Residues on buildings and de Rham cohomology of \(p\)-adic symmetric domains.
type: Item
instance of: scholarly article
title: Residues on buildings and de Rham cohomology of \(p\)-adic symmetric domains.
zbMATH Open document ID: 1103.14010
DOI: 10.1215/S0012-7094-01-10615-7
author: Ehud de Shalit
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: For a finite extension \(K\) of the field \(\mathbb{Q}_p\), Drinfeld's \(p\)-adic symmetric domain of dimension \(d\) over \(K\) is defined as the complement \({\mathcal X}\) in \(\mathbb{P}^d\) of the union of all the \(K\)-rational hyperplanes. Endowed with the natural structure of a rigid analytic space, this domain \({\mathcal X}\) is acted on by the group \(G:=\text{PGL}_{d+1}(K)\), and its central role in both the study of Shimura varieties and the representation theory of \(G\) has become manifest during the past three decades. However, in contrast to the classical real symmetric domains, \(p\)-adic symmetric domains are not simply connected in the étale topology, which makes their cohomology theories difficult to tackle. In a fundamental paper published in 1991, \textit{P. Schneider} and \textit{U. Stuhler} [Invent. Math. 105, 47--122 (1991; Zbl 0751.14016)] showed how to compute the cohomology of \({\mathcal X}\) for any cohomology theory satisfying certain additional axioms, including the rigid de Rham cohomology. Their approach established, for each \(0\leq k\leq d\), an isomorphism between \(H^k_{\text{dR}}({\mathcal X})\) and a certain space \(C^k\) of \(k\)-cochains on the Bruhat-Tits building \({\mathcal T}\) of \(G\). In the paper under review, the author re-examines this cohomological result by Schneider and Stuhler, together with a number of questions left open back then. In fact, the author developes a considerably modified, somewhat more explicit and combinatorial approach to computing the de Rham cohomology of \(p\)-adic symmetric domains. To this end, he first analyzes local systems on Bruhat-Tits buildings attached to hyperplane arrangements, describes the related harmonic analysis and the corresponding simplicial cohomology, and then uses this explicit description to develop a novel theory of residues on Bruhat-Tits buildings in arbitrary dimension. The notion of residue of a closed form along simplices in a Bruhat-Tits building, combined wit…
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 20G25
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 14G22
zbMATH DE Number: 1820775
zbMATH Keywords: rigid geometry
zbMATH Keywords: representations
zbMATH Keywords: Shimura varieties
zbMATH Keywords: Drinfeld modules
reviewed by: Werner Kleinert
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1820775