Polynomials realizing images of Galois representations of an elliptic curve

Abstract

The aim of the inverse Galois problem is to find extensions of a given field whose Galois group is isomorphic to a given group. In this article, we are interested in subgroups of $\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})$ where $n$ is an integer. We know that, in general, we can realize these groups as the Galois group of a given number field, using the torsion points on an elliptic curve. Specifically, a theorem of Reverter and Vila gives, for each prime $n$, a polynomial, depending on an elliptic curve, whose Galois group is $\mathrm{GL}_2(\mathbb{Z}/n\mathbb{Z})$. In this article, we generalize this theorem in several directions, in particular for $n$ not necessarily prime. We also determine a minimum for the valuations of the coefficients of the polynomials arising in our construction, depending only on $n$.