A combinatorial refinement of the Kronecker-Hurwitz class number relation

TL;DR

This paper introduces a refined version of the Kronecker-Hurwitz class number relation using a novel tessellation of the Euclidean plane into semi-infinite triangles labeled by PSL_2(Z), which may have broader applications.

Contribution

It provides a new combinatorial refinement of the classical class number relation through a geometric tessellation approach.

Findings

Refinement of the Kronecker-Hurwitz relation achieved

Tessellation of the Euclidean plane into semi-infinite triangles introduced

Potential applications in number theory and geometric group theory

Abstract

We give a refinement of the Kronecker-Hurwitz class number relation, based on a tesselation of the Euclidean plane into semi-infinite triangles labeled by $PSL_2(\mathbb{Z})$ that may be of independent interest.