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

Paper | 2010 | $45.00 / £30.95 | ISBN: 9780691142937
Cloth | 2010 | $78.50 / £55.00 | ISBN: 9780691142920
256 pp. | 6 x 9

eBook | 2010 | $45.00 | ISBN: 9781400835393

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Chapter 1 [PDF]

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.

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.

Review:

"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

Table of Contents:

Preface vii
Chapter 1: The fixed point formula 1
Chapter 2: The groups 31
Chapter 3: Discrete series 47
Chapter 4: Orbital integrals at p 63
Chapter 5: The geometric side of the stable trace formula 79
Chapter 6: Stabilization of the fixed point formula 85
Chapter 7: Applications 99
Chapter 8: The twisted trace formula 119
Chapter 9: The twisted fundamental lemma 157
Appendix: Comparison of two versions of twisted transfer factors 189
Bibliography 207
Index 215

Series:

• Annals of Mathematics Studies
  Phillip A. Griffiths, John N. Mather, and Elias M. Stein, Editors

Subject Area:

• Mathematics

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