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Finite descent obstructions and rational points on curves

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
Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II > Chair Mathematics II - 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