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An effective Chabauty–Kim theorem

Part of: Diophantine equations Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  14 May 2019

Jennifer S. Balakrishnan  and
Netan Dogra
Jennifer S. Balakrishnan
Affiliation:
Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA email [email protected]
Netan Dogra
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK email [email protected]
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Abstract

The Chabauty–Kim method allows one to find rational points on curves under certain technical conditions, generalising Chabauty’s proof of the Mordell conjecture for curves with Mordell–Weil rank less than their genus. We show how the Chabauty–Kim method, when these technical conditions are satisfied in depth 2, may be applied to bound the number of rational points on a curve of higher rank. This provides a non-abelian generalisation of Coleman’s effective Chabauty theorem.

Keywords

rational pointscurveseffective methods

MSC classification

Primary: 14G05: Rational points
Secondary: 11D45: Counting solutions of Diophantine equations
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 6 , June 2019 , pp. 1057 - 1075
Copyright
© The Authors 2019 

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Footnotes

Balakrishnan is supported in part by NSF grant DMS-1702196, the Clare Boothe Luce Professorship (Henry Luce Foundation), and Simons Foundation grant #550023.

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