On Langlands program, global fields and shtukas

N. M. Glazunov

Čebyševskij sbornik, Tome 21 (2020) no. 3, pp. 68-83.

Résumé

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. This communication is an introduction to the Langlands Program, global fields and to $D$-shtukas and finite shtukas (over algebraic curves) over function fields. At first recall that linear algebraic groups found important applications in the Langlands program. Namely, for a connected reductive group $ G $ over a global field $ K $, the Langlands correspondence relates automorphic forms on $ G $ and global Langlands parameters, i.e. conjugacy classes of homomorphisms from the Galois group $ {\mathcal Gal} ({\overline K} / K) $ to the dual Langlands group $ \hat G ({\overline {\mathbb Q}} _ p) $. In the case of fields of algebraic numbers, the application and development of elements of the Langlands program made it possible to strengthen the Wiles theorem on the Shimura-Taniyama-Weil hypothesis and to prove the Sato-Tate hypothesis. V. Drinfeld and L. Lafforgue have investigated the case of functional global fields of characteristic $ p> 0 $ ( V. Drinfeld for $ G = GL_2 $ and L. Lafforgue for $ G = GL_r, r $ is an arbitrary positive integer). They have proved in these cases the Langlands correspondence. Under the process of these investigations, V. Drinfeld introduced the concept of a $ F $-bundle, or shtuka, which was used by both authors in the proof for functional global fields of characteristic $ p> 0 $ of the studied cases of the existence of the Langlands correspondence. Along with the use of shtukas developed and used by V. Drinfeld and L. Lafforge, other constructions related to approaches to the Langlands program in the functional case were introduced. G. Anderson has introduced the concept of a $ t $-motive. U. Hartl, his colleagues and students have introduced and have explored the concepts of finite, local and global $ G $-shtukas. 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.

Détail

Keywords: Langlands correspondence, global field, Drinfeld module, shtuka, finite shtuka, local Anderson-module, cotangent complex, formal group.

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