Quaternionic $p$-adic continued fractions

TL;DR

This paper extends the theory of $p$-adic continued fractions to quaternion algebras over $\,\mathbb{Q}$, establishing properties, finiteness criteria, and implications for quadratic equations in this non-commutative setting.

Contribution

It introduces a quaternionic $p$-adic continued fraction framework, generalizing classical results to non-commutative quaternion algebras over $\,\mathbb{Q}$.

Findings

Characterization of elements with finite continued fractions in quaternionic setting

A criterion for finiteness based on quaternionic height

Implications for solutions of quadratic polynomial equations in quaternion algebras

Abstract

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 $B$.