Algebraic functions and class number formulas
Sushmanth J. Akkarapakam, Patrick Morton
TL;DR
The paper extends Deuring-type class-number relations to the prime $p=3$ by analyzing extended ring class fields $L_{ ext{O},9}$ over imaginary quadratic fields with $-d\equiv1\pmod{3}$, linking class numbers to Frobenius–inertia data via a cubic algebraic function. It constructs polynomials $R_n(x)$ encoding periodic points of a cubic algebraic function $\hat{F}(z)$, and a $3$-adic lift $F(z)$ whose dynamics mirror Frobenius automorphisms, enabling a precise counting of periodic points corresponding to class fields $\Sigma_{ ext{wp}_3'^2}\Omega_f$ (and variants) with degree $6h(-d)$. The main result is a manifest class-number formula: $\sum_{-d\in\mathfrak{D}_{n,3}} h(-d) = \frac{1}{3}\sum_{k|n} \mu(n/k)3^k$ for $n>1$, derived by factorizing $P_n(x)=\prod_{k|n}R_k(x)^{\mu(n/k)}$ into irreducibles whose degrees sum to $6\sum h(-d)$, matching the total degree of $P_n$ via the periodic-point correspondence. The framework is illustrated with explicit computations and examples (including $p=5$ and $p=7$), and the paper discusses conjectural extensions and limitations across primes, supported by numerical data and conjectural guidance for inertia fields. The work bridges class-field theory, dynamical systems of algebraic functions, and $3$-adic analysis to produce concrete arithmetic formulas with potential for broader applicability to small primes and their powers.
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 $L_{\mathcal{O},9}/K_d$.