Frequency of cyclic elliptic-curve groups

Definitions and mathematical statement

Let $q$ be a prime power and let $E/\mathbb F_q$ be an elliptic curve. Its rational points form a finite abelian group, which is cyclic if some point generates all of $E(\mathbb F_q)$. Determine how frequently this occurs for a random elliptic curve over $\mathbb F_q$.

The sampling law must be fixed. If $\mathcal E_q$ is the finite set of $\mathbb F_q$-isomorphism classes, one may use the uniform class distribution or the common automorphism-weighted distribution. The latter probability is \[ P_q^{\mathrm{wt}}= \frac{\sum_{[E]\in\mathcal E_q,\ E(\mathbb F_q)\ \mathrm{cyclic}}|\operatorname{Aut}_{\mathbb F_q}(E)|^{-1}} {\sum_{[E]\in\mathcal E_q}|\operatorname{Aut}_{\mathbb F_q}(E)|^{-1}}. \] Uniform sampling of nonsingular coefficients in a chosen Weierstrass model is another input convention requiring characteristic-dependent specifications. The source does not select a measure or specify whether an exact formula, an asymptotic, or uniform bounds are requested.

The summary attributes an at-least-$75\%$ lower bound and a strictly-less-than-$100\%$ conclusion to Vladut (1999), but supplies neither the ensemble nor the range/limiting quantifier, so these are not asserted for every field. Its fixed-curve-over-$\mathbb Q$, varying-prime density discussion is a distinct ensemble and includes unstated nonobstruction hypotheses for positive-density claims. It is not substituted for the random-curve question at fixed $q$.

Short English statement

How large is the proportion of elliptic curves over $\mathbb F_q$ with cyclic rational-point group? Fix the random-curve measure and the exact or asymptotic counting regime; fixed-curve prime-density results concern a different question.

Sources

• Igor Shparlinski, IPAM Workshop on Number Theory and Cryptography --- Open Problems (2006), attachment 6224, PDF pages 3 and 18, schedule source https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule; accessed 2026-07-29. The supplied provenance records a PDF/legacy-PPT visual audit.
• Related attributions supplied in the summary: Vladut (1999); Lang--Trotter connections; Cojocaru, Murty, and Duke (2001--2006), with ERH conditions; Bianchi--Tsimerman-type work; Freiberg--Kurlberg--Soberón--Vega; and A. Lucchini Arteche. The record supplies no full publication identifiers or precise scope for these references.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $8800036$, source code catalogue entry, Frequency of cyclic elliptic-curve groups. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.