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

The exact set of periodic points in $\bar{\mathbb{Q}}$ of the algebraic function $\widehat{F}(z)=(-1\pm \sqrt{1-z^4})\mathrm{/}{z}^2$ 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 ${\mathrm{\Omega }}_f$ 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 $F(z)=(-1+ \sqrt{1-z^4})/z^2$isaliftoftheFrobeniusautomorphismonthecoordinates$\pi$forwhich$|\pi|_2<1$,forany$d \equiv 7$(mod$8$), when considered as elements of the maximal unramified extension ${\mathsf{\text{K}}}_2$ 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 $\bar{\mathbb{Q}}$ are also determined.