Typical exponent of an elliptic-curve group

Definitions and mathematical statement

For an elliptic curve $E/\mathbb F_q$, let \[ e_q(E)=\min\{m\ge1:mP=0\text{ for every }P\in E(\mathbb F_q)\} \] be the exponent of its finite abelian group of rational points. Writing $E(\mathbb F_q)\cong\mathbb Z/d_1\mathbb Z\times\mathbb Z/d_2\mathbb Z$, with $d_1\mid d_2$, gives $e_q(E)=d_2$. The source asks whether this exponent is typically close to $q$.

The sampling and comparison are essential. One regime fixes the finite field and samples curves, then studies the resulting distributions as $q$ grows; another fixes an elliptic curve over $\mathbb Q$ and varies primes of good reduction. Sampling Weierstrass models, isomorphism classes or automorphism-weighted classes gives different statements. "Close" may mean ratio tending to $1$, a lower bound $q^{1-o(1)}$, or a weaker comparison. No one of these is specified by the excerpt.

The summary attributes an at-least-75-percent cyclicity result in a varying-curve regime to Vladut (1999). On cyclic instances $e_q(E)=\#E(\mathbb F_q)$, whose ratio to $q$ tends to $1$ by the Hasse bound. It separately attributes to Duke (2003) the estimate $e_p(E)\ge p^{3/4-\varepsilon}$ for almost all primes in a fixed-curve regime, and mentions ERH-conditional cyclicity work of Cojocaru--Murty--Duke. Exact sampling, curve hypotheses, constants and conditional statements are not supplied.

These reports do not specify one universal typical law. In particular a lower bound of exponent $3/4-\varepsilon$ is not an upper bound proving that typical exponents are below $p$, despite such language in the summary. Its "often yes but not always" conclusion is therefore retained only as a qualitative source assessment, not a resolution of an undefined probability/asymptotic question. The partial-progress status requires current primary-literature review.

Short English statement

Is the exponent of a typical elliptic-curve group nearly the field size? The answer must specify how curves or primes are sampled and how close is measured; the source reports results in distinct regimes without fixing a single asymptotic law.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $8800035$, 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 13: 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.
• Vladut (1999); Duke (2003); Cojocaru--Murty--Duke conditional cyclicity work, as cited by the supplied summary. No full titles or exact theorem hypotheses are included.