The Zariski closure of integral points on varieties parametrizing periodic continued fractions

Abstract

Let $R$ be the ring of $S$-integers in a number field $K$. Let $\mathcal{B}=\{\beta ,{\beta }^{\ast }\}$ be the multi-set of roots of a nonzero quadratic polynomial over $R$. There are varieties $V(\mathcal{B}{)}_{N,k}$ defined over $R$ parametrizing periodic continued fractions $[b_1,\dots ,b_N,\overline{a_1,\dots ,a_k}]$ for $\beta$ or ${\beta }^{\ast }$. We study the $R$-points on these varieties, finding contrasting behavior according to whether groups of units are infinite or not. If $R$ is the rational integers or the ring of integers in an imaginary quadratic field, we prove that the $R$-points of $V(\mathcal{B}{)}_{N,k}$ are not Zariski dense. On the other hand, suppose that $\beta \notin K\cup \{\infty \}$, $R^{\times }$ is infinite, and that there are infinitely many units in the (left) order $R_{\beta }$ of $\beta R+R\subseteq K(\beta )$ with norm to $K$ equal to $(-1{)}^k$. Then we prove that the $R$-points on $V(\mathcal{B}{)}_{1,k}$ are Zariski dense for $k\ge 8$ and the $R$-points on $V(\mathcal{B}{)}_{0,k}$ are Zariski dense for $k\ge 9$. We also prove that $V(\mathcal{B}{)}_{1,k}$ and $V(\mathcal{B}{)}_{0,k}$ are $K$-rational irreducible varieties for $k$ sufficiently large.

References in corpus (1)

• Periodic continued fractions over $S$-integers in number fields and Skolem's $p$-adic method