Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.07564 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910054959022080 |
|---|---|
| author | Roven, Sam Wang, Alexander |
| author_facet | Roven, Sam Wang, Alexander |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07564 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Hasse Principle for conic bundles over even degree extensions Roven, Sam Wang, Alexander Number Theory 14G05 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. |
| title | On the Hasse Principle for conic bundles over even degree extensions |
| topic | Number Theory 14G05 |
| url | https://arxiv.org/abs/2508.07564 |