There is a long history of equidistribution results on \(p^{\theta}\) modulo 1, where \(0<\theta<1\) is fixed and \(p\) runs over the primes. The paper under review studies equidistribution of \(\alpha p^{\theta}\), where \(\alpha>0\) and \(p\) is restricted to primes satisfying \(\sigma_p\in C\). Here, \(C\) is a union of conjugacy classes in \(G=\)Gal\((L|\mathbb{Q})\), \(L\) being a Galois extension of \(\mathbb{Q}\), and \(\sigma_p\) is the Frobenius automorphism associated with any prime in \(L\) lying above \(p\). More precisely, a conditional result is obtained for the quantity \[ \sharp\{x<p\le 2p : \delta_1\le \{\alpha p^{\theta}\}< \delta_2, \ \sigma_p\in C\} - (\delta_2-\delta_1)\cdot \frac{\sharp C}{\sharp G} \cdot \pi(x) \] with \(0\le \delta_1<\delta_2\le 1\), which depends on the parameters \(\alpha,\theta,\delta_1,\delta_2\) and \(\Delta\), assuming that the Dedekind zeta function \(\zeta_L(s)\) of \(L\) has no zeros in the half plane \(\operatorname{Re}(s)>1-\Delta\). (In particular, one may take \(\Delta=1/2\) under the Grand Riemann Hypothesis.) To this end, the authors adapt ideas of A. Balog [Arch. Math. 40, 434–440 (1983; Zbl 0517.10038)], J. C. Lagarias and A. M. Odlyzko [in: Algebr. Number Fields, Proc. Symp. London math. Soc., Univ. Durham 1975, 409–464 (1977; Zbl 0362.12011)]. As a consequence, taking a cyclotomic extension \(L\), a bound for \[ \sharp\{x<p\le 2x : \delta_1\le \{\alpha p^{\theta}\}<\delta_2, \ p\equiv a\bmod{q}\} \] for relatively prime \(a\) and \(q\) follows under a quasi-Riemann hypothesis for Dirichlet \(L\)-functions. As an application of their above result for general Galois extensions \(L\), the authors prove an asymptotic result on the density of primes \(p\) for which the trace of Frobenius \(a_E(p)\), \(E\) being an elliptic curve without complex multiplication over \(\mathbb{Q}\), satisfies a congruence condition \[ a_p(E)\equiv [2\sqrt{p}] \bmod{l}, \] where \(l\) is a prime restricted to a certain interval. (In this case, they take \(L\) to be the \(l\)-torsion field of \(E\).) From this, taking \(l\) as large as possible, they derive a bound for the number of primes \(p\) for which \(a_p(E)\) has the extreme value \(a_p(E)=[2\sqrt{p}]\), namely \[ \sharp \{x<p\le 2x : p\nmid N_E, \ a_p(E)=[2\sqrt{p}]\} \ll_E x^{19/18-2\Delta/9}(\log x)^{-1/9}, \] \(N_E\) being the conductor of \(E\).