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.)

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    $\begingroup$ Maybe the results on elliptic divisibility sequences could help en.wikipedia.org/wiki/Elliptic_divisibility_sequence. $\endgroup$ – François Brunault Nov 13 '17 at 8:13
  • $\begingroup$ @FrançoisBrunault: They do indeed. More specifically, an application of Siegel's integrality finiteness theorem as in section 2 of Silverman's Wieferich's criterion and the $abc$ conjecture is enough to give infinitely many $p_i$ (ineffectively). Much more can be said, as I tried to indicate in my answer. $\endgroup$ – Vesselin Dimitrov Nov 13 '17 at 8:30

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).

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    $\begingroup$ Thanks a lot! The Miri--Murty result is exactly the kind of statement that I was hoping would be true. (In dynamical terms, I expect that if $f:X \to X$ is a selfmap of a variety over a number/function field $K$ and $x \in X(K)$ is a point of infinite order, then there exists an integer $n>0$ and infinitely many primes $P_i$ of $K$ such that the reduction of $f^n(x)$ modulo $P_i$ is a periodic point.) $\endgroup$ – ulrich Nov 13 '17 at 9:56
  • $\begingroup$ I've been looking at the proof of the Miri--Murty theorem and have some doubts. They claim (bottom of p. 95) that there are many primes $p$ so that $|E(\mathbb{F}_p)|$ has only one "small" prime factor. However, $|E(\mathbb{Q})|$ could have more than one prime factor so I don't see how their claim could possibly be correct (unless they are assuming that $E(\mathbb{Q})$ is trivial, but this is not stated explicitly anywhere). $\endgroup$ – ulrich Nov 17 '17 at 4:50
  • $\begingroup$ Sorry, $E(\mathbb{Q})$ should have been $E(\mathbb{Q})_{tors}$. $\endgroup$ – ulrich Nov 17 '17 at 5:21
  • $\begingroup$ I don't have an access to their paper at the moment, but as $|E(\mathbb{Q})_{\mathrm{tors}}|$ is in any case uniformly bounded, surely such a condition can't make much of a difference? Could they intend e.g. only one small prime factor $q > 11$, or not dividing the order of the torsion subgroup? $\endgroup$ – Vesselin Dimitrov Nov 17 '17 at 5:28
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    $\begingroup$ I do want a stronger statement than what I had written: what I really need is that (for fixed $E$) given any integer $m > 0$, there exists an infinite set of primes $T_m$ so that the $m$-primary part of the order of $E(\mathbb{F}_p)$ is bounded above by a constant independent of $p \in T_m$. This follows immediately from Miri--Murty, but is of course much weaker. $\endgroup$ – ulrich Nov 19 '17 at 10:14

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