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arXiv · 2511.00583

Algebraic functions and class number formulas

Sushmanth J. Akkarapakam·Patrick Morton
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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 $L_{\mathcal{O},9}/K_d$.

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Sushmanth J. Akkarapakam, Patrick Morton. 2025-11-01. Algebraic functions and class number formulas. https://arxiv.org/abs/2511.00583

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