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2007 Finite descent obstructions and rational points on curves
Michael Stoll
Algebra Number Theory 1(4): 349-391 (2007). DOI: 10.2140/ant.2007.1.349


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.


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Michael Stoll. "Finite descent obstructions and rational points on curves." Algebra Number Theory 1 (4) 349 - 391, 2007.


Received: 24 January 2007; Revised: 23 October 2007; Accepted: 20 November 2007; Published: 2007
First available in Project Euclid: 20 December 2017

zbMATH: 1167.11024
MathSciNet: MR2368954
Digital Object Identifier: 10.2140/ant.2007.1.349

Primary: 11G30
Secondary: 11G10 , 14G05 , 14H30

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

Rights: Copyright © 2007 Mathematical Sciences Publishers


Vol.1 • No. 4 • 2007
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