Algebraic functions and class number formulas

Abstract

A class number formula is proved for extended ring class fields $L_{\mathcal{O},9}$ over imaginary quadratic fields $K_d = \mathbb{Q}(\sqrt{-d})$, in which the prime $p = 3$ splits, by determining the fields generated by the periodic points of a well-chosen algebraic function. The number of periodic points of a given period $n \ge 2$ for this algebraic function equals six times the sum of class numbers of imaginary quadratic orders $\textsf{R}_{-d}$, for which the Artin symbol for a prime ideal divisor $\wp_3$ in $K_d$ of $3$ has order $n$ in the Galois group of $F_d/K_d$, where $F_d$ is the inertia field of $\wp_3$ in $L_{\mathcal{O},9}/K_d$.