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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