Algebraic functions and class number formulas

2025/11/01 by Akkarapakam, Sushmanth J., Morton, Patrick
#11G15 #11S05 #11S31 #14H05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.00583

Abstract

A class number formula is proved for extended ring class fields L_O,9 over imaginary quadratic fields K_d = ℚ(√(-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 ≥ 2 for this algebraic function equals six times the sum of class numbers of imaginary quadratic orders \textsfR_-d, for which the Artin symbol for a prime ideal divisor \wp₃ 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 \wp₃ in L_O,9/K_d.

Citations

• A proof of Sugawara's conjecture on Hasse-Weber ray class invariants
• Periodic points of algebraic functions related to a continued fraction of Ramanujan

Related

• Class Number Formulas for Certain Biquadratic Fields
• The class number formula for imaginary quadratic fields
• Class Numbers of Orders in Quartic Fields
• Class Numbers via 3-Isogenies and Elliptic Surfaces
• Arithmetic properties of orders in imaginary quadratic fields