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arXiv · 2210.00659

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

Sushmanth J. Akkarapakam·Patrick Morton
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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.

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Sushmanth J. Akkarapakam, Patrick Morton. 2022-10-03. Periodic points of algebraic functions related to a continued fraction of Ramanujan. https://arxiv.org/abs/2210.00659

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