Compactification of the stacks of Drinfeld's shtukas
type: Item
instance of: scholarly article
title: Compactification of the stacks of Drinfeld's shtukas
zbMATH Open document ID: 1082.11036
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2005-02-23T00:00:00Z
review text: The study and success of Langlands' correspondence for \(\text{Gl}_r\) in positive characteristic are based on the study of the arithmetic and geometry of stacks of Drinfeld's shtukas; c.f. for \(r=2\), the works of V. G. Drinfeld, and for general \(r\) the works L. Lafforgue. Let \(X\) be a smooth, geometrically connected, projective algebraic curve over a finite field \({\mathbb F}_q\). A rank-\(r\) shtuka over \(S\) is a diagram of rank \(r\) vector bundles over \(X\times S\), \({\mathcal E} \hookrightarrow {\mathcal E}' \hookleftarrow {\mathcal E}''\overset \tau \simeq {\mathcal E}^{\sigma}\), where \({\mathcal E}'/{\mathcal E}\) and \({\mathcal E}'/{\mathcal E}''\) are supported for the graph of two morphisms \(0: S\to X\) and \(\infty :S\to X\). \(0\) and \(\infty\) define a morphism \( \text{Cht}^{r,d } \to {X\times X} \), \( \text{Cht}^{r,d}\) being the stack of these objects (\(\deg({\mathcal E} )=d\)). This stack is smooth but neither proper nor of type finite. To avoid these problems one considers a polygon, \(p\), and the stack of finite type, \(\text{Cht}^{r,d, \bar p\leq p}\), of shtukas with Harder-Narasimhan polygon \(\bar p\) and \(\bar p \leq p\), c.f. [\textit{L. Lafforgue}, ``Chtoucas de Drinfeld et conjecture de Ramanujan-Petersson'', Astérisque 243 (1997; Zbl 0899.11026)]. However, \(\text{Cht}^{r,d, \bar p\leq p}\), is not proper, L. Lafforgue introduces a compactification, \(\overline{\text{Cht}^{r,d, \bar p\leq p}}\), of these stacks. This is the stack of iterated shtukas. Grosso modo, an iterated shtuka is a shtuka, \({\mathcal E} \hookrightarrow {\mathcal E}' \hookleftarrow {\mathcal E}''\overset \tau\Leftarrow {\mathcal E}^\sigma\), where \(\tau\) is allowed to be a complete morphism. Verification of the properness of this stack is not direct and it takes several pages [cf. \textit{L. Lafforgue}, J. Am. Math. Soc. 11, No. 4, 1001--1036 (1998; Zbl 1045.11041) and ``Cours à l'Institut Tata sur les chtoucas de Drinfeld et la correspondence d…
Mathematics Subject Classification ID: 11G09
zbMATH DE Number: 2138003
zbMATH Keywords: shtukas
zbMATH Keywords: Langlands correspondence
author: Ngo Dac Tuan
MaRDI profile type: Publication
full work available at URL: j.crma.2004.11.006
OpenAlex ID: W2040884004
cites work: Variation of geometric invariant theory quotients. (With an appendix: ``An example of a thick wall'' by Nicolas Ressayre)
cites work: Une compactification des champs classifiant les chtoucas de Drinfeld
cites work: Geometric invariant theory and flips
DOI: 10.1016/J.CRMA.2004.11.006
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description: scientific article; zbMATH DE number 2138003