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

Hence for α ∈ K ∗ , the product formula states that∏‖α‖ υ = 1.υ∈M KIn particular, if υ is Archimedean, corresponding to a real or complex embedding σ ofK then⎧⎪⎨ |σ(α)||α| υ = |σ(α)| and ‖α‖ υ =⎪⎩ |σ(α)| 2if σ is realif σ is complex.If υ is finite and p is a prime ideal corresponding to υ, then for α ∈ K\{0} we have‖α‖ υ = N(p) − ordp(α) .For α ∈ K, the (absolute) logarithmic height h(α) is given byh(α) =1[K : Q]∑υ∈M Kd υ log max {1, |α| υ } =1[K : Q]∑υ∈M Klog max {1, ‖α‖ υ } .(3.2.1)The absolute logarithmic height of α is independent of the field K containing α. Notethat in the case K = Q this definition of height agrees with the definition of size givenat the beginning of this chapter.Let S be a finite set of places of K including all the infinite places. Set s = |S|(we exclude the case s = 1). We define the ring of S-integers of K asO S = {α ∈ K : |α| υ ≤ 1, υ /∈ S} ,and the group of S-units as the group of units of O S ,O ∗ S = {α ∈ K : |α| υ = 1, υ /∈ S} .The unit theorem of Dirichlet and Chevalley [30, Chapter V] states that the group O ∗ S isa free abelian group of rank s − 1. A set of generators for O ∗ Sis known as a system offundamental S-units in K. For example, let K = Q and S = {∞, 2, 3}. (By abuse ofnotation we write p instead of the place corresponding to the p-adic valuation.) Then24