Abstract

Let C : Y ² = a_nXⁿ + ⋯ + a₀ be a hyperelliptic curve with the a_i 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(Q). 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 = .

Keywords

curve, integral point, Jacobian, height, Mordell–Weil group, Baker's bound, Mordell–Weil sieve

Mathematical Subject Classification

Primary: 11G30

Secondary: 11J86

Authors