Equidistribution of $\alpha p^{\theta}$ with a Chebotarev condition and applications to extremal primes

TL;DR

This paper proves a distribution result for fractional parts of $eta p^ heta$ with primes under Chebotarev conditions and applies it to count primes with extremal Frobenius traces in elliptic curves, assuming a zero-free hypothesis.

Contribution

It establishes a joint distribution for fractional parts of $eta p^ heta$ for primes in Chebotarev sets and applies this to elliptic curve prime trace analysis.

Findings

Distribution of fractional parts of $eta p^ heta$ in Chebotarev sets is established.

Asymptotic count of primes with extremal Frobenius traces modulo large primes is derived.

Results depend on a zero-free region hypothesis for Dedekind zeta functions.

Abstract

We establish a joint distribution result concerning the fractional part of $\alpha p^\theta$ for $\theta \in (0,1), \ \alpha>0$, where $p$ is a prime satisfying a Chebotarev condition in a fixed finite Galois extension over $\mathbb{Q}$. As an application, for a fixed non-CM elliptic curve $E/\mathbb{Q}$, an asymptotic formula is given for the number of primes at the extremes of the Sato-Tate measure modulo a large prime $\ell$. These are precisely the primes $p$ for which the Frobenius trace $a_p(E)$ satisfies the congruence $a_p(E)\equiv [2\sqrt{p}] \bmod \ell$. We assume a zero-free region hypothesis for Dedekind zeta functions of number fields.