A Refined Version of the Lang-Trotter Conjecture

Abstract

Let E be an elliptic curve defined over the rational numbers and r a fixed integer. Using a probabilistic model consistent with the Chebotarev density theorem for the division fields of E and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to x which have Frobenius trace equal to r, where r is a fixed integer. However, as shown in this note, this asymptotic estimate cannot hold for allr in the interval with a uniform bound for the error term, because an estimate of this kind would contradict the Chebotarev density theorem as well as the Sato-Tate conjecture. The purpose of this note is to refine the Lang-Trotter conjecture, by taking into account the “semicircular law,” to an asymptotic formula that conjecturally holds for arbitrary integers r in the interval , with a uniform error term. We demonstrate consistency of our refinement with the Chebotarev density theorem for a fixed division field, and with the Sato-Tate conjecture. We also present numerical evidence for the refined conjecture.