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Tables of Contents for The Geometry of Some Special Arithmetic Quotients
Chapter/Section Title
Page #
Page Count
Introduction
1
15
Moduli spaces of PEL structures
15
21
Shimura's construction
15
7
Endomorphism rings
15
1
Abelian varieties with given endomorphisms
16
2
Arithmetic groups
18
2
PEL structures
20
2
Moduli spaces
22
7
Moduli functors
22
1
Examples
22
3
Coarse and fine moduli spaces
25
1
Moduli spaces of abelian varieties
26
3
Hyperbolic planes
29
7
The rational groups
29
2
Arithmetic groups
31
3
Modular subvarieties
34
2
Arithemtic quotients
36
30
Siegel modular varieties
37
6
The groups
37
1
Compactifications
38
1
Modular subvarieties
38
1
Commensurable subgroups
39
4
Picard modular varieties
43
3
The groups
43
2
Compactification
45
1
Modular subvarieties
46
1
Domains of type IVn
46
4
The groups
46
1
A four-dimensional family of K3-surfaces
47
3
Janus-like algebraic varieties
50
16
The arithmetic quotients
52
5
The theorem
57
4
A remarkable duality
61
5
Projective embeddings
66
42
The tetrahedron in P3
67
7
Arrangements defined by Weyl groups
67
1
Rank 4 arrangements
68
1
The tetrahedron
69
1
A birational transformation
69
2
Fermat covers associated with arrangements
71
1
The hypergeometric differential equation
72
2
The Segre cubic S3
74
10
Segre's cubic primal
74
2
A birational transformation
76
2
Uniformization
78
5
Moduli interpretation
83
1
The Igusa quartic I4
84
10
The quartic locus associated to a configuration of 15 lines
84
2
Igusa's results
86
3
Moduli interpretation
89
2
Birational transformations
91
2
The Siegel modular threefold of level 4
93
1
The Hessian varieties of S3 and I4
94
11
The Nieto quintic
94
4
Two birational transformations
98
2
Moduli interpretation
100
3
A conjecture
103
2
The Coble variety Y
105
3
The 27 lines on a cubic surface
108
60
Cubic surfaces
109
24
The classical approach
109
3
Equations
112
2
Special cubic surfaces
114
14
Del Pezzo surfaces
128
1
Coble's hexahedral form
129
4
Solving algebraic equations
133
15
Algebraic equations
133
1
Resolvents
134
1
The modular equation
135
4
Hermite, Kronecker and Brioschi
139
3
The geometric description
142
1
The icosahedron
143
3
Klein's solution
146
2
Solving the equation of 27th degree
148
20
The unitary reflection groups of order 25,920
150
5
Klein's suggestion
155
5
Coble's solution
160
8
The Burkhardt quartic
168
54
Burkhardt's quartic primal
169
3
Subloci on B4
172
13
The configuration of the 45 nodes
172
6
Curves on B
178
1
Steiner primes
178
1
Jordan primes
179
4
n--primes
183
1
x--primes
184
1
B4 is rational
184
1
B4 is self-Steinnerian
185
10
The Witting configuration in P3
187
3
The map Hess(B) → ST (B) = B
190
1
Level 3 structures on Kummer surfaces
190
5
The solution of the Burkhardt form problem
195
3
Ball quotients
198
4
Logarithmic Chern classes
198
2
Relative proportionality
200
2
Uniformization
202
15
Step I
203
4
Step II
207
10
Moduli interpretation
217
5
Abelian surfaces with a level 3 structure
217
1
Abelian fourfolds with complex multiplication
218
3
The dual variety
221
1
A Gem of the modular universe
222
33
The Weyl group W (E6)
223
11
Notations
223
2
Roots
225
1
Vectors
226
2
The arrangement defined by W (E6)
228
1
Special Loci
228
4
The configuration defined by the 36 points
232
2
Invariants
234
1
The invariant quintic
234
5
Equation
234
1
Singular locus
235
2
Resolution of singularities
237
1
I5 is rational
238
1
Hyperplane sections
239
5
Lower-dimensional intersections
239
1
Reducible hyperplane sections
239
2
Special hyperplane sections
241
1
Generic hyperplane sections
241
3
Tangent hyperplane sections
244
1
Projection from a triple point
244
7
The cuspidal model
244
2
Projection from a triple point
246
1
Double octics and quinctic hypersurfaces
247
2
The dual picture
249
2
I5 and cubic surfaces
251
1
The Picard group
251
1
I5 and cubic surfaces: combinatorics
251
1
The dual variety
252
3
Degree
252
1
Singular locus
253
1
Reducible hyperplane sections
254
1
Special hyperplane sections
254
1
Appendices
255
65
A Rational groups of hermitian type
255
27
A.1 Algebras
255
1
A.1.1 Hermitian forms over skew fields
255
2
A.1.2 Central simple algebras
257
1
A.1.3 Algebras with involution
257
2
A.1.4 Cyclic algebras
259
1
A.1.5 Quaternion algebras
260
2
A.2 Weil's correspondence
262
1
A.2.1 Over C
262
1
A.2.2 Over k
263
1
A.2.3 Lie algebras
263
1
A.2.4 Over R
264
1
A.3 Classification
265
1
A.3.1 Split orthogonal case
266
1
A.3.2 Non-split orthogonal case
267
2
A.3.3 Split symplectic case
269
1
A.3.4 Non-split symplectic case
269
1
A.3.5 Split unitary case
270
1
A.3.6 Non-split unitary case
271
1
A.4 Rational parabolics and boundary components
272
1
A.4.1 Boundary components
273
2
A.5 Orders and arithmetic groups
275
1
A.5.1 Orders in associative algebras
276
2
A.5.2 Arithmetic groups -- classical cases
278
4
B Some classical algebraic geometry
282
38
B.1 Invariants
282
1
B.1.1 Invariants and covariants
282
6
B.1.2 Invariants of finite groups actions
288
2
B.2 Symbolic notation
290
1
B.2.1 Notations
291
2
B.2.2 The fundamental theorem
293
2
B.2.3 Mixed concomitants
295
1
B.2.4 The transference principle of Clebsch
296
1
B.3 Binary forms
297
1
B.3.1 Invariants as functions of the roots
297
1
B.3.2 Quintics
298
2
B.3.3 Sextics
300
2
B.4 Cubic surfaces
302
1
B.4.1 Invariants and linear covariants
302
5
B.4.2 Hexahedral form
307
3
B.5 Some quartic surfaces
310
1
B.5.1 Kummer surfaces
310
1
B.5.2 Desmic surfaces
311
4
B.5.3 Symmetroids and Weddle surfaces
315
5
References
320
7
Index
327