Short-interval sector problems for CM elliptic curves

Panidapu, Apoorva; Thorner, Jesse

doi:10.2140/involve.2023.16.1

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Short-interval sector problems for CM elliptic curves

Apoorva Panidapu and Jesse Thorner

Vol. 16 (2023), No. 1, 1–12

DOI: 10.2140/involve.2023.16.1

Abstract

Let $E ∕ ℚ$ 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 (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_{\begin{array}{c} x < p \leq x + h \\ 𝜃_{p} \in I \end{array}} \log p \sim \frac{1}{2} 1_{π ∕ 2 \in I} + \frac{| I |}{2 π},

where $1_{π ∕ 2 \in I}$ equals 1 if $\frac{π}{2} \in I$ and $0$ otherwise. We also discuss an extension of this result to the distribution of the Fourier coefficients of holomorphic cuspidal CM newforms.

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Keywords

CM elliptic curves, equidistribution, Grossencharacter, $L$-function, zero-density estimate

Mathematical Subject Classification

Primary: 11M41

Milestones

Received: 9 May 2021

Revised: 24 April 2022

Accepted: 5 May 2022

Published: 14 April 2023

Communicated by Amanda Folsom

Authors

Apoorva Panidapu
Department of Mathematics and Statistics San Jose State University San Jose, CA United States

Jesse Thorner
Department of Mathematics University of Illinois Urbana, IL United States

Vol. 16, No. 1, 2023