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

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.