Quaternionic \(p\)-adic continued fractions

We develop a theory of \(p\)-adic continued fractions for a quaternion algebra \(B\) over \(\mathbb Q\) ramified at a rational prime \(p\). Many properties holding in the commutative case can be proven also in this setting. In particular, we focus our attention on the characterization of elements ha...

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Bibliographic Details
Published in:arXiv.org 2022-08
Main Authors: Capuano, Laura, Mula, Marzio, Terracini, Lea
Format: Article
Language:English
Subjects:
Fractions
Mathematical analysis
Polynomials
Quadratic equations
Quaternions
Online Access:Get full text
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Summary:We develop a theory of \(p\)-adic continued fractions for a quaternion algebra \(B\) over \(\mathbb Q\) ramified at a rational prime \(p\). Many properties holding in the commutative case can be proven also in this setting. In particular, we focus our attention on the characterization of elements having a finite continued fraction expansion. By means of a suitable notion of quaternionic height, we prove a criterion for finiteness. Furthermore, we draw some consequences about the solutions of a family of quadratic polynomial equations with coefficients in \(B\).
ISSN:2331-8422