Drinfeld modules with maximal Galois action on their torsion points

Abstract

To each Drinfeld module over a finitely generated field with generic characteristic, one can associate a Galois representation arising from the Galois action on its torsion points. Recent work of Pink and Rütsche has described the image of this representation up to commensurability. Their theorem is qualitative, and the objective of this paper is to complement this theory with a worked out example. In particular, we give examples of Drinfeld modules of rank 2 for which the Galois action on their torsion points is as large as possible. We will follow the approach that Serre used to give explicit examples of his openness theorem for elliptic curves. Using our specific examples, we will numerically test analogues of some well-known elliptic curve conjectures.