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arXiv · 2309.09938

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

Anish Ray
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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.

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Anish Ray. 2023-09-18. On the Constants of the Lang-Trotter Conjecture for CM Elliptic Curves. https://arxiv.org/abs/2309.09938

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