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

Abstract

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