Periodic points of algebraic functions and Deuring's class number formula

TL;DR

This paper characterizes the periodic points of a specific algebraic function in algebraic number fields, linking them to solutions of Fermat's equation in ring class fields and providing algebraic methods for their computation, thus connecting to Deuring's class number formula.

Contribution

It establishes a connection between periodic points of an algebraic function and solutions to Fermat's equation in ring class fields, offering new algebraic methods for computing these points and related class equations.

Findings

Periodic points correspond to solutions of Fermat's equation in ring class fields.

The algebraic function lifts the Frobenius automorphism in 2-adic fields.

Provides algebraic methods for computing class equations and periodic points.

Abstract

The exact set of periodic points in $\overline{\mathbb{Q}}$ of the algebraic function $\widehat{F}(z)=(-1\pm \sqrt{1-z^4})/z^2$ is shown to consist of the coordinates of certain solutions $(x,y)=(\pi, \xi)$ of the Fermat equation $x^4+y^4=1$ in ring class fields $\Omega_f$ over imaginary quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ of odd conductor $f$, where $-d \equiv 1$ (mod $8$). This is shown to result from the fact that the $2$-adic function $F(z)=(-1+ \sqrt{1-z^4})/z^2$ is a lift of the Frobenius automorphism on the coordinates $\pi$ for which $|\pi|_2<1$, for any $d \equiv 7$ (mod $8$), when considered as elements of the maximal unramified extension $\textsf{K}_2$ of the $2$-adic field $\mathbb{Q}_2$. This gives an interpretation of the case $p=2$ of a class number formula of Deuring. An algebraic method of computing these periodic points and the corresponding class equations $H_{-d}(x)$ is given that is applicable for small periods. The pre-periodic points of $\widehat{F}(z)$ in $\overline{\mathbb{Q}}$ are also determined.