On the Hasse Principle for conic bundles over even degree extensions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914072767758336 |
|---|---|
| author | Roven, Sam |
| author_facet | Roven, Sam |
| 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 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-Thélène, following Colliot-Thélène and Sansuc. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_10211 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Hasse Principle for conic bundles over even degree extensions Roven, Sam Number Theory Algebraic Geometry 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 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-Thélène, following Colliot-Thélène and Sansuc. |
| title | On the Hasse Principle for conic bundles over even degree extensions |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2203.10211 |