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

TL;DR

This paper introduces a practical algorithm for computing Galois conjugates and irreducible polynomials of special modular function values at CM points, refining previous methods with a new approach based on extended form class groups.

Contribution

The paper presents a simplified, explicit algorithm leveraging extended form class groups to improve the computation of Galois actions on modular function values at CM points.

Findings

Algorithm efficiently computes Galois conjugates of modular function values.

Refines previous methods by integrating extended form class groups.

Provides practical tools for explicit class field theory applications.

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.