Periodic continued fractions and elliptic curves over quadratic fields

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 $\sqrt{f(x)}$ 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 $n\notin \{13,15,17\}$. Moreover we give necessary and sufficient conditions for the existence of such continued fraction expansions with period $13,15$ or $17$ over $K$.

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