The isomorphism between Lubin-Tate and Drinfeld towers
type: Item
instance of: scholarly article
title: The isomorphism between Lubin-Tate and Drinfeld towers
zbMATH Open document ID: 1136.14001
author: Laurent Fargues
author: Alain Genestier
author: Vincent Lafforgue
published in: Progress in Mathematics
publication date: 2007-05-14T00:00:00Z
review text: Let \(F\) be a local field: if \(\text{ char}(F)=0\) then \(F\) is a finite extension of \({\mathbb Q}_p\), but if \(\text{char}(F)=p>0\) then \(F\cong{\mathbb F}_q((\pi))\) for a finite field \({\mathbb F}_q\). Suppose for the moment that \(F={\mathbb Q}\) and let \(W_F\) denote its Weil group. Let \(n\geq2 \) and let \(D\) be a division algebra of invariant \(\frac{1}{n}\) over \(F\). The Lubin-Tate tower and the Drinfeld tower are certain infinite towers of \(F\)-rigid analytic spaces endowed with actions of \(\text{GL}_n(F)\times D^{\times}\times W_F\) which are defined by moduli problems parametrizing certain \(p\)-divisible groups with additional structures. Similarly for any \(F\). For \(\text{char}(F)=0\) these towers are \(p\)-adic local analogs of Shimura varieties (which in their turn are towers of algebraic varieties over a number field, e.g. modular curves). They are special cases of moduli spaces for \(p\)-divisible groups as defined in full generality by \textit{M. Rapoport} and \textit{T. Zink} [``Period spaces for \(p\)-divisible groups.'' Ann. Math. Stud. 141. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)]. For \(\text{char}(F)=p>0\) these towers are local analogs of moduli spaces for Shtukas. The moduli problems and the actions of the group \(\text{GL}_n(F)\times D^{\times}\) are in a certain sense perpendicular: while \(\text{GL}_n(F)\) acts `vertically' (i.e. by Hecke correspondences) on the Lubin-Tate tower and `horizontally' (i.e. stabilizing each layer) on the Drinfeld tower, \(D^{\times}\) acts horizontally on the Lubin-Tate tower and vertically on the Drinfeld tower. It was expected for a long time, and was recently established for the Lubin-Tate tower by \textit{M. Harris} and \textit{R. Taylor} [``The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich.'' Ann. Math. Stud. 151, Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)], that the \(\ell\)-adic \((\ell\neq …
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 14L05
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11S31
Mathematics Subject Classification ID: 11R39
Mathematics Subject Classification ID: 14G22
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 5152723
zbMATH Keywords: Lubin-Tate tower
zbMATH Keywords: Drinfeld tower
zbMATH Keywords: local Langlands correspondence
zbMATH Keywords: rigid space
zbMATH Keywords: formal scheme
zbMATH Keywords: period space
zbMATH Keywords: Rapoport-Zink-space
zbMATH Keywords: formal group
zbMATH Keywords: moduli space
zbMATH Keywords: etale cohomology
reviewed by: Elmar Große-Klönne
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5152723