A Herbrand-Ribet theorem for function fields

We prove a function field analogue of the Herbrand-Ribet theorem on cyclotomic number fields. The Herbrand-Ribet theorem can be interpreted as a result about cohomology with $μ_p$-coefficients over the splitting field of $μ_p$, and in our analogue both occurrences of $μ_p$ are replaced with the $\mathfrak{p}$-torsion scheme of the Carlitz module for a prime $\mathfrak{p}$ in $\F_q[t]$.

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