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Remark on fundamental groups and effective Diophantine methods for hyperbolic curves

In: Number Theory, Analysis and Geometry

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  • Minhyong Kim

    (University of Oxford, Mathematical Instititute
    Pohang University of Science and Technology, Department of Mathematics)

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.

Suggested Citation

  • Minhyong Kim, 2012. "Remark on fundamental groups and effective Diophantine methods for hyperbolic curves," Springer Books, in: Dorian Goldfeld & Jay Jorgenson & Peter Jones & Dinakar Ramakrishnan & Kenneth Ribet & John Tate (ed.), Number Theory, Analysis and Geometry, edition 127, pages 355-368, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-1260-1_16
    DOI: 10.1007/978-1-4614-1260-1_16
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