Jayce Getz • Mark Goresky
r
Hilbert Modular Forms
with Coefficients in
Intersection Homology
and Quadratic Base
Change
!i
Contents
1 Introduction
1.1
An observation of Serre
1.2
Notational conventions
1.3
The setting
1.4
First main theorem
1.5
Definition of IH*E (Xo(c)) and7X E ( m ) . . . „
1.6
Second main theorem
1.7
Explicit cycles
1.8
Finding cycles dual to families of automorphic forms
1.9
Comments on related literature
1.10 Comparison with Zagier's formula
1.11 Outline of the book
1.12 Problematic primes
Acknowledgement
: . . .
1
3
4
6
8
9
11
13
14
16
17
18
18
2 Review of Chains and Cochains
2.1
Cell complexes and orientations
2.2
Subanalytic sets and stratifications
2.3
Sheaves and the derived category
2.4
The sheaf of chains
2.5
Homology manifolds
2.6
Cellular Borel-Moore chains
2.7
Algebraic cycles
21
22
23
24
25
26
28
3 Review of Intersection Homology and Cohomology
3.1
The sheaf of intersection chains
3.2
The sheaf of intersection cochains
3.3
Homological stratifications
3.4
Products in intersection homology and cohomology
3.5
Finite mappings
3.6
Correspondences
29
30
32
35
37
38
x
Contents
4 Review of Arithmetic Quotients
4.1
The setting
„
4.2
Baily-Borel compactification
4.3
L2 differential forms
4.4
Invariant differential forms
4.5
Hecke correspondences for discrete groups
4.6
Mappings induced by a Hecke correspondence
4.7
The reductive Borel-Serre compactification
4.8
Saper's theorem
4.9
Modular cycles
4.10 Integration
5
6
Generalities on Hilbert Modular Forms and Varieties
5.1
Hilbert modular Shimura varieties
5.2
Hecke congruence groups
5.3
Weights
5.4
Hilbert modular forms
5.5
Cohomological normalization . . .
5.6
Hecke operators
5.7
The Petersson inner product
5.8
Newforms
5.9
Fourier series
5.10 Killing Fourier coefficients
5.11 Twisting
5.12 L-functions
5.12.1 The standard L-function
5.12.2 Rankin-Selberg L-functions
5.12.3 Adjoint L-functions
5.12.4 Asai L-functions . .
5.13 Relationship with Hida's notation
Automorphic Vector Bundles and Local Systems
6.1
Generalities on local systems
6.2
Classical description of automorphic vector bundles
6.2.1 Representations of F
6.2.2 Representations of
6.2.3 Flat vector bundles
6.2.4 Orbifold local systems
6.3
Classical description of automorphy factors
6.4
Adelic automorphic vector bundles
6.4.1 Definitions
41
42
43
46
47
48
49
50
51
53
58
60
61
62
65
66
68
71
72
74
76
81
82
83
84
85
89
'
-. . . .
92
94
94
95
95
96
97
98
98
Contents
6.5
6.6
x 6.7
6.8
6.9
6.10
6.11
xi
6.4.2 Flat bundles
6.4.3 Orbifold bundles . .f
Representations of GL2
Representations of G = Res£/QGL2
The section PZ
The local system £(k, Xo)
Adelic geometric description of automorphic forms
Differential forms
Action of the component group
99
100
100
101
102
103
105
107
109
7 The Automorphic Description of Intersection Cohomology
7.1
The local system £(K , \O )
7.2
The automorphic description of intersection cohomology
7.3
Complex conjugation
7.4
Atkin-Lehner operator
7.5
Pairings of vector bundles
7.6
Generalities on Hecke correspondences
"
7.7
Hecke correspondences in the Hilbert modular case
7.8
Atkin-Lehner-Hecke compatibility
7.9
Integral coefficients
H2
114
116
117
122
126
129
131
132
8 Hilbert Modular Forms with Coefficients in a Hecke Module
8.1
Notation
8.2
Base change for the Hecke algebra
8.3
Hilbert modular forms with coefficients in a Hecke module ....
8.4
Hilbert modular forms with coefficients in intersection homology
8.5
Proof of Theorem 8.3
8.6
The Fourier coefficients of [7(m), $7)XE]///,
136
136
141
143
144
146
9 Explicit Construction of Cycles
9.1
Notation for the quadratic extension L / E
9.2
Canonical section over the diagonal
9.3
Homological properties of Z Q ( C E )
9.4
The twisting correspondence
9.5
Twisting the cycle Z 0( C E )
151
152
157
160
165
10 The Full Version of Theorem 1.3
10.1 Statement of results
10.2 Rankin-Selberg integrals
167
170
xii
Contents
11 Eisenstein Series with Coefficients in Intersection Homology
11.1 Eisenstein series
v
11.2 Invariant classes revisited
11.3 Definition of the V X E (m)
11.4 Statement and proof of Theorem 11.2
179
180
181
181
Appendices
A Proof of Proposition 2.4
A.l
Cellular cosheaves
A.2
Proof of Lemma 2.3
A.3 Proof of Proposition 2.4
183
184
185
B Recollections on Orbifolds
B.l
Effective actions
B.2
Definitions
B.3 Refinement
B.4 Stratification
B.5 Sheaves and cohomology
B.6
Differential forms .
B.7 Groupoids
187
190
192
193
194
196
197
C Basic
C.l
C.2
C.3
C.4
199
200
201
203
Adelic Facts
Adeles and ideles
Characters of L \A L
Characters of GLi(L)\GL!(Al)
Haar measure on the adeles
D Fourier Expansions of Hilbert Modular Forms
D.l Statement of the theorem
D.2 Fourier analysis on GL2(L)\GL2(Al)
D.3 Whittaker models
D.4 Decomposition of Wh
D.5 Computing W^ and Who
D.6 Final steps
205
206
207
208
209
212
E Review of Prime Degree Base Change for GL2
E.l
Automorphic forms and automorphic representations
E.2
Hecke operators
E.3 Agreement of L-functions
E!4
Langlands functoriality
E.5
Prime degree base change for GLi
E.6
Conductors of admissible representations of GL2
214
218
221
222
228
229
Contents
E.7
E.8
xiii
The archimedean places
Global base change
f.
232
234
Bibliography
237
Index of Notation
249
Index of Terminology
253 '