Splitting of almost ordinary abelian surfaces in families and the $S$-integrality conjectures

Let $A$ be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic $p$. Suppose $\mathrm{\Delta }$ is an infinite set of positive integers, such that $\left(\frac{m}{p}\right)=1$ for $\mathrm{\forall }m\in \mathrm{\Delta }$. If $A$ does not admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of $A$ has endomorphism ring containing $\mathbb{Z}[x]\mathrm{/}(x^2-m)$ for some $m\in \mathrm{\Delta }$. This implies that there are infinitely many places modulo which $A$ is not simple, generalizing the main result of arXiv:1812.11679 to the non-ordinary case. As an another application, we also generalize the $S$-integrality theorem for elliptic curves over number fields, as proved in arXiv:math/0509485, to the setting of abelian surfaces over global function fields.