Pair correlation statistics for Sato-Tate sequences

TL;DR

This paper studies the statistical distribution of pairs of angles derived from Hecke eigenvalues associated with various modular forms, providing new insights into their correlation behavior across different settings.

Contribution

It introduces the average pair correlation function for Hecke angles in small intervals, extending analysis to Hilbert modular forms and forms on hyperbolic 3-spaces.

Findings

Derived average pair correlation functions for Hecke angles.

Extended correlation analysis to Hilbert modular forms.

Analyzed correlation statistics for modular forms on hyperbolic 3-spaces.

Abstract

We investigate the pair correlation statistics for sequences arising from Hecke eigenvalues with respect to spaces of primitive modular cusp forms. We derive the average pair correlation function of Hecke angles lying in small subintervals of $[0,1]$. The averaging is done over non-CM newforms of weight $k$ with respect to $\Gamma_0(N).$ We also derive similar statistics for Hilbert modular forms and modular forms on hyperbolic 3-spaces.