Numerical Invariants of Totally Imaginary Quadratic $\mathbb{Z}[\sqrt{p}]$-orders

TL;DR

This paper classifies certain imaginary quadratic orders over real quadratic orders with prime discriminants, computes their numerical invariants, and discusses applications to quaternion algebra class number calculations.

Contribution

It provides a complete classification and explicit computation of invariants for totally imaginary quadratic orders over specific real quadratic orders, extending understanding of their structure.

Findings

Classified all proper totally imaginary quadratic orders with index greater than one.

Computed class numbers, indices, and local embeddings for these orders.

Facilitated calculations of class numbers of totally definite quaternion algebras.

Abstract

Let $A$ be a real quadratic order of discriminant $p$ or $4p$ with a prime $p$. In this paper we classify all proper totally imaginary quadratic $A$-orders $B$ with index $w(B)=[B^\times: A^\times]>1$. We also calculate numerical invariants of these orders including the class number, the index $w(B)$ and the numbers of local optimal embeddings of these orders into quaternion orders. These numerical invariants are useful for computing the class numbers of totally definite quaternion algebras.