Distribution of elliptic-curve group structures

Definitions and mathematical statement

Let $q$ be a prime power and let $E/{\mathbb{F}}_q$ be an elliptic curve. Its rational-point group has the form \[ E({\mathbb{F}}_q)\cong{\mathbb{Z}}/m{\mathbb{Z}}\times{\mathbb{Z}}/n{\mathbb{Z}},\qquad m\mid n, \] with $m\mid q-1$ and $|mn-(q+1)|\leq2\sqrt q$. These are necessary structural constraints, not an assertion that every pair satisfying them is realized.

For example, under uniform sampling of ${\mathbb{F}}_q$-isomorphism classes one can define \[ C_q(m,n)=\#\{[E]_{{\mathbb{F}}_q}:E({\mathbb{F}}_q)\cong{\mathbb{Z}}/m{\mathbb{Z}}\times{\mathbb{Z}}/n{\mathbb{Z}}\},\qquad P_q(m,n)=\frac{C_q(m,n)}{\sum_{a,b}C_q(a,b)}. \] The source asks to study this type of distribution and identify its correct nonuniform law. It does not prescribe uniform isomorphism-class sampling: weighting each curve by $1/|\operatorname{Aut}_{{\mathbb{F}}_q}E|$, sampling equation coefficients, or conditioning on ordinary/supersingular curves gives different measures. The display is one explicit candidate model.

A limiting law also needs an asymptotic regime and a normalization of the changing groups: $q$ through primes, all prime powers, or fixed-characteristic extensions are different questions. Fixed-field exact counts, asymptotic frequencies, and pointwise uniform estimates are distinct targets. No such regime, error term or conjectural density is supplied, so the request is a distribution programme rather than one specified limiting theorem.

Short English statement

Determine how elliptic-curve rational-point group types are distributed, using a specified sampling measure and finite-field asymptotic regime. Exact counts, cyclic frequencies and a limiting nonuniform law are separate questions.

Sources

• Local UnsolvedMath registry, permanent ID 8800034, exact source code catalogue entry, "Distribution of elliptic-curve group structures"; attributed to Igor Shparlinski, no proposed year supplied. Registry status partially solved, classification PARTIAL-PROGRESS.
• IPAM Workshop on Number Theory and Cryptography---Open Problems (2006), attachment 6224, PDF page 9, https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule. Supplied pdf-and-legacy-ppt visual audit accessed July 29, 2026; provenance records 17 PDFs/two PPT decks inspected with repeated builds excluded. No PDF/PPT was accessed here.
• The summary names Lenstra (1987) for an "essentially uniform" Hasse-interval order statement and Vladut (1999) for an at-least-75-percent cyclic statement. No sampling, limiting regime or exact hypotheses are supplied, so neither phrase is adopted as an unqualified fixed-$q$ probability theorem.
• "On group structures realized by elliptic curves over a finite field", arXiv:1003.3000, is supplied for exact structure counts and realized-type counts, without authors. The summary also names David--Smith for conditional asymptotics using primes in short arithmetic progressions and unspecified Canadian Journal of Mathematics follow-up bounds, without full references.
• The supplied assessment reports substantial progress but no settled general nonuniform limiting law. Provenance flags current primary-literature status and redistribution terms NEEDS_REVIEW.