On optimal embeddings and trees

TL;DR

This paper uses Bruhat-Tits trees to analyze optimal embeddings of commutative orders into quaternion algebras, determining conjugacy classes of Eichler orders and computing local embedding numbers, extending prior work.

Contribution

It extends previous research by classifying conjugacy classes of Eichler orders and calculating local embedding numbers using Bruhat-Tits trees.

Findings

Classified conjugacy classes of Eichler orders for optimal embeddings.

Computed local embedding numbers for quaternion algebras.

Extended previous results on embeddings of orders.

Abstract

We apply the theory of Bruhat-Tits trees to the study of optimal embeddings of two and three dimensional commutative orders into quaternion algebras. Specifically, we determine how many conjugacy classes of global Eichler orders in a quaternion algebra yield optimal representations of such orders. This completes the previous work by C. Maclachlan, who considered only Eichler orders of square free level and integral domains as sub-orders. The same technique is used in the second part of this work to compute local embedding numbers, extending previous results by J. Brzezinski.