Uniform bounds for integral points on families of elliptic curves (Q5960982)

Let \(K\) be a number field, \(E\) a fixed minimal elliptic surface over \(\mathbb{P}^1\) with section \(\Theta\), all defined over \(K\) and a Weierstrass equation for \(E\) defined over \(O_{K,S}\), the ring of \(S\)-integers of \(K\). The author gives conditions under which the number of \(O_{K,S}\)-integral points on the (nonsingular) fiber \(F_t\), \(t\in K\), is bounded by a constant \(N\) depending only on the Weierstrass equation of \(E\) and on \(O_{K,S}\). NEWLINENEWLINENEWLINEThe proof uses the general method pioneered by \textit{L. Caporaso, J. Harris} and \textit{B. Mazur} [J. Am. Math. Soc. 10, 1-35 (1997; Zbl 0872.14017)] proving that Lang's Conjecture implies the boundedness of the number of rational points on a curve of genus \(g>1\) defined over \(K\), where the bound depends only on \(g\) and \(K\), and this method has already been used by Abramovich and Pacelli.