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Recently, the second author has associated a finiteF  q    [  T  ]    ‐module   H   to the Carlitz module over a finite extension ofF  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 ofF  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.

Arithmetic of characteristic p special L ‐values