Periodic points of algebraic functions related to a continued fraction of Ramanujan

Abstract

A continued fraction $v(\tau )$ of Ramanujan is evaluated at certain arguments in the field $K=\mathbb{Q}(\sqrt{-d})$, with $-d\equiv 1$ (mod $8$), in which the ideal $(2)={\wp }_2{\wp }_2^{\prime }$ is a product of two prime ideals. These values of $v(\tau )$ are shown to generate the inertia field of ${\wp }_2$ or ${\wp }_2^{\prime }$ in an extended ring class field over the field $K$. The conjugates over $\mathbb{Q}$ of these same values, together with $0,-1\pm \sqrt{2}$, are shown to form the exact set of periodic points of a fixed algebraic function $\hat{F}(x)$, independent of $d$. These are analogues of similar results for the Rogers-Ramanujan continued fraction.

43 pages