Short-interval sector problems for CM elliptic curves

Let E/Q be an elliptic curve that has complex multiplication (CM) by an imagi-nary quadratic field K. For a prime p, there exists θ_p ∈ [0, π] such that p + 1 − #E(F_p) = 2^√ p cos θ_p. Let x > 0 be large, and let I ⊆ [0, π] be a subinterval. We prove that if δ > 0 and θ > 0 are fixed numbers such that δ + θ <⁵24^,x1−δ ≤ h ≤ x, and | I | ≥ x^−θ, then 1 h ∑ x<p≤x+h θ_p ∈I log p ∼¹2¹π/2∈I +^|I| 2π^, where 1_π/2∈I equals 1 if^π ∈ I and 0 otherwise. We also discuss an extension of 2 this result to the distribution of the Fourier coefficients of holomorphic cuspidal CM newforms.