Research

Lower Bounds on the Number of Fields With Galois Groups GL(2, ell)

 

A conjecture of Malle predicts an asymptotic for the number of fields with discriminant bounded by X with a given Galois group, G. The conjecture has been proven in certain cases, but remains open in general. In this thesis, we consider the case when G is the general linear group of dimension 2 over a finite field, GL(2, ell). It is known that number fields with this Galois group can arise from the torsion points of rational elliptic curves. The precise Galois group of the field of torsion points of an elliptic curve is the image of its Galois representation. Elliptic curves whose representation is not surjective, meaning their Galois group is not GL(2,ell), can be parameterized by modular curves. By studying the rational points on these modular curves, we give a lower bound on the number of these fields with bounded discriminant X. Additionally, we discuss future improvements we hope to make.