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 $x^2 + 1 =2y^4$.
Categories: math.NT math.AG