THAINE'S THEOREM IN FUNCTION FIELD

Let F be a finite real abelian extension of a global function field k with G = Gal(F/k). Assume that F is an extension field of the Hilbert class field $K_e$ of k and is contained in a cyclotomic function field $K_n$. Let $\ell$ be any prime number not dividing $ph_k{\mid}G{\mid}$. In this paper, we show that if $\theta{\in}\mathbb{Z}[G]$ annihilates the Sylow $\ell$-subgroup of ${\mathcal{O}}^{\times}_F/{\mathcal{C}}_F$, then (q-1)$\theta$ annihilates the Sylow $\ell$-subgroup of ${\mathcal{Cl}}_F$.