On the formal Carlitz module

Abstract

Let $L/{\mathbb{F}}_q(\theta )$ be a finite extension, and let $P$ be a monic irreducible polynomial in ${\mathbb{F}}_q[\theta ]$. We show that the value at 1 of the Carlitz–Goss $P$-adic zeta function attached to $L/{\mathbb{F}}_q(\theta )$ is intimately connected to the module of Taelman's units (for the Carlitz module) associated with the finite extension $L$.