The Carlitz shtuka

TL;DR

This paper explores algebraic constructions of unit and class modules in the Carlitz module context, relating them to Ext modules in shtukas, advancing the understanding of function field analogues of classical number theory structures.

Contribution

It provides new algebraic constructions of unit and class modules associated with the Carlitz module, connecting them to Ext modules in shtukas for the first time.

Findings

New algebraic constructions of unit and class modules

Relationship established between these modules and Ext modules in shtukas

Enhanced understanding of function field analogues of classical number theory concepts

Abstract

Recently we have used the Carlitz exponential map to define a finitely generated submodule of the Carlitz module having the right properties to be a function field analogue of the group of units in a number field. Similarly, we constructed a finite module analogous to the class group of a number field. In this short note more algebraic constructions of these "unit" and "class" modules are given and they are related to Ext modules in the category of shtukas.