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(s): Sophie Morel
Series: Annals of Mathematics Studies 173
Publisher: Princeton University Press
Year: 2010
Language: English
Pages: 231
Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 6
Preface......Page 8
1.1 Shimura varieties......Page 14
1.2 Local systems and Pink?ˉs theorem......Page 17
1.3 Integral models......Page 19
1.4 Weighted cohomology complexes and intersection complex......Page 23
1.5 Cohomological correspondences......Page 28
1.6 The fixed point formulas of Kottwitz and Goresky-Kottwitz-MacPherson......Page 31
1.7 The fixed point formula......Page 36
2.1 Definition of the groups and of the Shimura data......Page 44
2.2 Parabolic subgroups......Page 46
2.3 Endoscopic groups......Page 48
2.4 Levi subgroups and endoscopic groups......Page 54
3.1 Notation......Page 60
3.3 Transfer......Page 62
3.4 Calculation of certain ......Page 69
4.1 A Satake transform calculation (after Kottwitz)......Page 76
4.2 Explicit calculations for unitary groups......Page 77
4.3 Twisted transfer map and constant terms......Page 84
5.2 Normalization of the transfer factors......Page 92
5.3 Fundamental lemma and transfer conjecture......Page 93
5.4 A result of Kottwitz......Page 94
6.1 Preliminary simplifications......Page 98
6.2 Stabilization of the elliptic part, after Kottwitz......Page 101
6.3 Stabilization of the other terms......Page 102
7.1 Stable trace formula......Page 112
7.2 Isotypical components of the intersection cohomology......Page 116
7.3 Application to the Ramanujan-Petersson conjecture......Page 123
8.1 Nonconnected groups......Page 132
8.2 The invariant trace formula......Page 138
8.3 Stabilization of the invariant trace formula......Page 143
8.4 Applications......Page 148
8.5 A simple case of base change......Page 162
9.1 Notation......Page 170
9.2 Local data......Page 171
9.3 Construction of local data......Page 181
9.4 Technical lemmas......Page 192
9.5 Results......Page 199
A.1 Comparison of ......Page 202
A.2 Relation between inv ......Page 208
A.3 Matching for (G,H)-regular elements......Page 214
Bibliography......Page 220
Index......Page 228
