Elliptic-curve orders with a large prime divisor

Definitions and mathematical statement

For a prime power $q$ and an elliptic curve $E/\mathbb F_q$, put $N_q(E)=\#E(\mathbb F_q)$. For an integer $N>1$, let $P^+(N)$ be its largest prime divisor, with $P^+(1)=1$. The source asks to quantify curves for which $P^+(N_q(E))$ is large.

For example, if $\mathcal E_q$ is the finite set of $\mathbb F_q$-isomorphism classes, one possible counting function is \[ A(q,Y)=\#\{[E]\in\mathcal E_q:P^+(N_q(E))\ge Y\},\qquad Y>1. \] Determine useful sharp bounds or asymptotics, or the corresponding proportion, for the intended curve-counting model and threshold $Y=Y(q)$. This is an explicit possible formalization: the source does not choose isomorphism-class counting rather than Weierstrass-model or automorphism-weighted counting, and does not define "large". Thresholds such as $q^\theta$ or proximity to $q$ give different questions. Nor is the field-asymptotic regime specified.

The Hasse bound places $N_q(E)$ in $[q+1-2\sqrt q,q+1+2\sqrt q]$, but the distribution of curve orders within this interval is part of the counting problem. The supplied summary mentions Lenstra-type distribution estimates and Ford--Shparlinski-type work on large prime factors, reporting partial quantitative results and some near-$q$ prime-divisor claims. It provides no exact thresholds, proportions, hypotheses or references, so no universal near-$q$ guarantee or optimal exponent is inferred. The 2006 framing and later partial-progress assessment remain source reports pending current primary-literature review.

Short English statement

Count elliptic curves whose finite-field group order has a large prime factor. A precise result must specify the curve-sampling measure, prime-factor threshold and field-asymptotic regime; the source gives only a broad quantitative research target.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $8800042$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06. Attributed to Igor Shparlinski.
• IPAM, Workshop on Number Theory and Cryptography---Open Problems (2006), attachment 6224, PDF page 21: https://www.ipam.ucla.edu/programs/workshops/workshop-i-number-theory-and-cryptography-open-problems/?tab=schedule. Supplied PDF/legacy-PPT visual audit accessed 2026-07-29; it reports 19 inspected surviving attachments.
• Lenstra and Ford--Shparlinski-type estimates are mentioned by the summary without full publication details or exact quantitative statements.