Periodic continued fractions and elliptic curves over quadratic fields

TL;DR

This paper classifies when the continued fraction expansion of square roots of certain quartic polynomials over quadratic fields is periodic, explicitly listing cases and providing conditions for specific periods like 13, 15, or 17.

Contribution

It explicitly characterizes all quartic polynomials over quadratic fields with periodic continued fractions and determines conditions for specific periods not occurring over rationals.

Findings

Period lengths are limited to a specific finite set.

Explicit polynomials are provided for certain periods.

Necessary and sufficient conditions are established for periods 13, 15, and 17.

Abstract

Let $f(x)$ be a square free quartic polynomial defined over a quadratic field $K$ such that its leading coefficient is a square. If the continued fraction expansion of $\displaystyle \sqrt{f(x)}$ is periodic, then its period $n$ lies in the set \[\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,22,26,30,34\}.\] We write explicitly all such polynomials for which the period $n$ occurs over $K$ but not over $\Q$ and $\displaystyle n\not\in\{ 13,15,17\}$. Moreover we give necessary and sufficient conditions for the existence of such continued fraction expansions with period $13, 15$ or $17$ over $K$.