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

TL;DR

This paper explores the Galois module structure of a finite module associated with the Carlitz module over function fields, deriving analogues of classical cyclotomic number field results and addressing conjectures in the function field setting.

Contribution

It provides the first detailed analysis of the Galois module structure of the Carlitz module's associated finite module in cyclotomic extensions, including analogues of class number formulas and conjecture refutations.

Findings

Derived function field analogues of the p-adic class number formula.

Proved a version of Mazur-Wiles theorem for function fields.

Refuted Anderson's Kummer-Vandiver-type conjecture in this context.

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.