The twisted Euclidean algorithm: applications to number theory and geometry. (English) Zbl 1467.11127
The author studies orders in quaternion algebras, generalizing the concept of a Euclidean ring and the Euclidean algorithm that has been introduced in his earlier work [A. Sheydvasser, J. Number Theory 204, 41–98 (2019; Zbl 1454.11206)] and has been motivated by work on circle packings [K. E. Stange, Trans. Am. Math. Soc. 370, No. 9, 6169–6219 (2018; Zbl 1393.52013)].
As a consequence he proves (Theorem 1.1) that orders in quaternion algebras, which are Euclidean (in his extended sense) must have class number \(1\), enumerates their isomorphism classes over definite, rational quaternion algebras, and relates the algorithm to the construction of Dirichlet domains for Kleinian subgroups of the isometry group of hyperbolic 4-space. The paper closes with a section of open problems.
As a consequence he proves (Theorem 1.1) that orders in quaternion algebras, which are Euclidean (in his extended sense) must have class number \(1\), enumerates their isomorphism classes over definite, rational quaternion algebras, and relates the algorithm to the construction of Dirichlet domains for Kleinian subgroups of the isometry group of hyperbolic 4-space. The paper closes with a section of open problems.
Reviewer: Alexander Hulpke (Fort Collins)
MSC:
| 11Y40 | Algebraic number theory computations |
| 11R52 | Quaternion and other division algebras: arithmetic, zeta functions |
| 20G30 | Linear algebraic groups over global fields and their integers |
| 51M10 | Hyperbolic and elliptic geometries (general) and generalizations |
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