Continued Fractions and the Classification Problem for Elliptic Fields Over Quadratic Fields of Constants

The theory of periodicity of functional continued fractions has found deep applications to the problem of finding and constructing fundamental units and $S$-units, the problem of describing points of finite order on elliptic curves, and the torsion problem in Jacobians of hyperelliptic curves. Functional continued fractions are also of interest from the point of view of arithmetic applications, in particular, to solving norm equations or Pell-type functional equations. In this paper, given any quadratic number field $K$, all square-free fourth-degree polynomials $f(x) \in K[x]$ are described such that $\sqrt{f}$ has periodic continued fraction expansion in the field $K((x))$ of formal power series and the elliptic field $L=K(x)(\sqrt{f})$ has a fundamental $S$-unit of degree $m$, $2 \le m \le 12$, $m \ne 11$, where the set $S$ consists of two conjugate valuations defined on $L$ and related to the uniformizing element $x$ of the field $K(x)$.