Polynomials realizing images of Galois representations of an elliptic curve

TL;DR

This paper generalizes a theorem relating elliptic curves and Galois groups, providing explicit polynomials for a wider class of groups and analyzing their coefficient valuations.

Contribution

It extends Reverter and Vila's theorem to composite n and determines minimal valuations of the associated polynomials' coefficients.

Findings

Generalization of the polynomial construction for non-prime n

Explicit bounds on coefficients' valuations

Broader realization of Galois groups via elliptic curves

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 GL(2,Z/nZ) 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 (2000) gives, for each prime n, a polynomial, depending on an elliptic curve, whose Galois group is GL(2,Z/nZ). In this article, we generalize this theorem in several directions, in particular for n non necessarily prime. We also determine a minimum for the valuations of the coefficients of the polynomials arising in our construction, depending only on n.