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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.07564 |
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Table of Contents:
- Let $k$ be a number field and let $π\colon X \rightarrow\mathbb{P}_k^1$ 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-Thélène, following Colliot-Thélène and Sansuc.