The semi-simple zeta function of quaternionic Shimura varieties
type: Item
instance of: scholarly article
title: The semi-simple zeta function of quaternionic Shimura varieties
zbMATH Open document ID: 1009.11044
author: Harry Reimann
published in: Lecture Notes in Mathematics
publication date: 1997-05-29T00:00:00Z
review text: This valuable monograph is most adequately described by its preface and parts of the author's introduction. The aim of this monograph is the study of the Shimura variety \(\text{Sh}_{D,C}\) attached to the multiplicative group of a quaternion division algebra \(D\) over a totally real number field \(F\) and a sufficiently small open compact subgroup \(C\) of \(D^\times (\mathbb{A}_f)\). For every rational prime \(p\) which is unramified in \(F\) and such that \(C= C_p\times C^p\) where \(C_p\) is a maximal compact subgroup of \(D^\times (\mathbb{Q}_p)\) and \(C^p\) is a subgroup of \(D^\times (\mathbb{A}_f^p)\), the semisimple local zeta-function \(Z_p^{ss} (\text{Sh}_{D,C},s)\) is expressed as a product of integral powers of semisimple local \(L\)-functions of automorphic representations of \(D^\times (\mathbb{A})\). In an appendix a mistake occuring in the constructions of Langlands and Rapoport needed to state their conjecture on the reduction of a Shimura variety in the most general case is corrected, and the conjecture is restated in a slightly stronger form. It is proved for \(\text{Sh}_{D,C}\) up to a question concerning the reduction of Abelian varieties of CM-type, which is discussed in another appendix. If, however, either the quaternion algebra \(D\) is totally indefinite or the condition for the presence of good reduction which is part of the conjecture is satisfied, the proof for \(\text{Sh}_{D,C}\) is complete. In the latter of these two cases we obtain the expected expression for the local factor of the Hasse-Weil zeta-function in terms of local factors of automorphic \(L\)-functions. We now give a short survey of the contents of the present monograph. Its first part comprising \S\S 1-7 is devoted to the construction and detailed study of a reduction of the Shimura variety attached to \(D^\times\) and \(C\) at a certain place \(\wp\). Apart from the totally indefinite case, this variety is not of PEL type. In \S 1 we recall the main idea of its constr…
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 14G35
zbMATH DE Number: 1015090
zbMATH Keywords: Shimura variety
zbMATH Keywords: semisimple local zeta-function
zbMATH Keywords: semisimple local \(L\)-functions
zbMATH Keywords: automorphic representations
zbMATH Keywords: Hasse-Weil zeta-function
reviewed by: Olaf Ninnemann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0093995
OpenAlex ID: W1530939748
DOI: 10.1007/BFB0093995
description: scientific article; zbMATH DE number 1015090