Abstract
Let k \mathbf {k} be a global function field, and let A \mathbf {A} be the elements of k \mathbf {k} regular outside a fixed place ∞ \infty . Let ϕ : A → K { τ } \phi :\mathbf {A}\to K\{\tau \} be a Drinfeld module of generic characteristic and rank n n . For a prime ℘ \wp of K K of good reduction, let F ℘ \mathbb {F}_\wp be the residue field at ℘ \wp , and let χ A ( ϕ ( F ℘ ) ) \chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big ) be the Euler-Poincaré characteristic of F ℘ \mathbb {F}_\wp viewed as an A \mathbf {A} -module. We determine the normal order of the number of distinct prime ideals of A \mathbf {A} dividing χ A ( ϕ ( F ℘ ) ) \chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big ) , denoted by ω A ( χ A ( ϕ ( F ℘ ) ) ) \omega _\mathbf {A}\Big (\chi _\mathbf {A}\big (\phi (\mathbb {F}_\wp )\big )\Big ) , as ℘ \wp runs over primes of K K of degree x x with a specified splitting behaviour. Furthermore, let a ∈ K a\in K be non-torsion for ϕ \phi , and let f a ( ℘ ) f_a(\wp ) be the Euler-Poincaré characteristic of the submodule of ϕ ( F ℘ ) \phi (\mathbb {F}_\wp ) generated by a a modulo ℘ \wp . We also consider the problem of determining the distribution of ω A ( f a ( ℘ ) ) \omega _\mathbf {A}\big (f_a(\wp ) \big ) as ℘ \wp runs over primes of K K of degree x x with a specified splitting behaviour. Note that we do not make restrictions on A \mathbf {A} , ϕ \phi , or its endomorphism ring E n d K {sep} ( ϕ ) \rm {End}_{K^\textrm {{sep}}}(\phi ) .
Accepted Version
Published Version
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