Finite descent obstructions and rational  points on curves

Stoll, Michael

doi:10.2140/ant.2007.1.349

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Finite descent obstructions and rational points on curves

Michael Stoll

Vol. 1 (2007), No. 4, 349–391

DOI: 10.2140/ant.2007.1.349

Abstract

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 $Ш (k, A)$ 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.

An errata was posted on 20 March 2017 in an online supplement.

Keywords

rational point, descent obstruction, covering, twist, torsor under finite group scheme, Brauer–Manin obstruction

Mathematical Subject Classification 2000

Primary: 11G30

Secondary: 14G05, 11G10, 14H30

Supplementary material

Errata posted on 20 March 2017

Milestones

Received: 24 January 2007

Revised: 23 October 2007

Accepted: 20 November 2007

Published: 1 November 2007

Authors

Michael Stoll
School of Engineering and Science Jacobs University Bremen P.O.Box 750561 28725 Bremen Germany
http://www.faculty.jacobs-university.de/mstoll/

Vol. 1, No. 4, 2007