Frobenius distributions of Drinfeld modules over finite fields

Abstract

We express the weighted class number of Drinfeld A A -modules of rank two with given characteristic polynomial over the finite field F p = A / p {\mathbb {F}} _{\mathfrak {p}}=A/{\mathfrak {p}} ( p ∈ Spec ⁡ A ({\mathfrak {p}} \in \operatorname {Spec}A , where A = F q [ T ] ) A=\mathbb {F} _q[T]) as an infinite product of local terms. Some auxiliary results of independent interest about characteristic polynomials of Drinfeld modules are given.