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.)



Orders of reductions of rational points on elliptic curves

> 
> 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](http://homepages.math.uic.edu/~cojocaru/cojocaru-CNTA-2004.pdf), as well as [this paper](https://link.springer.com/article/10.1007/s00208-008-0249-9) 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).


