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

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 $\mathbb{Z}[1\mathrm{/}p]$. 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.