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Integral points on
hyperelliptic curves
Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll and Szabolcs Tengely |
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Vol. 2 (2008), No. 8, 859–885
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Abstract |
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Let be a hyperelliptic curve with the rational integers, , and the polynomial on the right-hand side irreducible. Let be its Jacobian. We give a completely explicit upper bound for the integral points on the model , provided we know at least one rational point on and a Mordell–Weil basis for . We also explain a powerful refinement of the Mordell–Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on the genus hyperelliptic models and . |
Keywords
curve, integral point, Jacobian, height, Mordell–Weil
group, Baker's bound, Mordell–Weil sieve
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Mathematical Subject Classification 2000
Primary: 11G30
Secondary: 11J86
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Milestones
Received: 28 January 2008
Revised: 2 September 2008
Accepted: 12 September 2008
Published: 23 November 2008
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Authors |
| Yann Bugeaud | |
| Université Louis Pasteur U. F. R. de mathématiques 7, rue René Descartes 67084 Strasbourg Cedex France |
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| http://www-irma.u-strasbg.fr/~bugeaud/ | |
| Maurice Mignotte | |
| Université Louis Pasteur U. F. R. de mathématiques 7, rue René Descartes 67084 Strasbourg Cedex France |
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| Samir Siksek | |
| Institute of Mathematics University of Warwick Coventry CV4 7AL United Kingdom |
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| http://www.warwick.ac.uk/~maseap/ | |
| Michael Stoll | |
| Mathematisches Institut Universität Bayreuth 95440 Bayreuth Germany |
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| http://www.mathe2.uni-bayreuth.de/stoll/ | |
| Szabolcs Tengely | |
| Institute of Mathematics, University
of Debrecen Number Theory Research Group, Hungarian Academy of Sciences P.O.Box 12 4010 Debrecen Hungary |
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| http://www.math.klte.hu/~tengely/ | |
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