degree extensions under certain conditions and over most quadratic extensions assuming Schinzel's hypothesis.

Contribution

It establishes new cases where the Hasse principle holds for conic bundles over even degree extensions, extending previous results using Brauer-Manin obstruction and fibration methods.

Findings

Hasse principle holds over even degree extensions with four singular fibers and non-empty adelic points.

Conditional proof that the Hasse principle holds over all but finitely many quadratic extensions assuming Schinzel's hypothesis.

Brauer-Manin obstruction vanishes in the studied cases, enabling the application of fibration techniques.

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 with $X(\mathbb{A}_k)\neq \emptyset$ 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-Th\'el\`ene, following Colliot-Th\'el\`ene and Sansuc.