Periodic continued fractions and elliptic curves over quadratic fields. (September 2016) - Sadek, Mohammad
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 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 ∉ 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 .