Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, III

TL;DR

This paper characterizes all periodic points of a specific algebraic function linked to the Rogers-Ramanujan continued fraction, revealing their algebraic nature and implications for class numbers in quadratic fields, especially for prime 5.

Contribution

It determines all periodic points of the algebraic function related to the Rogers-Ramanujan continued fraction and proves a conjecture connecting ring class fields with periodic points for prime 5.

Findings

Periodic points include specific algebraic numbers and conjugates of Rogers-Ramanujan values.

New connections between periodic points and class numbers of quadratic fields.

Proof of a conjecture relating ring class fields to periodic points for p=5.

Abstract

All the periodic points of a certain algebraic function related to the Rogers-Ramanujan continued fraction $r(\tau)$ are determined. They turn out to be $0, \frac{-1 \pm \sqrt{5}}{2}$, and the conjugates over $\mathbb{Q}$ of the values $r(w_d/5)$, where $w_d$ is one of a specific set of algebraic integers, divisible by the square of a prime divisor of 5, in the field $K_d=\mathbb{Q}(\sqrt{-d})$, as $-d$ ranges over all negative quadratic discriminants for which $\left(\frac{-d}{5}\right) = +1$. This yields new insights on class numbers of orders in the fields $K_d$. Conjecture 1 of Part I is proved for the prime $p=5$, showing that the ring class fields over fields of type $K_d$ whose conductors are relatively prime to $5$ coincide with the fields generated over $\mathbb{Q}$ by the periodic points (excluding -1) of a fixed $5$-adic algebraic function.