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Short-interval sector problems for CM elliptic curves

Apoorva Panidapu and Jesse Thorner

Vol. 16 (2023), No. 1, 1–12
Abstract

Let E be an elliptic curve that has complex multiplication (CM) by an imaginary quadratic field K. For a prime p, there exists 𝜃p [0,π] such that

p + 1 #E(𝔽p) = 2pcos 𝜃p.

Let x > 0 be large, and let I [0,π] be a subinterval. We prove that if δ > 0 and 𝜃 > 0 are fixed numbers such that δ + 𝜃 < 5 24, x1δ h x, and |I| x𝜃 , then

1 h x<px+h 𝜃pI log p 1 21π2I+|I| 2π,

where 1π2I equals 1 if π 2 I and 0 otherwise. We also discuss an extension of this result to the distribution of the Fourier coefficients of holomorphic cuspidal CM newforms.

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