#### Vol. 1, No. 4, 2007

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors' Addresses Editors' Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
Finite descent obstructions and rational points on curves

### Michael Stoll

Vol. 1 (2007), No. 4, 349–391
##### 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 $A\left(k\right)$ is finite and 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