Lang--Trotter Conjecture for CM Elliptic Curves

TL;DR

This paper investigates the Lang--Trotter conjecture for CM elliptic curves, providing unconditional upper bounds and conditional explicit asymptotics for the distribution of primes with specific Frobenius traces, based on classical residue laws and conjectures.

Contribution

It offers the first unconditional upper bounds and a conditional explicit asymptotic formula for the Lang--Trotter conjecture in the CM case, utilizing classical residue reciprocity laws.

Findings

Unconditional upper bounds for ,r(x) confirming part of the conjecture.

Conditional explicit asymptotic formula based on Hardy--Littlewood conjecture.

Summary of classical residue laws and reciprocity laws for quadratic, cubic, and quartic residues.

Abstract

Given an elliptic curve $E$ over $\mathbb{Q}$ and non-zero integer $r$, the Lang--Trotter conjecture predicts a striking asymptotic formula for the number of good primes $p\leqslant x$, denoted by $\pi_{E,r}(x)$, such that the Frobenius trace of $E$ at $p$ is equal to the given integer $r$. We focus on the CM case in this memoir, and show how to realize the following two goals: (1) to give an unconditional estimate for $\pi_{E,r}(x)$, which confirms the upper bound part of the conjecture up to a constant multiple; (2) to give a conditional explicit asymptotic formula for $\pi_{E,r}(x)$ based on the Hardy--Littlewood conjecture on primes represented by quadratic polynomials. For completeness, we also summarize classical results on quadratic, cubic and quartic residues, as well as the corresponding reciprocity laws. This part should be of independent interests and could provide useful materials for more junior readers. We also highlight some possible extensions of the arguments in this memoir that may work for other statistical problems of CM elliptic curves.