On Parshin-Arakelov theorem and uniformity of $S$-integral sections on elliptic surfaces

Abstract

Let $f \colon X \to B$ be a complex elliptic surface and let $\DD \subset X$ be an integral divisor dominating $B$. It is well-known that the Parshin-Arakelov theorem implies the Mordell conjecture over complex function fields by a beautiful covering trick of Parshin. In this article, we construct a similar map in the context of $(S, \DD)$-integral points on elliptic curves over function fields to obtain a new proof of certain uniform finiteness results on the number of $(S, \DD)$-integral points. A second new proof is also given by establishing a uniform bound on the canonical height by means of the tautological inequality. In particular, our construction provides certain uniform quantitative informations on the set-theoretic intersection of curves with the singular divisor in the compact moduli space of stable curves.