On the equations zm = F(x, y) and Axp + Byq = Czr

Abstract

We investigate integer solutions of the superelliptic equation (Equation presented) where F is a homogeneous polynomial with integer coefficients, and of the generalized Fermat equation (Equation presented) where A, B and C are non-zero integers. Call an integer solution (x, y, z) to such an equation proper if gcd(x, y, z) = 1. Using Faltings’ Theorem, we shall give criteria for these equations to have only finitely many proper solutions. We examine (1) using a descent technique of Kummer, which allows us to obtain, from any infinite set of proper solutions to (1), infinitely many rational points on a curve of (usually) high genus, thus contradicting Faltings’ Theorem (for example, this works if F(t, 1) = 0 has three simple roots and m ≥ 4). We study (2) via a descent method which uses unramified coverings of P1 \ {0, 1, ∞} of signature (p, q, r), and show that (2) has only finitely many proper solutions if l/p + l/q + 1/r < 1. In cases where these coverings arise from modular curves, our descent leads naturally to the approach of Hellegouarch and Frey to Fermat’s Last Theorem. We explain how their idea may be exploited for other examples of (2). We then collect together a variety of results for (2) when 1/p + 1/q + 1/r ≥ 1. In particular, we consider ‘local-global’ principles for proper solutions, and consider solutions in function fields. © 1995 Oxford University Press.