Counting and equidistribution in quaternionic Heisenberg groups

TL;DR

This paper explores the connection between quaternionic hyperbolic geometry and arithmetic counting, providing new formulas and theorems for rational points in quaternionic spaces and their distribution.

Contribution

It introduces a Mertens counting formula and a Neville equidistribution theorem for rational points in quaternionic hyperbolic spaces, advancing understanding of their arithmetic and geometric properties.

Findings

Proved a Mertens counting formula for rational points in quaternionic hyperbolic spaces.

Established a Neville equidistribution theorem for rational points in quaternionic Heisenberg groups.

Enhanced the link between quaternionic geometry and arithmetic counting methods.

Abstract

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension $2$. We prove a Mertens counting formula for the rational points over a definite quaternion algebra $A$ over $\mathbb Q$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over $A$ in quaternionic Heisenberg groups.