Rational points on hyperelliptic curves and an explicit Weierstrass preparation theorem

Abstract

 By using the so-called elliptic curve Chabauty method, N. Bruin [1], V. Flynn and J. Wetherell [6] have extended Chabauty's method to some cases where the rank of the Jacobian may not be less than the genus. The main tool in these methods is a theorem of Strassman on p-adic zeros of power series in one variable, and is applicable only if certain Jacobians are of rank less than or equal to 1. In the present paper, we give an explicit generalization of Strassman's theorem to several variables, enabling us to treat cases where the rank is greater than 1. We apply this to find all the rational points on a hyperelliptic curve of rank and genus equal to 4.