Elliptic curves with smooth group order

Definitions and mathematical statement

Let $p$ be prime and let $E/\mathbb F_p$ be an elliptic curve, with finite group order $N_E=\#E(\mathbb F_p)$. For $y\geq2$, an integer $N\geq1$ is $y$-smooth if every prime divisor of $N$ is at most $y$. Equivalently, $P^+(N)\leq y$, where $P^+(N)$ is its largest prime factor and $P^+(1)=1$.

Prove a sufficiently strong lower bound for the number of curves with $P^+(N_E)\leq y$ in the intended smoothness regime. For a specified probability law $\nu_p$ on elliptic-curve isomorphism classes, a proportion is \[ S(p,y)=\Pr_{E\sim\nu_p}\left[P^+(\#E(\mathbb F_p))\leq y\right]. \] One may instead count coefficient models or use automorphism-weighted class counts; the source does not choose among them.

The words "smooth" and "sufficiently many" leave the essential quantitative problem unspecified: the bound $y=y(p)$, the desired lower bound on $S(p,y)$ or a raw count, and uniformity over primes are absent. A statement for every sufficiently large prime, one for almost all primes, and an average-over-primes statement are different targets. No unrestricted positive-density claim is invented. The supplied summary reports nonoptimal smoothness estimates from elliptic-curve/Hasse-interval and smooth-number methods, but gives no exact theorem or benchmark.

Short English statement

Show that many elliptic curves over a prime field have group orders with only small prime factors. Fix the smoothness cutoff, curve-counting measure, and quantitative lower bound and prime-uniformity requirements.

Sources

• Igor Shparlinski, IPAM Workshop on Number Theory and Cryptography --- Open Problems (2006), attachment 6224, PDF page 21, 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 in the supplied summary: Lenstra, Hasse-interval distribution methods; Luca--Shparlinski, smooth-order counting; Gaudry, Hess, and Smart, related Weil-descent/index-calculus smoothness. No complete publication identifiers or precise hypotheses are supplied.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $8800041$, source code catalogue entry, Elliptic curves with smooth group order. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.