Short-interval sector problems for CM elliptic curves

TL;DR

This paper investigates the distribution of angles associated with CM elliptic curves over primes in short intervals, establishing asymptotic formulas under certain conditions, and extends results to Fourier coefficients of CM newforms.

Contribution

It provides new asymptotic results for the distribution of Frobenius angles in short intervals for CM elliptic curves and extends these findings to CM newforms.

Findings

Asymptotic distribution formula for Frobenius angles in short intervals

Conditions on interval length and interval position for the results to hold

Extension of distribution results to Fourier coefficients of CM newforms

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 $\delta+\theta<\frac{5}{24}$, $x^{1-\delta}\leq h\leq x$, and $|I|\geq x^{-\theta}$, then \[ \frac{1}{h}\sum_{\substack{x < p \le x+h \\ \theta_p \in I}}\log{p}\sim \frac{1}{2}\mathbf{1}_{\frac{\pi}{2}\in I}+\frac{|I|}{2\pi}, \] where $\mathbf{1}_{\frac{\pi}{2}\in I}$ 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.