Finite descent obstructions and rational points on curves

By Michael Stoll. 2006

126 citations recorded by the source corpus.

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 X(k, A) has no nontrivial divisible elements, 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 ≥ 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 in the only one on curves.

Subfields: Geometry and Topology, Mathematical Physics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/a945897bcd541bfd576a9d59a9d335deabcb1338

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In Geometry and Topology

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Citations in the corpus

Integral points on hyperelliptic curves (2008, methodology): “The Mordell–Weil sieve is a technique that can be used to show the non-existence of rational points on a curve (for example [11], [13]), or to help determine the set of rational points in conjunction with the method of Chabauty (for example [10]); for connections to the Brauer–Manin obstruction see,”