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 ${\theta }_p\in [0,\pi ]$ such that $p+1-\#E({\mathbb{F}}_p)=2\sqrt{p}\cos {\theta }_p$. Let $x>0$ be large, and let $I\subseteq [0,\pi ]$ be a subinterval. We prove that if $\delta >0$ and $\theta >0$ are fixed numbers such that $\delta +\theta <\frac{5}{24}$, $x^{1-\delta }\le h\le x$, and $|I|\ge x^{-\theta }$, 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 $1_{\frac{\pi }{2}\in I}$ equals 1 if $\frac{\pi }{2}\in I$ and $0$ otherwise. We also discuss an extension of this result to the distribution of the Fourier coefficients of holomorphic cuspidal CM newforms.

10 pages, significant revision after referee comments

References in corpus (2)

• Sato-Tate Distributions
• Primes and prime ideals in short intervals