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

Abstract

Abstract Inspired by the existence of multiple alternative definitions of continued fraction expansions for elements in $$\mathbb {Q}_p$$ Q p , we study the p -adic convergence of periodic continued fractions with partial quotients in $$\mathbb {Z}[1/p]$$ Z [ 1 / p ] from a geometric point of view. 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, describing the zero and one-dimensional cases by combining tools from algebraic geometry, arithmetic, and the theory of Pell equations and of linear recurrences.