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 question seems tangentially related.

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    $\begingroup$ I guess you want to consider the average of $n_p$ over $p<X$ (the question title says "average" but the question body does not). One can give an upper bound of the form $cp $ by some $c<1$ depending on $E$ by considering the various reasons that a small prime $\ell$ could divide the order of the index and noting they can all be detected by Chebotarev (either there is an $\ell$-torsion point and an $\ell$th root of $P$ or two independent $\ell$-torsion points) and then going to prime powers and composites. Presumably someone has conjectured this upper bound is sharp, this seems hard to prove. $\endgroup$ – Will Sawin Apr 18 at 3:28
  • $\begingroup$ Thanks, yes. I do want the average. That's along the lines of what me and my friend were thinking but we wanted to know if anyone had considered this problem before trying to think seriously about it. $\endgroup$ – Asvin Apr 18 at 3:30
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    $\begingroup$ There has been work for the reductions of a fixed subgroup of $E(\mathbb{Q})$, see Akbary, Ghioca, Murty V. Kumar, Reductions of points on elliptic curves, Math. Ann. 347 (2010), no. 2, 365–394. But for their result they need to assume the rank is $>18$. The lower bound is like $p/f(p)$ for any function $f$ tending to infinity arbitrarily slowly. For your second question I don't know. $\endgroup$ – François Brunault Apr 18 at 6:45
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    $\begingroup$ @reuns See this paper of Kurlberg and Pomerance: arxiv.org/abs/1108.5209 $\endgroup$ – Asvin Apr 18 at 21:09
  • $\begingroup$ Thank you, I think it deserves a discussion, how it works for $\sum_{p\le X}\text{order}(g\bmod p) $ for some fixed integer $g$ and how it fails for $\sum_{p\le X}n_p $ $\endgroup$ – reuns Apr 19 at 0:59

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., C. R., Math., Acad. Sci. Paris 337, No. 11, 689-692 (2003). ZBL1048.11045.

Kowalski, E., Analytic problems for elliptic curves, J. Ramanujan Math. Soc. 21, No. 1, 19-114 (2006). ZBL1144.11069.

Freiberg, Tristan; Pollack, Paul, The average of the first invariant factor for reductions of CM elliptic curves mod (p), Int. Math. Res. Not. 2015, No. 21, 11333-11350 (2015). ZBL1398.11088.

Cojocaru, Alina Carmen, Primes, elliptic curves and cyclic groups, 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.

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    $\begingroup$ This is all very interesting, but it answers a different question than the one asked. $\endgroup$ – Chris Wuthrich Apr 18 at 9:53
  • $\begingroup$ @ChrisWuthrich It is fairly related in that $e_p$ is an upper bound for the order of any element so lower bounds for the order of the reduction of an element imply lower bounds for $e_p$ / upper bounds for $d_p$. // In the third paragraph, one of the "CM"s should be "non-CM". $\endgroup$ – Will Sawin Apr 18 at 13:54
  • $\begingroup$ I think the $d_p$ in the second paragraph should be $e_p$? Thanks! This is very interesting. $\endgroup$ – Asvin Apr 18 at 15:57
  • $\begingroup$ @WillSawin The second CM should be non-CM. Corrected. Thanks $\endgroup$ – 2734364041 Apr 18 at 16:07
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    $\begingroup$ @Asvin I made the second paragraph a bit more precise. This should address your comment. $\endgroup$ – 2734364041 Apr 18 at 16:08

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