Geometry of Integral Binary Hermitian Forms

TL;DR

This paper extends Conway's geometric approach from binary quadratic forms over Q to integral binary hermitian forms over quadratic imaginary fields, revealing new geometric structures called 'oceans' in hyperbolic 3-space.

Contribution

It introduces a novel geometric framework for understanding integral binary hermitian forms via 'oceans' in hyperbolic 3-space, generalizing Conway's 'river' concept.

Findings

Indefinite forms define 'oceans' in hyperbolic 3-space.

The geometric structures relate to Bianchi groups and their actions.

Provides a new perspective on the classification of hermitian forms.

Abstract

We generalize Conway's approach to integral binary quadratic forms on Q to study integral binary hermitian forms on quadratic imaginary extensions of Q. In Conway's case, an indefinite form that doesn't represent 0 determines a line ("river") in the spine T associated with SL(2,Z) in the hyperbolic plane. In our generalization, such a form determines a plane ("ocean") in Mendoza's spine associated with the corresponding Bianchi group SL(2,A) in hyperbolic 3-space.