Criteria for p-ordinarity of families of elliptic curves over infinitely many number fields

Abstract: Let $K_i$ be a number field for all $i \in \mathbb{Z}_{> 0}$ and let $\mathcal{E}$ be a family of elliptic curves containing infinitely many members defined over $K_i$ for all $i$. Fix a rational prime $p$. We give sufficient conditions for the existence of an integer $i_0$ such that, for all $i > i_0$ and all elliptic curve $E \in \mathcal{E}$ having good reduction at all $\mathfrak{p} \mid p$ in $K_i$, we have that $E$ has good ordinary reduction at all primes $\mathfrak{p} \mid p$.