Video library: G. V. Fedorov, On the periodicity of continued fractions in elliptic fields

VIDEO LIBRARY


А.A.Karatsuba's 80th Birthday Conference in Number Theory and Applications May 27, 2017 11:35, Moscow, Department of Mechanics and Mathematics, Lomonosov Moscow State University

On the periodicity of continued fractions in elliptic fields G. V. Fedorov^ab ^a Institute for System Research, Russian Academy of Sciences ^b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract: The paper [1] raised the question of the finiteness of the number squarefree polynomials $f \in \mathbb{Q}[h]$ of fixed degree with periodic continued fraction expansion of $\sqrt{f\mathstrut}$ in the field $\mathbb{Q}((h))$, for which the fields $\mathbb{Q}(h)(\sqrt{f\mathstrut})$ are not isomorphic to one another and to fields of the form $\mathbb{Q}(h)(\sqrt{ch^{n} + 1})$, where $c \in \mathbb{Q}^{\ast}$, $n \in \mathbb{N}$. In the joint paper, V.P. Platonov and G.V. Fedorov [2] obtained a positive answer to this question for elliptic fields $\mathbb{Q}(h)(\sqrt{f\mathstrut})$, $\deg f = 3$. The report will present the results of the article [2]. [1] V.P. Platonov, G.V. Fedorov, On the periodicity of continued fractions in elliptic fields. Doklady Mathematics. 2017 (to appear). [2] V.P. Platonov, G.V. Fedorov, On the periodicity of continued fractions in hyperelliptic fields. Doklady Mathematics. 2017 (to appear). Language: English