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arXiv · 2208.03983

Quaternionic $p$-adic continued fractions

Laura Capuano·Marzio Mula·Lea Terracini
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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 $B$.

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Laura Capuano, Marzio Mula, Lea Terracini. 2022-08-08. Quaternionic $p$-adic continued fractions. https://arxiv.org/abs/2208.03983

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