Endoscopy for GSp(4) and the Cohomology of Siegel Modular Threefolds
The geometry of modular curves and the structure of their cohomology groups have been a rich source for various number-theoretical applications over the last decades. Similar applications may be expected from the arithmetic of higher dimensional modular v
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1968
Rainer Weissauer
Endoscopy for GSp(4) and the Cohomology of Siegel Modular Threefolds
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Rainer Weissauer Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 69120 Heidelberg Germany [email protected]
ISBN: 978-3-540-89305-9 e-ISBN: 978-3-540-89306-6 DOI: 10.1007/978-3-540-89306-6 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008940147 Mathematics Subject Classification (2000): 11Fxx, 11Gxx, 11Exx, 32Nxx, 57M99 c Springer-Verlag Berlin Heidelberg 2009 ° This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper springer.com
Preface
This volume grew out of a series of preprints which were written and circulated between 1993 and 1994. Around the same time, related work was done independently by Harder [40] and Laumon [62]. In writing this text based on a revised version of these preprints that were widely distributed in summer 1995, I finally did not pursue the original plan to completely reorganize the original preprints. After the long delay, one of the reasons was that an overview of the results is now available in [115]. Instead I tried to improve the presentation modestly, in particular by adding cross-references wherever I felt this was necessary. In addition, Chaps. 11 and 12 and Sects. 5.1, 5.4, and 5.5 were added; these were written in 1998. I will give a more detailed overview of the content of the different chapters below. Before that I should mention that the two main results are the proof of Ramanujan’s conjecture for Siegel modular forms of genus 2 for forms which are not cuspidal representations associated with parabolic subgroups (CAP representations), and the study of the endoscopic lift for the group GSp(4). Both topics are formulated and proved in the first five chapters assuming the stabilization of the trace formula. All the remaining technical results, which are necessary to obtain the stabilized trace formula, are presented in the remaining chapters. Chapter 1 gathers results on the cohomology of Siegel modular threefolds that are used in later chapters, notably in Chap. 3. At the beginning of Chap. 1, important facts f
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