On Langlands program, global fields and shtukas

TL;DR

This paper surveys key developments in the Langlands program, global fields, and shtukas, highlighting their influence on algebra and number theory, and discusses recent approaches to $D$-shtukas and finite shtukas.

Contribution

It provides a selective overview of significant results and recent approaches in the study of shtukas within the context of the Langlands program.

Findings

Results on Langlands program over algebraic number fields.

Approaches to $D$-shtukas and finite shtukas by Hartl and colleagues.

Connections between shtukas and the internal development of $G$-shtukas theory.

Abstract

The purpose of this paper is to survey some of the important results on Langlands program, global fields, $D$-shtukas and finite shtukas which have influenced the development of algebra and number theory. It is intended to be selective rather than exhaustive, as befits the occasion of the 80-th birthday of Yakovlev, 75-th birthday of Vostokov and 75-th birthday of Lurie. Under assumptions on ground fields results on Langlands program have been proved and discussed by Langlands, Jacquet, Shafarevich, Parshin, Drinfeld, Lafforgue and others. In this review article, we first present results on Langlands program and related representation over algebraic number fields. Then we briefly present approaches by U. Hartl, his colleagues and students to the study of $ D $ --shtukas and finite shtukas. These approaches and our discussion relate to the Langlands program as well as to the internal development of the theory of $ G $-shtukas.