Finiteness criteria and uniformity of integral sections in some families of abelian varieties

TL;DR

This paper establishes finiteness criteria and uniform bounds for integral sections in families of abelian varieties over Riemann surfaces, linking geometric divisors to Diophantine finiteness properties.

Contribution

It provides new sufficient conditions on divisors for finiteness of integral sections and introduces uniform bounds in certain trivial families of abelian varieties.

Findings

Finiteness of integral sections under specific divisor conditions.

A uniform bound on the number of integral sections in trivial families.

A numerical criterion for finiteness in trivial abelian surface families.

Abstract

Let $A$ be 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 a certain effective horizontal divisor $\DD \subset \mathcal{A}$. We give a sufficient condition on the divisor $\DD$ for the finiteness of the set of $(S, \DD)$-integral sections for every finite subset $S \subset B$. These integral sections $\sigma$ correspond to rational points in $A(K)$ which satisfy the geometric condition $f ( \sigma(B) \cap \DD)\subset S$. This notion is the geometric variant of integral solutions of a system of \emph{Diophantine equations}. When $\mathcal{A}= A_0 \times B$ for some complex abelian variety $A_0$, we also give a certain uniform bound on the number of $(S, \DD)$-integral sections. For trivial families of abelian surfaces, a numerical criterion on $\DD$ for the finiteness of $(S, \DD)$-integral sections is obtained.