Squarefree Group Orders in Elliptic-Curve Reductions

Definitions and mathematical statement

Let $E/{\mathbb{Q}}$ be an elliptic curve satisfying $\operatorname{End}_{{\mathbb{Q}}}(E)\cong{\mathbb{Z}}$, where $\operatorname{End}_{{\mathbb{Q}}}(E)$ is the ring of endomorphisms defined over ${\mathbb{Q}}$. For a rational prime $p$ of good reduction, let $E_p/{\mathbb{F}}_p$ denote its reduction and put $N_p=\#E_p({\mathbb{F}}_p)$. A positive integer $n$ is squarefree if $\ell^2\nmid n$ for every prime $\ell$. Write $\pi(x)=\#\{p\le x:p\text{ is prime}\}$ and \[ S_E(x)=\#\{p\le x:p\text{ is a prime of good reduction for }E,\ N_p\text{ is squarefree}\}. \] The supplied question asks whether, for every such $E$, there exists an explicit constant $\delta(E)$, depending on $E$, such that \[ S_E(x)\sim\delta(E)\pi(x)\qquad(x\to\infty). \] Here $f(x)\sim g(x)$ means that $f(x)/g(x)\to1$. Restriction to primes of good reduction makes the reduction notation well defined; the source does not explicitly state this restriction.

Two qualifications remain unresolved. First, the supplied literature assessment calls the stated hypothesis the non-CM case, whereas the condition $\operatorname{End}_{{\mathbb{Q}}}(E)\cong{\mathbb{Z}}$ does not itself impose the geometric non-CM condition $\operatorname{End}_{\overline{{\mathbb{Q}}}}(E)\cong{\mathbb{Z}}$. The displayed hypothesis is retained literally. Second, the source gives neither a positivity hypothesis nor a convention for $\delta(E)=0$. With the usual meaning of $\sim$, the displayed assertion requires $\delta(E)>0$; a possibly zero density statement $S_E(x)=\delta(E)\pi(x)+o(\pi(x))$ would be a different formulation. No choice between these intended meanings, or definition of what counts as an explicit constant, is supplied.

Short English statement

For an elliptic curve over ${\mathbb{Q}}$ with the stated rational endomorphism ring, the question asks for an explicit asymptotic proportion of good primes whose reductions have squarefree group order. The intended non-CM hypothesis and treatment of a possible zero proportion require clarification.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003271$, source problem code catalogue entry, title Squarefree Group Orders in Elliptic-Curve Reductions; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied. The proposed year is recorded as 2016.
• Supplied report citation: Analytic Number Theory (2017), Oberwolfach Reports, pp. 2975--3030, DOI 10.4171/owr/2016/53: https://doi.org/10.4171/owr/2016/53. The background describes these reports as documenting mathematical workshops and their open-problem discussions. The supplied 2017 title date and 2016 DOI/proposed-year information are retained without reconciliation.
• Supplied literature source, "OWR 53 report text": https://oa.tib.eu/renate/bitstreams/7113322d-c5e3-40f5-817e-0d93cf4769d3/download.
• Supplied literature source, "Wong 2024": https://arxiv.org/abs/2408.05904.
• Supplied literature source, "Zarhin 2024": https://doi.org/10.1090/spmj/1483.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes support from average results and advances in the CM case, and says that the stated case remains open. This is supplied dated curation derived from the OWR record, not an independently verified assessment. Its description of the rational endomorphism hypothesis as non-CM requires clarification. Time-sensitive status and cited claims require independent verification.