Vol. 2, No. 8, 2008

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Integral points on hyperelliptic curves

Yann Bugeaud, Maurice Mignotte, Samir Siksek, Michael Stoll and Szabolcs Tengely

Vol. 2 (2008), No. 8, 859–885
Abstract

Let C : Y 2 = anXn + + a0 be a hyperelliptic curve with the ai rational integers, n 5, and the polynomial on the right-hand side irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(). 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 2 hyperelliptic models Y 2 Y = X5 X and Y 2 = X 5 .

Keywords
curve, integral point, Jacobian, height, Mordell–Weil group, Baker's bound, Mordell–Weil sieve
Mathematical Subject Classification 2000
Primary: 11G30
Secondary: 11J86
Milestones
Received: 28 January 2008
Revised: 2 September 2008
Accepted: 12 September 2008
Published: 23 November 2008
Authors
Yann Bugeaud
Université Louis Pasteur
U. F. R. de mathématiques
7, rue René Descartes
67084 Strasbourg Cedex
France
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
Samir Siksek
Institute of Mathematics
University of Warwick
Coventry CV4 7AL
United Kingdom
http://www.warwick.ac.uk/~maseap/
Michael Stoll
Mathematisches Institut
Universität Bayreuth
95440 Bayreuth
Germany
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
http://www.math.klte.hu/~tengely/