On the Constants of the Lang-Trotter Conjecture for CM Elliptic Curves

TL;DR

This paper verifies the equivalence of the Lang-Trotter and Hardy-Littlewood conjectures for 20 CM elliptic curves by analyzing explicit constants in their asymptotics, advancing understanding of prime distributions in these contexts.

Contribution

It confirms the conjectured equality of constants for 20 CM elliptic curves, establishing the conjecture's validity in these cases and linking two major conjectures in number theory.

Findings

Verified the constant equality for 20 CM elliptic curves.

Established the equivalence of the two conjectures for these curves.

Provided explicit constants in the asymptotic formulas.

Abstract

In 2021, Daqing Wan and Ping Xi studied the equivalence of the Lang-Trotter conjecture for CM elliptic curves and the Hardy-Littlewood conjecture for primes represented by a quadratic polynomial. Wan and Xi provided an alternative description of the Lang-Trotter conjecture under the Hardy-Littlewood conjecture. They obtained an explicit constant $\overline{\omega}_{E,r}$ in the asymptotics of the Lang-Trotter conjecture. They further conjectured that this particular constant would be equal to the constant $C_{E,r}$ in the asymptotics of the original Lang-Trotter conjecture. In this paper, we verify the same for $20$ CM elliptic curves, which also establishes the equivalence of the Lang-Trotter Conjecture and the Hardy-Littlewood Conjecture with respect to $r$, for these CM elliptic curves.