Morel On the Cohomology of Certain Non-Compact Shimura Varieties

Course Book
ISBN: 978-1-4008-3539-3
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 232 Seiten

Reihe: Annals of Mathematics Studies

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.

Autoren/Hrsg.

Morel, Sophie

Fachgebiete

• Mathematik | Informatik Mathematik Topologie
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie
• Mathematik | Informatik Mathematik Algebra Zahlentheorie

Inhaltsverzeichnis

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

Über Autor(innen)

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.