Generalized integral points on abelian varieties and the Geometric Lang-Vojta conjecture

TL;DR

This paper introduces a new geometric height concept for integral sections of abelian varieties over function fields, leading to finiteness and growth results that support the Geometric Lang-Vojta conjecture.

Contribution

It develops a hyperbolic-homotopic height framework to study integral points, providing new finiteness and growth results beyond algebraic methods.

Findings

Finiteness of large unions of integral points under certain conditions

Polynomial growth of integral points in specified settings

New evidence supporting the Geometric Lang-Vojta conjecture

Abstract

Let $A$ be an abelian variety over the function field $K$ of a compact Riemann surface $B$. Fix a model $f \colon \mathcal{A} \to B$ of $A/K$ and an effective horizontal divisor $\mathcal{D} \subset \mathcal{A}$. We study $(S, \mathcal{D})$-integral sections $\sigma$ of $\mathcal{A}$ where $S \subset B$ is arbitrary. These sections $\sigma$ are algebraic and satisfy the geometric condition $f(\sigma(B) \cap \mathcal{D})\subset S$. Developing the idea of Parshin, we formulate a hyperbolic-homotopic height of such sections as a substitute for intersection theory to establish new results concerning the finiteness and the polynomial growth of large unions of $(S, \mathcal{D})$-integral points where $S$ is only required to be finite in a thin analytic open subset of $B$. Such results are out of reach of purely algebraic methods and imply new evidence and interesting phenomena to the Geometric Lang-Vojta conjecture.