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An effective Chabauty–Kim theorem

Published online by Cambridge University Press:  14 May 2019

Jennifer S. Balakrishnan
Affiliation:
Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA email jbala@bu.edu
Netan Dogra
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK email dogra@maths.ox.ac.uk

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.

Type
Research Article
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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