On the Cohomology of Certain Non-Compact Shimura Varieties (AM-173)

Sophie Morel

Published by Princeton University Press 2010

On the Cohomology of Certain Non-Compact Shimura Varieties (AM-173)

Sophie Morel

Volume 173 in the series Annals of Mathematics Studies

https://doi.org/10.1515/9781400835393

Overview
Contents

About this book

This book studies the intersection cohomology of the Shimura varieties associated to unitary groups of any rank over Q. In general, these varieties are not compact. The intersection cohomology of the Shimura variety associated to a reductive group G carries commuting actions of the absolute Galois group of the reflex field and of the group G(Af) of finite adelic points of G. The second action can be studied on the set of complex points of the Shimura variety. In this book, Sophie Morel identifies the Galois action--at good places--on the G(Af)-isotypical components of the cohomology.

Morel uses the method developed by Langlands, Ihara, and Kottwitz, which is to compare the Grothendieck-Lefschetz fixed point formula and the Arthur-Selberg trace formula. The first problem, that of applying the fixed point formula to the intersection cohomology, is geometric in nature and is the object of the first chapter, which builds on Morel's previous work. She then turns to the group-theoretical problem of comparing these results with the trace formula, when G is a unitary group over Q. Applications are then given. In particular, the Galois representation on a G(Af)-isotypical component of the cohomology is identified at almost all places, modulo a non-explicit multiplicity. Morel also gives some results on base change from unitary groups to general linear groups.

Author / Editor information

Sophie Morel is a member in the School of Mathematics at the Institute for Advanced Study in Princeton and a research fellow at the Clay Mathematics Institute.

Reviews

This book is a research monograph, yet the author takes care in recalling in detail the relevant notation and previous results instead of just referring to the literature. Also, explicit calculations are given, making the book readable not only for experts but also for interested advanced students.---Eva Viehmann, Mathematical Reviews

Topics

Frontmatter Publicly Available Download PDF	i
Contents Publicly Available Download PDF	v
Preface Requires Authentication Unlicensed Licensed Download PDF	vii
Chapter 1. The fixed point formula Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 2. The groups Requires Authentication Unlicensed Licensed Download PDF	31
Chapter 3. Discrete series Requires Authentication Unlicensed Licensed Download PDF	47
Chapter 4. Orbital integrals at p Requires Authentication Unlicensed Licensed Download PDF	63
Chapter 5. The geometric side of the stable trace formula Requires Authentication Unlicensed Licensed Download PDF	79
Chapter 6. Stabilization of the fixed point formula Requires Authentication Unlicensed Licensed Download PDF	85
Chapter 7. Applications Requires Authentication Unlicensed Licensed Download PDF	99
Chapter 8. The twisted trace formula Requires Authentication Unlicensed Licensed Download PDF	119
Chapter 9. The twisted fundamental lemma Requires Authentication Unlicensed Licensed Download PDF	157
Appendix. Comparison of two versions of twisted transfer factors Requires Authentication Unlicensed Licensed Download PDF	189
Bibliography Requires Authentication Unlicensed Licensed Download PDF	207
Index Requires Authentication Unlicensed Licensed Download PDF	215

(Deutsch)