Arithmetic of characteristic p special L-values

Abstract

Recently the second author has associated a finite $\F_q[T]$-module $H$ to the Carlitz module over a finite extension of $\F_q(T)$. This module is an analogue of the ideal class group of a number field. In this paper we study the Galois module structure of this module $H$ for `cyclotomic' extensions of $\F_q(T)$. We obtain function field analogues of some classical results on cyclotomic number fields, such as the $p$-adic class number formula, and a theorem of Mazur and Wiles about the Fitting ideal of ideal class groups. We also relate the Galois module $H$ to Anderson's module of circular units, and give a negative answer to Anderson's Kummer-Vandiver-type conjecture. These results are based on a kind of equivariant class number formula which refines the second author's class number formula for the Carlitz module.