Minimal Polynomials of Singular Moduli

TL;DR

This paper presents a method to algebraically compute minimal polynomials of singular moduli on modular curves, including Shimura curves, using known norm algorithms, and applies these to calculate algebraic $abc$-ratios.

Contribution

It introduces a novel approach to determine minimal polynomials of singular moduli algebraically, especially for Shimura curves where analytical methods are unavailable.

Findings

Successfully computed minimal polynomials below a discriminant threshold.

Determined algebraic $abc$-ratios for singular moduli.

Demonstrated the effectiveness of norm algorithms in algebraic computations.

Abstract

Given a properly normalized parametrization of a genus-0 modular curve, the complex multiplication points map to algebraic numbers called singular moduli. In the classical case, the maps can be given analytically. However, in the Shimura curve cases, no such analytical expansion is possible. Fortunately, in both cases there are known algorithms for algebraically computing the rational norms of the singular moduli. We demonstrate a method of using these norm algorithms to algebraically determine the minimal polynomial of the singular moduli below a discriminant threshold. We then use these minimal polynomials to compute the algebraic $abc$-ratios for the singular moduli.