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To each Drinfeld module over a finitely generated field with genericcharacteristic, one can associate a Galois representation arising from theGalois action on its torsion points. Recent work of Pink and R\"utsche hasdescribed the image of this representation up to commensurability. Theirtheorem is qualitative, and the objective of this paper is to complement thistheory with a worked out example. In particular, we give examples of Drinfeldmodules of rank 2 for which the Galois action on their torsion points is aslarge as possible. We will follow the approach that Serre used to give explicitexamples of his openness theorem for elliptic curves. Using our specificexamples, we will numerically test analogues of some well-known elliptic curveconjectures.

Drinfeld modules with maximal Galois action on their torsion points