Quadratic Chabauty:p-adic heights and integral points on hyperelliptic curves

Abstract

AbstractWe give a formula for the component atpof thep-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for findingp-adic approximations top-integral points on such curves when the Mordell–Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of suchp-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell–Weil group tensored withℚ$\mathbb{Q}$.