Lubin–Tate extensions and Carlitz module over a projective line: an explicit connection

Abstract

In this article we consider different approaches for constructing maximal abelian extensions for local and global geometric fields. The Lubin–Tate theory plays key role in the maximal abelian extension construction for local geometric fields. In the case of global geometric fields, Drinfeld modules are of particular interest. In this paper we consider the simpliest special case
of Drinfeld modules for projective line which is called the Carlitz module.
In the introduction, we provide motivation and a brief historical background on the topics covered in the work.
In the first and second sections we provide brief information about Lubin–Tate modules and Carlitz module.
In the third section we present two main results:
• an explicit connection between the local and global field theory in the geometric case for projective line over finite field: it is proved that the extension tower of Carlitz module induces the tower of the Lubin–Tate extensions.
• a connection between Artin maps of extensions of a function field of an arbitrary projective smooth irreducible curve and extensions of completions of local rings at closed points of
this curve.
In the last section we formulate different open problems and interesting directions for further research, which include generalization first result for an arbitrary smooth projective curve over a finite field and consideration Drinfeld modules of higher rank.