S-INTEGRAL POINTS ON HYPERELLIPTIC CURVES
Homero R. Gallegos-Ruiz · International Journal of Number Theory · 2011
Let C : Y2 = an Xn + ⋯ + a0 be a hyperelliptic curve with the ai rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. Let S be a finite set of rational primes. We give a completely explicit upper bound for the size of the S-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 use a refinement of the Mordell–Weil sieve which, combined with the upper bound, is capable of determining all the S-integral points. The method is illustrated by determining the S-integral points on the genus 2 hyperelliptic model Y2 - Y = X5 - X for the set S of the first 22 primes.