Lang and Trotter's conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $E/\mathbb Q$ be an elliptic curve without complex multiplication, meaning $\operatorname{End}_{\overline{\mathbb Q}}(E)=\mathbb Z$. Let $N_E$ be its conductor. For a good prime $p\nmid N_E$ define $a_p(E)=p+1-\#E(\mathbb F_p)$, counting the point at infinity. For $p\ge5$, good reduction is supersingular precisely when $a_p(E)=0$. Put \[ S_E(X)=\#\{p\text{ prime}:5\le p<X,\ p\nmid N_E,\ a_p(E)=0\}. \] Writing $f(X)\sim g(X)$ means $f(X)/g(X)\to1$.

Statement recorded in the supplied source.

Does $S_E(X)$ satisfy the Lang--Trotter prediction \[S_E(X)\sim C_{E,0}\frac{\sqrt X}{\log X}\] for the nonnegative constant $C_{E,0}$ associated to $E$? As in the cited formulation, when $C_{E,0}=0$ this statement means that $S_E(X)$ is bounded. When $C_{E,0}>0$, it asks for the usual asymptotic and hence bounds above and below by positive multiples of $\sqrt X/\log X$. [S2]

Short English statement

Does the number of supersingular good primes of a non-CM rational elliptic curve have the Lang--Trotter square-root-over-logarithm asymptotic, using the source convention that a zero leading constant means bounded count?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400056 (AMR-093-0056). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 David Zywina, Bounds for the Lang--Trotter conjectures, arXiv:1508.07682v1 (2015) §1.1, Conjecture 1.1(a), p. 1. https://arxiv.org/pdf/1508.07682.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Lang and Trotter's conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.