Vol. 1, No. 4, 2007

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Finite descent obstructions and rational points on curves

Michael Stoll

Vol. 1 (2007), No. 4, 349–391

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 Ck maps nontrivially into an abelian variety Ak 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.
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

Received: 24 January 2007
Revised: 23 October 2007
Accepted: 20 November 2007
Published: 1 November 2007
Michael Stoll
School of Engineering and Science
Jacobs University Bremen
P.O.Box 750561
28725 Bremen