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

Abstract

The exact set of periodic points in Q of the algebraic function ˆ F(z) = (−1±p1 − z4)/z2 is shown to consist of the coordinates of certain solutions (x, y) = ( , ) of the Fermat equation x4+y4 = 1 in ring class fields f over imaginary quadratic fields K = Q(p−d) of odd conductor f, where −d = dKf2 1 (mod 8). This is shown to result from the fact that the 2-adic function F(z) = (−1 + p1 − z4)/z2 is a lift of the Frobenius automorphism on the coordinates for which | |2 < 1, for any d 7 (mod 8), when considered as elements of the maximal unramified extension K2 of the 2-adic field Q2. 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 ˆ F(z) in Q are also determined.