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

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 $a_p(E)\equiv [2\sqrt{p}]\mathrm{mod}\ell$. We assume a zero-free region hypothesis for Dedekind zeta functions of number fields.