Hilbert's Irreducibility, Modular Forms, and Computation of Certain Galois Groups

TL;DR

This paper explores the use of Hilbert's irreducibility theorem and modular forms to study Galois groups and splitting fields of rational functions, with computational methods implemented in MAGMA and SAGE.

Contribution

It introduces new computational techniques for analyzing Galois groups of rational functions on modular curves using Hilbert's irreducibility and modular forms.

Findings

Effective algorithms for Galois group computation

Application of modular forms to Galois theory problems

Implementation of methods in MAGMA and SAGE

Abstract

In this paper we discuss applications of our earlier work in studying certain Galois groups and splitting fields of rational functions in $\mathbb Q\left(X_0(N)\right)$ using Hilbert's irreducibility theorem and modular forms. We also consider computational aspect of the problem using MAGMA and SAGE.