S-integral points on hyperelliptic curves Homero Renato Gallegos ...

Section 3.8 explains how the variant of the Mordell–Weil sieve in [15] can be used to effectivelysieve up to the bounds obtained in Section 3.6 and thus completely determinethe set of S-<strong>integral</strong> <strong>points</strong> on the curve.3.1 DescentConsider a (non-empty) finite set of rational primes S and the S-<strong>integral</strong> <strong>points</strong> on theaffine model of the hyperelliptic curve (3.0.1), where the polynomial on the right-handside is irreducible. By appropriate scaling, one transforms the problem of finding theS-<strong>integral</strong> <strong>points</strong> on (3.0.1) to that of finding the S-<strong>integral</strong> <strong>points</strong> on a model of the formay 2 = x n + b n−1 x n−1 + · · · + b 0 , (3.1.1)where a and b 0 , . . . , b n−1 are integers, with a ≠ 0. We will denote the polynomial onthe right-hand side by f.Let α be a root of f. In a similar way to that in [15] we will show that for everyS-<strong>integral</strong> point (x, y) on the model (3.1.1) we havex − α = κξ 2 ,where κ, ξ ∈ K = Q(α) and κ is an algebraic integer that comes from a finite computableset. We follow ideas from [15], but we adjust the proofs for the S-<strong>integral</strong> case.In particular, the finite computable set is larger than that in [15] in one case.We will suppose that the Mordell–Weil group J(Q) of the curve C is known. Themethod depends on whether the degree of f is odd or even.3.1.1 The Odd Degree CaseSchaefer [37, Lemma 2.2] proved that when the polynomial f has odd degree, every degree0 divisor defined over Q is linearly equivalent to a divisor of the form ∑ mi=1 (P i−∞),where the set {P 1 , . . . , P m } is stable under the action of the Galois group Gal( ¯Q/Q),20