Remark on fundamental groups and effective Diophantine methods for hyperbolic curves

Minhyong Kim ()
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Minhyong Kim: University of Oxford, Mathematical Instititute

A chapter in Number Theory, Analysis and Geometry, 2012, pp 355-368 from Springer

Abstract: Abstract In a letter from Grothendieck to Faltings, it was suggested that a positive answer to the section conjecture should imply finiteness of points on hyperbolic curves over number fields. In this paper, we point out instead the analogy between the section conjecture and the finiteness conjecture for the Tate-Shafarevich group of elliptic curves. That is, the section conjecture should provide a terminating algorithm for finding all rational points on a hyperbolic curve equipped with a rational point.

Keywords: Diophantine geometry; fundamental group; Diophantine decidability (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4614-1260-1_16

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DOI: 10.1007/978-1-4614-1260-1_16

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