The class-module rank and Fitting-ideal conjecture for Carlitz twists

Let $Z(x,n)$ be written uniquely as

$$Z(x,n)=\zeta_C^*(n)(x-1)^{h_n}+\mathcal{O}((x-1)^{h_n+1})$$

in $A[\![x-1]\!]$, with $h_n\geq 0$ and $\zeta_C^*(n)\in A$. Let $\operatorname{Cl}(\underline{A}(n))$ denote the class module of the Carlitz twist $\underline{A}(n)$. Class-module rank and Fitting-ideal conjecture. For all integers $n$, $h_n$ equals the rank of $\operatorname{Cl}(\underline{A}(n))$. For non-positive $n$, $\zeta_C^*(n)$ generates the Fitting ideal of $\operatorname{Cl}(\underline{A}(n))$. The rank assertion extends the preceding rank theorem, while the Fitting-ideal assertion concerns the torsion of the class module for negative twists and was suggested by computations; the authors were unable to explain it by elementary means.