Arithmetic of characteristic p special L-values

Recently, the second author has associated a finite Fq[T]-module H to the Carlitz module over a finite extension of Fq(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 Fq(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.