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Weight filtrations on Selmer schemes and the effective Chabauty–Kim method

Part of: (Co)homology theory Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  20 June 2023

L. Alexander Betts
L. Alexander Betts*
Affiliation:
Department of Mathematics, Harvard University, Science Center Room 325, 1 Oxford Street, Cambridge, MA 02138, USA abetts@math.harvard.edu
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Abstract

We develop an effective version of the Chabauty–Kim method which gives explicit upper bounds on the number of $S$-integral points on a hyperbolic curve in terms of dimensions of certain Bloch–Kato Selmer groups. Using this, we give a new ‘motivic’ proof that the number of solutions to the $S$-unit equation is bounded uniformly in terms of $\#S$.

Keywords

étale fundamental grouprational points

MSC classification

Primary: 14G05: Rational points
Secondary: 14F35: Homotopy theory; fundamental groups 14G20: Local ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 159 , Issue 7 , July 2023 , pp. 1531 - 1605
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Copyright
© 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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