Lang–Trotter conjecture on primitive points for elliptic curves

The problem concerns how often reductions of points on elliptic curves satisfy the primitive-point condition predicted by Lang and Trotter. No retrieved source gives a proof, counterexample, or resolution of the full conjecture.

Known results

• Verzobio (2020) proved, in a related setting, that for non-torsion $P$ and suitable $Q$, the counting function satisfies $\#N_{P,Q}(x)\gg\sqrt{\log x}$.
• For prime-order torsion $Q$, he obtained $\displaystyle \lim_{x\to\infty}\frac{\#N_{P,Q}(x)}{\sqrt{\log x}}\geq\frac{1}{\sqrt{2\widehat h(P)}}$.
• These are partial lower bounds, not the full Lang–Trotter asymptotic.

2026 journal correction

A correction by Alexandre Benoist and Antonella Perucca, titled “Two natural variants of the Lang-Trotter conjecture on primitive points for elliptic curves,” was published online. The retrieved record does not state what was corrected, so its mathematical implications cannot be assessed.

Current status (as of September 2026): Partial lower bounds are known, but the full conjecture remains open and the 2026 correction has no assessable mathematical effect from the retrieved record.