Periodic continued fractions over $S$-integers in number fields and Skolem’s $p$-adic method
Volume 197 / 2021
Abstract
We generalize the classical theory of periodic continued fractions (PCFs) over ${\mathbf Z}$ to rings $\mathcal O \subseteq \mathbf C $ of $S$-integers in a number field. If it exists, the limit of a PCF $P=[b_1,\ldots ,b_N,\overline {a_1,\ldots ,a_k}]$ of type $(N,k)$ over $\mathcal O $ satisfies a quadratic polynomial in $\mathcal O [x]$. Let ${\mathcal B}$ be the multi-set of roots of such a polynomial. Using the continuant polynomials of Wallis and Euler we define an affine variety $V:= V({\mathcal B})_{N,k}$ generically of dimension $N+k-2$ whose $\mathcal O $-points $V(\mathcal O )$ are equivalent to such PCFs with potential limits in ${\mathcal B}$. Finding the integral points $V(\mathcal O )$ is related to factoring into elementary matrices in $\text {SL}_2(\mathcal O )$. We give an algorithm to determine if a PCF converges and, if so, to compute its limit.
As an example we generalize the prototypical PCF $\sqrt {2}=[1,\overline {2}]$ to $\alpha _n := 2\cos (2\pi /2^{n+2})$ given recursively by $\alpha _0=0$, $\alpha _{n+1}=\sqrt {2+\alpha _n}$, which lie in the tower of quadratic extensions forming the ${\mathbf Z}_2$-extension of $\mathbf {Q}$: $F_n=\mathbf {Q}(\alpha _n)=\mathbf {Q}(\zeta _{2^{n+2}})^+$ with integers $\mathcal O _n={\mathbf Z}[\alpha _n]$. The problem is 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 ${\mathcal B}_{n+1}:= \{\alpha _{n+1}, -\alpha _{n+1}\}$. For the three types $(N,k)=(0,3), (1,2), (2,1)$ where $V({\mathcal B})_{N,k}$ is a curve, Siegel’s theorem implies $V({\mathcal B})_{N,k}(\mathcal O )$ is finite for generic ${\mathcal B}$ and in particular for our ${\mathcal B}_{n+1}$ over $\mathcal O _n$. We find all the $\mathcal O _n$-points on $V({\mathcal B}_{n+1})_{N,k}$ for $n=0,1$ and $N+k\le 3$. When $n=1$ and $N+k=3$, we make extensive use of Skolem’s $p$-adic method for $p=2$, including its application to Ljunggren’s equation $x^2 + 1 =2y^4$.
