Sectorial equidistribution of the roots of $x^2 + 1$ modulo primes

TL;DR

This paper extends the equidistribution results of roots of x^2 + 1 modulo primes to subsequences with angular restrictions, using advanced spectral analysis on higher-dimensional arithmetic quotients.

Contribution

It introduces a new approach involving Poincare series on $SL_2(\mathbb{R})$ quotients and addresses challenges in nonspherical spectral analysis.

Findings

Subsequences of roots remain equidistributed under angular restrictions.

Develops bounds for nonspherical Eisenstein series.

Utilizes a non-spherical Selberg inversion formula.

Abstract

The equation $x^2 + 1 = 0\mod p$ has solutions whenever $p = 2$ or $4n + 1$. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. That the roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec from 1995. We show that a subsequence of the roots of the equation remains equidistributed even when one adds a restriction on the primes which has to do with the angle in the plane formed by their corresponding representation as a sum of squares. Similar to Duke, Friedlander and Iwaniec, we reduce the problem to the study of certain Poincare series, however, while their Poincare series were functions on an arithmetic quotient of the upper half plane, our Poincare series are functions on arithmetic quotients of $SL_2(\mathbb{R})$, as they have a nontrivial dependence on their Iwasawa $\theta$-coordinate. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, which posed a few new challenges. A couple of notable ones were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula.