Short-interval sector problems for CM elliptic curves

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve of conductor $N$ that has complex multiplication by an imaginary quadratic field $K$. For a prime $p\nmid N$, define $\theta_p \in [0, \pi]$ to be the angle for which $p+1-\#E(\mathbb{F}_p) = 2\sqrt{p} \cos \theta_p$, and let $I\subseteq[0,\pi]$ be a subinterval. Let $x>0$ be large. We prove that if $\delta>0$ and $\delta'>0$ are fixed numbers such that $$ \delta+\delta'<\frac{5}{24},\qquad |I|\geq x^{-\delta'},\qquad h\geq x^{1-\delta}, $$ then \[ \sum_{\substack{x < p \le x+h \theta_p \in I}}\log{p}\sim \Big(\frac{1}{2}\mathbf{1}_{\frac{\pi}{2}\in I}+\frac{|I|}{2\pi}\Big)h, \] where $\mathbf{1}_{\frac{\pi}{2}\in I}$ equals 1 if $\pi/2\in I$ and $0$ otherwise. We also discuss an extension of this result to the distribution of the Fourier coefficients of holomorphic cuspidal newforms with complex multiplication.