Hyperbolic angles from Heegner points

Giacomo Cherubini; Alessandro Fazzari

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arXiv · 2206.08282

Hyperbolic angles from Heegner points

Giacomo Cherubini·Alessandro Fazzari

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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.

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Giacomo Cherubini, Alessandro Fazzari. 2022-06-16. Hyperbolic angles from Heegner points. https://arxiv.org/abs/2206.08282

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