## Title data

Stoll, Michael:

**Finite descent obstructions and rational points on curves.**

*In:* Algebra & Number Theory.
Vol. 1
(2007)
Issue 4
.
- pp. 349-391.

ISSN 1937-0652

DOI: https://doi.org/10.2140/ant.2007.1.349

## Abstract in another language

Let k be a number field and X a smooth projective k-variety. In this paper, we study the information obtainable from descent via torsors under finite k-group

schemes on the location of the k-rational points on X within the adelic points. Our main result is that if a curve C/k maps nontrivially into an abelian variety A/k such that A(k) is finite and (k, A) has no nontrivial divisible element, then the information coming from finite abelian descent cuts out precisely the rational points of C. We conjecture that this is the case for all curves of genus at least 2. We relate finite descent obstructions to the Brauer-Manin obstruction; in particular, we prove that on curves, the Brauer set equals the set cut out by

finite abelian descent. Our conjecture therefore implies that the Brauer-Manin obstruction against rational points is the only one on curves.

## Further data

Item Type: | Article in a journal |
---|---|

Refereed: | Yes |

Keywords: | rational point; descent obstruction; covering; twist; torsor under finite group scheme; Brauer-Manin obstruction |

Institutions of the University: | Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) > Chair Mathematics II (Computer Algebra) - Univ.-Prof. Dr. Michael Stoll Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics |

Result of work at the UBT: | No |

DDC Subjects: | 500 Science > 510 Mathematics |

Date Deposited: | 02 Feb 2015 15:47 |

Last Modified: | 02 Feb 2015 15:47 |

URI: | https://eref.uni-bayreuth.de/id/eprint/6220 |