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

and for any place υ ∈ M Klog‖α‖ υ ≤ [K : Q] h(α).Proof. The first equality is an immediate consequence of the definition of absolute logarithmicheight and the product formula. For the second estimate note that for nonnegativex 1 , . . . , x nmax{1, x 1 x 2 · · · x n } ≤ max{1, x 1 } · · · max{1, x n }.We also havemax{1, x 1 + · · · + x n } ≤ n max{1, x 1 } · · · max{1, x n },which implies the third estimate. Finally, the last inequality is also an immediate consequenceof the definition of absolute logarithmic height.Lemma 3.2.2. Let K be a number field of degree d and S a finite set of places of Kincluding the infinite places. Denote by s the cardinality of S. Let ε ∈ K ∗ be a S-unit.Let η ∈ M K be a place of K making ‖ε‖ η minimal. Thenh(ε) ≤ − s d log‖ε‖ η.Proof. Note ‖ε‖ υ = 1 for all υ /∈ S so we can choose η ∈ S. Then 0 < ‖ε‖ η ≤ 1. Now,h(ε) = h(ε −1 ) = 1 ∑max{log‖ε −1 ‖ υ , 0}dυ∈M K≤ 1 ∑log‖ε −1 ‖ η = − s dd log‖ε‖ η,υ∈Sand we have now proved the lemma.Lemma 3.2.3. Let K be a number field of degree d and S a finite set of places of Kincluding the infinite places. Let ε be a S-unit. Then∑|log‖ε‖ υ | = 2d h(ε).υ∈S26