Linnik's Theorem for Sato-Tate Laws on Elliptic Curves with Complex Multiplication

TL;DR

This paper establishes an effective upper bound on the least prime for which the Frobenius angle of a CM elliptic curve falls within a specified interval, extending Linnik's Theorem to the setting of Sato-Tate laws.

Contribution

It proves a Linnik-type bound for the least prime with Frobenius angle in a given interval for CM elliptic curves, with explicit constants and effective computability.

Findings

Bound on the least prime p in terms of conductor N_E and interval width

Effective and explicit constants in the bound

Analogue of Linnik's Theorem for Sato-Tate distributions

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication (CM), and for each prime $p$ of good reduction, let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius. By the Hasse bound, $a_E(p) = 2\sqrt{p} \cos \theta_p$ for a unique $\theta_p \in [0, \pi]$. In this paper, we prove that the least prime $p$ such that $\theta_p \in [\alpha, \beta] \subset [0, \pi]$ satisfies \[ p \ll \left(\frac{N_E}{\beta - \alpha}\right)^A, \] where $N_E$ is the conductor of $E$ and the implied constant and exponent $A > 2$ are absolute and effectively computable. Our result is an analogue for CM elliptic curves of Linnik's Theorem for arithmetic progressions, which states that the least prime $p \equiv a \pmod q$ for $(a,q)=1$ satisfies $p \ll q^L$ for an absolute constant $L > 0$.