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 "almost all" of the $d_p$'s are "large" (where "almost all" and "large" are 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 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.

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.