Abstract
Let X be a smooth, projective curve defined over a finite field F r , and ∞ a (closed) point of X. Let κ be the function field of X and A the elements of κ regular everywhere except possibly at ∞. Let ϕ : A → K { τ } be a Drinfeld module defined over K, where ι : A → K is an injective map that identifies K as a finite extension of κ. Suppose that the rank of ϕ is n ⩾ 2 . For a place ℘ of good reduction for ϕ, reducing the coefficients of ϕ modulo ℘ equips the residue field F ℘ with an A-module structure. We establish that the number of places ℘ of K of good reduction for ϕ, of degree x, and such that ϕ ( F ℘ ) is a cyclic A-module, has a natural density. Furthermore, this density is positive if and only if there are infinitely many primes ℘ such that ϕ ( F ℘ ) is cyclic as an A-module. We do not make further restrictions on either A or the ring of endomorphisms End K sep ( ϕ ) .
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