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

2023/09/18 by Ray, Anish
#11F80 (Secondary) #11G05 (Primary) 11G15 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2309.09938

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 ω_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.

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