Arithmetical invariants of local quaternion orders
Volume 186 / 2018
Acta Arithmetica 186 (2018), 143-177
MSC: 16H10, 11R27, 11S45.
DOI: 10.4064/aa170601-13-8
Published online: 14 September 2018
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 $\rho(R^\bullet) = \sup\{k/l : u_1\cdots u_k = v_1 \cdots v_l$ 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 non-commutative orders in central simple algebras, such results have only been known for hereditary orders and for a few individual examples.
