A simplified algorithmic realization of Galois actions on special values of modular functions

2025/06/17 by Ja Kyung Koo, Koo, Ja Kyung, Dong Hwa Shin +3
Computer Science · #11F03 #11R65 #Digital Image Processing Techniques #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Primary 11R37 #Rough Sets and Fuzzy Logic #Secondary 11E12

paper · pdf · doi:10.48550/arxiv.2506.14139

openalex publication_date 2025/06/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

We propose an explicit and practical algorithm for computing Galois conjugates and irreducible polynomials for special values of modular functions evaluated at CM points associated with imaginary quadratic orders. Our approach builds upon the theory of extended form class groups developed by Jung et al., offering a refinement of earlier methods by Stevenhagen and Cho, respectively.

Citations

• Inverse limits of CM points on certain Shimura varieties
• Class fields and form class groups for solving certain quadratic Diophantine equations
• Arithmetic properties of orders in imaginary quadratic fields
• On some p-adic Galois representations and form class groups

Related

• Modular forms applied to the computational inverse Galois problem
• Certification of modular Galois representations
• Computing modular Galois representations
• The Explicit Hypergeometric-Modularity Method II
• Computing Galois representations and equations for modular curves X_H(ℓ)