Suppose $E$ is an elliptic curve over $\mathbb Q$ and $x \in E(\mathbb Q)$ is not torsion. We can reduce $x \pmod p$ for a prime $p$ of good reduction and it will have some order $n\_p$ in the group $E(\mathbb F\_p)$. Has there been any work on the asympotitcs of the average of $n\_p$ for $p < X$ as $X \to \infty$?


More generally, suppose $x,y \in E(\mathbb Q)$ are two linearly independent sections and let them generate subgroups $G\_x(p),G\_y(p) \subset E(\mathbb F\_p)$ for a prime of good reduction. Have the asymptotics of the average of $G\_x(p)\cap G\_y(p)$ been studied?


[This](https://mathoverflow.net/questions/285934/orders-of-reductions-of-rational-points-on-elliptic-curves) question seems tangentially related.



What's the average order of the reduction of a section of an elliptic curve
Let $E/\mathbb{Q}$ be an elliptic curve. There exist positive integers $d\_p$ and $e\_p$, with $d\_p|e\_p$, such that group $E(\mathbb{F}\_p)$ is isomorphic to $\mathbb{Z}/d\_p\mathbb{Z} \times \mathbb{Z}/e\_p\mathbb{Z}$. Kowalski conjectured that there exists a constant $c\_E>0$ such that $\sum\_{p\leq x}d\_p\sim c\_E f\_E(x)$, where $f\_E(x)=x$ if $E$ has CM and $f\_E(x) = \mathrm{Li}(x)$ otherwise.


Before Kowalski's paper, Duke showed that if $f$ is any function with $\lim\_{t\to\infty}f(t)=\infty$, then for almost all primes $p$, $E(\mathbb{F}\_p)$ contains a cyclic group of order $p/f(p)$ (where "almost all" is quantified in his paper). He used GRH for certain Dedekind zeta functions to handle the non-CM case, but not the CM case.


Freiberg and Pollack unconditionally proved an $\asymp$ version of Kowalski's conjecture in the CM case. We appear to be far off from such a result in the non-CM case.


Cojocaru has extensively studied the distribution of $p$ such that $E(\mathbb{F}\_p)$ is cyclic, in which case $d\_p=1$. Again, progress towards GRH is usually key for the non-CM case, but not the CM case.


*Duke, William*, [**Almost all reductions modulo (p) of an elliptic curve have a large exponent.**](http://dx.doi.org/10.1016/j.crma.2003.10.006), C. R., Math., Acad. Sci. Paris 337, No. 11, 689-692 (2003). [ZBL1048.11045](https://zbmath.org/?q=an:1048.11045).


*Kowalski, E.*, Analytic problems for elliptic curves, J. Ramanujan Math. Soc. 21, No. 1, 19-114 (2006). [ZBL1144.11069](https://zbmath.org/?q=an:1144.11069).


*Freiberg, Tristan; Pollack, Paul*, [**The average of the first invariant factor for reductions of CM elliptic curves mod (p)**](http://dx.doi.org/10.1093/imrn/rnv027), Int. Math. Res. Not. 2015, No. 21, 11333-11350 (2015). [ZBL1398.11088](https://zbmath.org/?q=an:1398.11088).


*Cojocaru, Alina Carmen*, [**Primes, elliptic curves and cyclic groups**](http://dx.doi.org/10.1090/conm/740/14901), Bucur, Alina (ed.) et al., Analytic methods in arithmetic geometry. Arizona winter school 2016, the University of Arizona, Tucson, AZ, USA, March 12–16, 2016. Providence, RI: American Mathematical Society (AMS); Montreal: Centre de Recherches Mathématiques (CRM). Contemp. Math. 740, 1-69 (2019). [ZBL1452.11069](https://zbmath.org/?q=an:1452.11069).


