Arithmetic of characteristic p special L-values (with an appendix by V. Bosser)

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.

(v2: several corrections in section 9; v3: minor corrections, improved exposition; v4: minor corrections; v5 minor corrections)

References in corpus (4)

• Values of certain L-series in positive characteristic
• Special L-values of Drinfeld modules
• A Dirichlet unit theorem for Drinfeld modules
• A Herbrand-Ribet theorem for function fields

Cited by in corpus (4)

• Arithmetic of positive characteristic L-series values in Tate algebras
• Log-algebraic identities on Drinfeld modules and special L-values
• An Equivariant Tamagawa Number Formula for Drinfeld Modules and Applications
• An Integral Digit Derivative Basis for Carlitz Prime Power Torsion Extensions