I am looking for references where the following (or similar questions) have been studied:
Let $K$ be a number field or a function field in one variable over a finite field and let $E$ be an elliptic curve (or more generally, an abelian variety) over $K$. If $x \in E(K)$ is a point of infinite order then the order of its reduction modulo a good prime tends to infinity with the order of the residue field.
Are there any results that are known about the prime factorisation of the order of the reduction of $x$? For example, is it known that there is an infinite sequence of rational primes $p_i$ and primes $P_i$ of (the ring of integers of) $K$ such that $p_i$ divides the order of the reduction of $x$ modulo $P_i$?
I would also be interested in similar statements for the order of the group of rational points on the reduction of any elliptic curve $E$ modulo primes of $K$.
(I expect that much stronger results should be true, but don't know the literature in this area.)