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Journal für die reine und angewandte Mathematik

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Volume 2016, Issue 720


Quadratic Chabauty: p-adic heights and integral points on hyperelliptic curves

Jennifer S. Balakrishnan
  • Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom of Great Britain and Northern Ireland
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/ Amnon Besser / J. Steffen Müller
Published Online: 2014-07-08 | DOI: https://doi.org/10.1515/crelle-2014-0048


We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-adic approximations to p-integral points on such curves when the Mordell–Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of such p-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell–Weil group tensored with .

Dedicated to the memory of Robert F. Coleman (1954–2014)


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About the article

Received: 2013-04-15

Revised: 2014-04-04

Published Online: 2014-07-08

Published in Print: 2016-11-01

Funding Source: National Science Foundation

Award identifier / Grant number: DMS-1103831

Funding Source: European Research Council

Award identifier / Grant number: 204083

Funding Source: Engineering and Physical Sciences Research Council

Award identifier / Grant number: I020519/1

Funding Source: Deutsche Forschungsgemeinschaft

Award identifier / Grant number: KU 2359/2-1

The first author was supported by NSF grant DMS-1103831. The second author’s stay at Oxford was funded by ERC grant 204083 and by EPSRC grant I020519/1. The third author was supported by DFG grant KU 2359/2-1.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 720, Pages 51–79, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2014-0048.

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