Algebraic functions and class number formulas

TL;DR

This paper establishes a class number formula for extended ring class fields over imaginary quadratic fields by analyzing the periodic points of a specific algebraic function, linking these points to class numbers and Galois group properties.

Contribution

It introduces a novel class number formula involving algebraic functions and periodic points, connecting them to Galois group elements and class numbers in imaginary quadratic fields.

Findings

Number of periodic points relates to class numbers of quadratic orders.

Periodic points correspond to fields generated by algebraic functions.

The formula links algebraic dynamics with class field theory.

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 $L_{\mathcal{O},9}/K_d$.