Binary hermitian forms and optimal embeddings

TL;DR

This paper explores the relationship between embeddings of quadratic orders into quaternionic orders and integral binary hermitian forms, establishing a discriminant-preserving bijection in specific cases.

Contribution

It introduces a new correspondence linking quadratic order embeddings with binary hermitian forms, preserving discriminants, in certain algebraic settings.

Findings

Discriminant-preserving bijection established

Correspondence between embeddings and hermitian forms in specific cases

Enhances understanding of algebraic structures involving quadratic and quaternionic orders

Abstract

Fix a quadratic order over the ring of integers. An embedding of the quadratic order into a quaternionic order naturally gives an integral binary hermitian form over the quadratic order. We show that, in certain cases, this correspondence is a discriminant preserving bijection between the isomorphism classes of embeddings and integral binary hermitian forms.