On Iwasawa Theory over Function Fields

Let $k_{\infty}$ be a $\Z_p^d$-extension of a global function field $k$ of characteristic $p$. Let $\Cl_{k_{\infty},p}$ be the $p$ completion of the class group of $k_{\infty}$. We prove that the characteristic ideal of the Galois module $\Cl_{k_{\infty},p}$ is generated by the Stickelberger element of Gross which calculates the special values of $L$ functions.