Equidistribution of αp^θ with a Chebotarev condition and applications to extremal primes

We establish a joint distribution result concerning the fractional part of (Formula presented.) for (Formula presented.), where p is a prime satisfying a Chebotarev condition in a fixed finite Galois extension over (Formula presented.). As an application, for a fixed non-CM elliptic curve (Formula presented.), 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 (Formula presented.) satisfies the congruence (Formula presented.). We assume a zero-free region hypothesis for Dedekind zeta functions of number fields.