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

Abstract

We generalize the classical theory of periodic continued fractions (PCFs) over ${\mathbf Z}$ to rings ${\mathcal O}$ of $S$-integers in a number field. Let ${\mathcal B}=\{\beta, {\beta^*}\}$ be the multi-set of roots of a quadratic polynomial in ${\mathcal O}[x]$. We show that PCFs $P=[b_1,\ldots,b_N,\bar{a_1\ldots ,a_k}]$ of type $(N,k)$ potentially converging to a limit in ${\mathcal B}$ are given by ${\mathcal O}$-points on an affine variety $V:=V({\mathcal B})_{N,k}$ generically of dimension $N+k-2$. We give the equations of $V$ in terms of the continuant polynomials of Wallis and Euler. The integral points $V({\mathcal O})$ are related to writing matrices in $\textrm{SL}_2({\mathcal O})$ as products of elementary matrices. We give an algorithm to determine if a PCF converges and, if so, to compute its limit. Our standard example generalizes the PCF $\sqrt{2}=[1,\bar{2}]$ to the ${\mathbf Z}_2$-extension of ${\mathbf Q}$: $F_n={\mathbf Q}(\alpha_n)$, $\alpha_{n}:=2\cos(2\pi/2^{n+2})$, with integers ${\mathcal O}_n={\mathbf Z}[\alpha_n]$. We want to find the PCFs of $\alpha_{n+1}$ over ${\mathcal O}_{n}$ of type $(N,k)$ by finding the ${\mathcal O}_{n}$-points on $V({\mathcal B}_{n+1})_{N,k}$ for ${\mathcal B}_{n+1}:=\{\alpha_{n+1}, -\alpha_{n+1}\}$. There are three types $(N,k)=(0,3), (1,2), (2,1)$ such that the associated PCF variety $V({\mathcal B})_{N,k}$ is a curve; we analyze these curves. For generic ${\mathcal B}$, Siegel's theorem implies that each of these three $V({\mathcal B})_{N,k}({\mathcal O})$ is finite. We find all the ${\mathcal O}_n$-points on these PCF curves $V({\mathcal B}_{n+1})_{N,k}$ for $n=0,1$. When $n=1$ we make extensive use of Skolem's $p$-adic method for $p=2$, including its application to Ljunggren's equation $x^2 + 1 =2y^4$.