Ramified descent
Fuente:
arXiv
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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866915887600107520 |
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| author | Demeio, Julian Lawrence |
| author_facet | Demeio, Julian Lawrence |
| contents | We investigate the "ramified descent problem": which adelic points of a smooth geometrically connected variety $X$ defined over a number field $K$ can be approximated by points that lift to a (twist of a) given ramified cover? We show that the natural descent set corresponding to the problem defines an obstruction to Hasse Principle and weak approximation. Furthermore, we introduce a Brauer-Manin obstruction to the problem. This obstruction can be purely transcendental (and non-trivial) even for abelian covers, which answers in the negative a question posed by Harari at a 2019 workshop. Moreover, the counterexample we produce is also an explicit example of transcendental obstruction to weak approximation for a quotient $SL_n/G$, with $G$ constant metabelian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_00843 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Ramified descent Demeio, Julian Lawrence Algebraic Geometry Number Theory 14G05, 14G12, 11G35 We investigate the "ramified descent problem": which adelic points of a smooth geometrically connected variety $X$ defined over a number field $K$ can be approximated by points that lift to a (twist of a) given ramified cover? We show that the natural descent set corresponding to the problem defines an obstruction to Hasse Principle and weak approximation. Furthermore, we introduce a Brauer-Manin obstruction to the problem. This obstruction can be purely transcendental (and non-trivial) even for abelian covers, which answers in the negative a question posed by Harari at a 2019 workshop. Moreover, the counterexample we produce is also an explicit example of transcendental obstruction to weak approximation for a quotient $SL_n/G$, with $G$ constant metabelian. |
| title | Ramified descent |
| topic | Algebraic Geometry Number Theory 14G05, 14G12, 11G35 |
| url | https://arxiv.org/abs/2112.00843 |