Abstract
The exact set of periodic points in \(\overline{\mathbb {Q}}\) of the algebraic function \({\hat{F}}(z)=(-1\pm \sqrt{1-z^4})/z^2\) 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 \(-d =d_K f^2 \equiv 1\) (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 \({\hat{F}}(z)\) in \(\overline{\mathbb {Q}}\) are also determined.
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Morton, P. Periodic points of algebraic functions and Deuring’s class number formula. Ramanujan J 50, 323–354 (2019). https://doi.org/10.1007/s11139-018-0120-x
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DOI: https://doi.org/10.1007/s11139-018-0120-x