# Arithmetical invariants of local quaternion orders

## TL;DR

This paper investigates the arithmetical properties of orders in quaternion algebras over DVRs, focusing on invariants like elasticity, distances, and catenary degrees, and provides characterizations of their finiteness.

## Contribution

It characterizes when the elasticity of local quaternion orders is finite and establishes finiteness results for distances and catenary degrees in this setting.

## Key findings

- Finiteness of elasticity is characterized.
- Distances and catenary degrees are finite for these orders.
- Results extend understanding beyond hereditary orders and specific examples.

## Abstract

Let $D$ be a DVR, let $K$ be its quotient field, and let $R$ be a $D$-order in a quaternion algebra $A$ over $K$. The elasticity of $R^\bullet$ is $\rho(R^\bullet) = \sup\{\, k/l : u_1\cdots u_k = v_1 \cdots v_l \text{ with $u_i$, $v_j$ atoms of $R^\bullet$ and $k$, $l \ge 1$} \,\}$ and is one of the basic arithmetical invariants that is studied in factorization theory. We characterize finiteness of $\rho(R^\bullet)$ and show that the set of distances $\Delta(R^\bullet)$ and all catenary degrees $\mathsf c_\mathsf d(R^\bullet)$ are finite. In the setting of noncommutative orders in central simple algebras, such results have only been understood for hereditary orders and for a few individual examples.