Stark units in positive characteristic

TL;DR

This paper explores Stark units in positive characteristic, linking them to Anderson's series, and extends log-algebraicity results for Drinfeld modules of various ranks.

Contribution

It introduces a new connection between Stark units and Anderson's equivariant series, generalizes log-algebraicity theorems, and provides alternative proofs using shtukas.

Findings

Module of Stark units derived from Anderson's series

Generalized Taelman-style class formula

Extended log-algebraicity theorem for higher rank Drinfeld modules

Abstract

We show that the module of Stark units associated to a sign-normalized rank one Drinfeld module can be obtained from Anderson's equivariant $A$-harmonic series. We apply this to obtain a class formula \`a la Taelman and to prove a several variable log-algebraicity theorem, generalizing Anderson's log-algebraicity theorem. We also give another proof of Anderson's log-algebraicity theorem using shtukas and obtain various results concerning the module of Stark units for Drinfeld modules of arbitrary rank.