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

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.

arXiv commentThis version of paper is completely reworked. The role of Hilbert's irreducibility theorem in computation of Galois groups is emphasized. The case of $X_0(72)$ is given in details