Hyperbolic angles from Heegner points

TL;DR

This paper investigates the distribution of lattice points on hyperbolic circles centered at Heegner points, showing equidistribution of angles for a density one subset of radii and connecting this to algebraic integers and a hyperbolic circle problem.

Contribution

It establishes a novel link between lattice points on hyperbolic circles and algebraic integers, providing explicit formulations and new bounds using sieve methods.

Findings

Angles of lattice points equidistribute on the unit circle for a density one subset of radii.

Connection between lattice points and algebraic integers with special norm forms.

Explicit formulation of the hyperbolic circle problem as a shifted convolution sum.

Abstract

We study lattice points on hyperbolic circles centred at Heegner points of class number one. Our main result is that, on a density one subset of radii tending to infinity, the angles of such points equidistribute on the unit circle. To prove this, we establish a connection between lattice points and algebraic integers in the associated field having norm of a special form and satisfying a congruence condition. As a by-product of this, we obtain an explicit formulation of the classical hyperbolic circle problem as a shifted convolution sum for the function that counts the number of algebraic integers with given norm. Along the way, we also prove a lower bound for shifted B-numbers, which is done by sieve methods.