Cyclicity of Drinfeld modules

Abstract

Let X be a smooth, projective curve defined over a finite field Fr, 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 EndKsep (φ).