Drin'feld shtukas and Langlands correspondence (following Laurent Lafforgue) (Q2736122)

This paper decribes the main ingredients invented by Drinfeld and recently used by \textit{L. Lafforgue} [Invent. Math. 147, No. 1, 1-241 (2002; Zbl 1038.11075)] in order to prove the global Langlands conjecture for \(\text{GL}_r\) over function fields. NEWLINENEWLINENEWLINELet \(X\) be a smooth projective curve over a finite field \(\mathbb{F}_q\), \(F\) its function field, and \(\mathbb{A}_F\) the corresponding ring of adèles. The first part of the paper recalls basic facts about automorphic forms on \(\text{GL}_r(\mathbb{A}_F)\), Galois representations, and the global Langlands conjecture. After that, the author defines the moduli stack \(\text{Cht}^r\) of Drinfeld shtukas of rank \(r\) as well as open substacks of truncated shtukas \(\text{Cht}^{r;\leq p}\) where \(p:[0,r]\rightarrow \mathbb{R}\) is a truncation parameter. A major technical difficulty is the fact that quotient substacks of finite type \(\text{Cht}^{r;\leq p}/a^{\mathbb{Z}}\) (for a fixed \(a\in\mathbb{A}_F^{\times}\), \(\deg(a)>0\)) are not stabilized by Hecke correspondences. In order to overcome this difficulty, Lafforgue constructed a compactification of \(\text{Cht}^{r;\leq p}/a^{\mathbb{Z}}\) and extended Hecke correspondences to the compactified stack \(\overline{\text{Cht}^{r;\leq p}}/a^{\mathbb{Z}}\). The second part of the paper describes these constructions of Lafforgue.