Lang–Trotter conjecture for Frobenius traces and fields

Establish the Lang–Trotter asymptotics for a non-CM elliptic curve E over Q: for every fixed integer t and squarefree integer Δ≥1, prove that there are constants C(t,E) and C(Δ,E)>0 such that the counting functions π_E(x,t) and π_E(x,Q(√−Δ)) satisfy the conjectured asymptotic formulas as x tends to infinity.

Background

The paper studies the distribution of Frobenius traces and Frobenius fields attached to the reductions of a non-CM elliptic curve E/Q. For a prime p of good reduction, the Frobenius trace is a_p(E), while the Frobenius field is Q(π_p(E_p)), an imaginary quadratic field determined by the characteristic polynomial X²−a_p(E)X+p.

The conjecture asserts precise asymptotic formulas for the number of primes with a prescribed trace t and, separately, with a prescribed imaginary quadratic Frobenius field Q(√−Δ). The paper focuses on the latter counting problem and proves unconditional lower and upper bounds in restricted families, but does not establish the conjectured asymptotic formula.

References

\begin{conjecture}[Lang--Trotter conjecture] Let $E/Q$ be a non-CM elliptic curve. Then there exist constants $C(t, E)$ and $C(\Delta, E)>0$, depending only on E and t, and on E and \Delta, respectively, such that

\pi_E(x, t)\sim C(t, E)\frac{x{\frac{1}{2}{\log x}, \quad \pi_E(x, Q(\sqrt{-\Delta}))\sim C(\Delta, E)\frac{x{\frac{1}{2}{\log x}

as $x\to \infty$. \end{conjecture}

— Infinitely many primes with a fixed Frobenius field for an elliptic curve over $\mathbb{Q}$  (2608.17639 - Wang, 18 Aug 2026) in Conjecture (Lang–Trotter conjecture), Section 1 (Introduction)