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

TL;DR

This paper develops new methods to establish uniform finiteness results for $(S, ext{divisor})$-integral points on elliptic surfaces over function fields, extending classical theorems and providing quantitative bounds.

Contribution

It constructs a novel map for $(S, ext{divisor})$-integral points and offers a new proof of uniform finiteness results using height bounds and tautological inequalities.

Findings

Established uniform finiteness of $(S, ext{divisor})$-integral points on elliptic surfaces.

Provided a uniform bound on the canonical height of integral points.

Gained quantitative insights into intersections with the singular divisor in moduli space.

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.