Orders of reductions of rational points on elliptic curves 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.)
 A: 
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 $P_i$?

Yes, a weak and ineffective form of this at least follows from Siegel's integrality finiteness theorem. See the above comments for a reference.
A lot more can be said. Conditionally on the GRH for Dedekind zeta functions,  Miri and Murty proved  that $|E(\mathbb{F}_P)|$ has at most $16$ prime factors (counting multiplicities!) for $\gg_E X / (\log{X})^2$ of the primes $P$ of norm $N(P) \leq X$. This they did by adapting Chen's method for his almost twin primes theorem; note that the problem of getting infinitely many prime orders of $E(\mathbb{F}_P)$ could be regarded as an elliptic variant of the twin prime problem. See Theorem 5 with the original reference [25] in this paper of Cojocaru, as well as this paper of hers for an unconditional proof in the CM case that $|E(\mathbb{F}_P)|$ is essentially squarefree infinitely often (with the right density in fact).
