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

TL;DR

This paper explores the algebraic and dynamical properties of Ramanujan's continued fraction values at special arguments, revealing their role in generating class fields and identifying periodic points of a related algebraic function.

Contribution

It establishes a connection between Ramanujan's continued fraction values, class field theory, and the periodic points of a specific algebraic function, extending previous results for Rogers-Ramanujan fractions.

Findings

Values generate inertia fields in extended ring class fields.

Conjugates form the complete set of periodic points of the algebraic function.

Results are independent of the parameter d in the quadratic field.

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 $(2) = \wp_2 \wp_2'$ is a product of two prime ideals. These values of $v(\tau)$ are shown to generate the inertia field of $\wp_2$ or $\wp_2'$ 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.