Integral Points on Hyperelliptic Curves

Y. Bugeaud, M. Mignotte, S. Siksek, M. Stoll, Sz. Tengely

Published 2008-01-29, updated 2010-03-16Version 4

We give a completely explicit upper bound for integral points on (standard) affine models of hyperelliptic curves, provided we know at least one rational point and a Mordell-Weil basis of the Jacobian. 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 a two genus 2 hyperelliptic curves with Mordell--Weil Jacobian ranks of 3 and 6.