Arithmetic of function field units

Abstract

Abstract. We prove a "discrete analogue" for Taelman's class modules of certain Conjectures formulated by R. Greenberg for cyclotomic fields.

Stark units in positive characteristic

“…When (the genus 0 case), various works have revealed the importance of certain units in the study of special values of the Goss ‐functions at . To give a simple example, the Carlitz module is considered to play the role of the multiplicative group over , and Anderson showed that the images through the Carlitz exponential of some special units give algebraic elements which are the equivalent of the circular units.…”

“…Recently, Taelman introduced the module of units attached to any Drinfeld module and proved a class formula which states that the special value of the Goss ‐function attached to a Drinfeld module at is the product of a regulator term arising from the module of units and an algebraic term arising from a certain class module. Also, deformations of Goss ‐series values in Tate algebras are investigated by Pellarin and two of the authors . For higher dimensional versions of Drinfeld modules, we refer the reader to .…”

Log‐algebraic identities on Drinfeld modules and special L‐values

“…It is important to mention that Anglès and Tavares Ribeiro considered the ‐deformed Drinfeld module of a given Drinfeld ‐module defined over the ring of integers of a finite extension of , and they established an ‘equivariant’ log‐algebraicity result for in [, Theorem 2] (we note that the exponential of is not the same as the exponential of , but they are closely related). The proof of their Theorem 2 is based on equivariant class module formulas [, Proposition 4], and it differs from Anderson's original methods that we study in this paper. We thank Anglès for clarifying these and related issues.…”

“…We thank Anglès for clarifying these and related issues. We also thank him for sharing his ideas with us about the possibility of the connection between [, Theorem 2] and our Theorem , which would require additional work beyond the scope of this paper. We further refer the reader to [, Corollary 3], where the authors use [, Theorem 2] to recover Anderson's original log‐algebraicity result in the case of the Carlitz module.…”

“…Fixing a Drinfeld A-module φ that is defined over A (see (2)), Taelman [25] defined its associated special L-value L(φ/A), as in (6). One notes that L(φ/A) is identical to the special value at s = 0 of the Goss L-function denoted by L(φ ∨ , 0), arising from the compatible system of the Galois representations on the dual of the Tate module of φ, when the Drinfeld module φ has everywhere good reduction.…”