We derive upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K , as a function of the degree [L : K ]. Our main tool is the adelic openness of the image of Galois representations, due to Serre, Pink and Rütsche. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.

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Let Z = X 1 × · · · × X n be a product of Drinfeld modular curves. We characterize those algebraic subvarieties X ⊂ Z containing a Zariski-dense set of CM points, i.e. points corresponding to n-tuples of Drinfeld modules with complex multiplication (and suitable level structure). This is a characteristic p analogue of a special case of the André-Oort conjecture.