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 ${\text{R}}_{-d}$, 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 $L_{\mathcal{O},9}/K_d$.

44 pages, 4 tables