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arXiv · 2402.00739

p-Adically convergent loci in varieties arising from periodic continued fractions

Laura Capuano·Marzio Mula·Lea Terracini·Francesco Veneziano
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Abstract

Inspired by several alternative definitions of continued fraction expansions for elements in $\mathbb Q_p$, we study $p$-adically convergent periodic continued fractions with partial quotients in $\mathbb Z[1/p]$. To this end, following a previous work by Brock, Elkies, and Jordan, we consider certain algebraic varieties whose points represent formal periodic continued fractions with period and preperiod of fixed lengths, satisfying a given quadratic equation. We then focus on the $p$-adically convergent loci of these varieties, characterizing the zero and one-dimensional cases.

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Laura Capuano, Marzio Mula, Lea Terracini, Francesco Veneziano. 2026-07-10. p-Adically convergent loci in varieties arising from periodic continued fractions. https://arxiv.org/abs/2402.00739

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