Thue equations and the method of Chabauty-Coleman

Abstract

Let OK be any domain with field of fractions K . Let F(x, y) ∈ OK [x, y] be a homogeneous polynomial of degree n, coprime to y, and assumed to have unit content (i.e., the coefficients of F generate the unit ideal in OK ). Assume that gcd(n, char(K )) = 1. Let h ∈ OK and assume that the polynomial hzn − F(x, y) is irreducible in K[x, y, z]. We denote by X F,h/K the nonsingular complete model of the projective plane curve CF,h/K defined by the equation hzn − F(x, y) = 0. We shall assume in this article that g(X F,h) ≥ 2. When K is a number field, Mordell’s Conjecture (now Faltings’ Theorem) implies that |X F,h(K )| < ∞. Caporaso, Harris, and Mazur ([CHM, 1.1]) have shown that if Lang’s conjecture for varieties of general type is true, then for any number field K , the size |X(K )| of the set of K -rational points of any curve X/K of genus g(X) ≥ 2 can be bounded by a constant depending only on g(X). Prior to the paper [CHM], Mazur and others had asked whether |X(K )| can be bounded by a constant depending only on g(X) and the Mordell-Weil rank of X/K over K (that is, the rank of the group J(K ) of K -rational points of the jacobian J/K of X/K ). These farreaching questions are totally open. As we shall recall in Sect. 1, the method of Chabauty-Coleman sometimes yields a bound for |X F,h(K )| depending only on g(X F,h) when it is known in advance that the Mordell-Weil rank of X F,h/K is small. Unfortunately, the Chabauty-Coleman method does not yield a bound for |X F,h(K )| independent of the coefficients of hzn − F(x, y) for all curves of the form X F,h . It does, however, produce such a nice bound for the number of primitive integral solutions of F(x, y) = h, as we now explain. Let K = Q and OK = Z. A classical Thue equation is an equation F(x, y) = h where F(x, 1) does not have repeated roots. Thue showed in 1909 that such an equation has finitely many solutions (x, y) ∈ Z2 if n ≥ 3.