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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 primesatisfying 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 theextremes of the Sato-Tate measure modulo a large prime $\ell$. These areprecisely the primes $p$ for which the Frobenius trace $a_p(E)$ satisfies thecongruence $a_p(E)\equiv [2\sqrt{p}] \bmod \ell$. We assume a zero-free regionhypothesis for Dedekind zeta functions of number fields.

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