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

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.