On arithmetic lattices in the plane

TL;DR

This paper studies the distribution and properties of arithmetic lattices in the plane, introducing a height function to estimate the number of classes with bounded height and exploring related invariants.

Contribution

It introduces a natural height function on similarity classes of arithmetic lattices and provides asymptotic estimates for their counts, including semi-stable and well-rounded classes.

Findings

Asymptotic estimates for the number of similarity classes with bounded height

Characterization of semi-stable and well-rounded classes

Discussion of properties of the associated $j$-invariant

Abstract

We investigate similarity classes of arithmetic lattices in the plane. We introduce a natural height function on the set of such similarity classes, and give asymptotic estimates on the number of all arithmetic similarity classes, semi-stable arithmetic similarity classes, and well-rounded arithmetic similarity classes of bounded height as the bound tends to infinity. We also briefly discuss some properties of the $j$-invariant corresponding to similarity classes of planar lattices.