S-INTEGRAL POINTS ON HYPERELLIPTIC CURVES

Let C : Y(2) = a(n)X(n) + ... + a(0) be a hyperelliptic curve with the a(i) 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 Y(2) - Y = X(5) - X for the set S of the first 22 primes.