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.
Mathematics Subject Classification (2010): 11G30; 14H25
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Acknowledgements
The author was supported in part by a grant from the National Science Foundation and a visiting professorship at RIMS. He is grateful to Kazuya Kato, Shinichi Mochizuki, and Akio Tamagawa for a continuing stream of discussions on topics related to this paper, and for their generous hospitality during the Fall of 2006.
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Dedicated to the memory of Serge Lang
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Kim, M. (2012). Remark on fundamental groups and effective Diophantine methods for hyperbolic curves. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_16
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DOI: https://doi.org/10.1007/978-1-4614-1260-1_16
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