Unramified extensions and geometric $\mathbb{Z}_p$-extensions of global   function fields
  We study on finite unramified extensions of global function fields (function fields of one valuable over a finite field). We show two results. One is an extension of Perret's result about the ideal class group problem. Another is a construction of a geometric $\mathbb{Z}_p$-extension which has a certain property. 
Tsuyoshi Itoh