Periodic continued fractions and elliptic curves over quadratic fields. (September 2016)

• Record Type: Journal Article
• Authors: Sadek, Mohammad
• Is Part Of: Journal of symbolic computation. Volume 76(2016)
• Page Start: 200
• Page End: 218
• Publication Date: 2016-09
• Subjects: 11A55 -- 11J70; Elliptic curves -- Continued fractions -- Quadratic fields; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.01.003
• Languages: English
• ISSNs: 0747-7171

Abstract

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 .