Short-interval sector problems for CM elliptic curves

Abstract

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