p-Adically convergent loci in varieties arising from periodic continued fractions
Laura Capuano, Marzio Mula, Lea Terracini, Francesco Veneziano
TL;DR
This work develops a $p$-adic analogue of Brock–Elkies’ PCF framework by studying $p$-adically convergent periodic continued fractions with partial quotients in $\mathbb{Z}[1/p]$ through the PCF varieties $V(F)_{N,k}$. A central convergence criterion reduces convergence to finitely many $p$-adic inequalities on the periodic block, enabling explicit descriptions of the $p$-adic convergent loci for small types $(N,k)$ and linking them to quadratic and Pell-type equations. The paper provides complete classifications for types $(0,1)$, $(1,1)$, and $(0,2)$, and analyzes more delicate cases $(2,1)$, $(1,2)$, and $(0,3)$, including pure radical cases and finiteness results via generalized Pell equations and Siegel-type arguments. It further constructs explicit families of convergent $p$-adic PCFs for $d=a^2+1$ and for negative Pell settings, and presents several type $(1,3)$ examples, including $\sqrt{p^2+1}$ expansions. Overall, the results illuminate the arithmetic and geometric structure governing $p$-adic convergence of PCFs and provide concrete, computable instances connecting continued fractions, Pell theory, and Diophantine analysis in the $p$-adic context.
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.