Introduction .................................................. vii
0.1 Background and motivation ................................ vii
0.2 Subject matter of this book ............................... xv
0.3 Organization of this book ................................ xxi
0.4 Acknowledgments ........................................ xxiii
Chapter 1 p-divisible groups ................................... 1
1.1 Definitions ................................................ 1
1.2 Classification ............................................. 3
Chapter 2 The Honda-Tate classification ........................ 5
2.1 Abelian varieties over finite fields ....................... 5
2.2 Abelian varieties over  p .................................. 6
Chapter 3 Tate modules and level structures ................... 13
3.1 Tate modules of abelian varieties ......................... 13
3.2 Virtual subgroups and quasi-isogenies ..................... 14
3.3 Level structures .......................................... 15
3.4 The Tate representation ................................... 15
3.5 Homomorphisms of abelian schemes .......................... 16
Chapter 4 Polarizations ....................................... 19
4.1 Polarizations ............................................. 19
4.2 The Rosati involution ..................................... 22
4.3 The Weil pairing .......................................... 22
4.4 Polarizations of B-linear abelian varieties ............... 23
4.5 Induced polarizations ..................................... 23
4.6 Classification of weak polarizations ...................... 24
Chapter 5 Forms and involutions ............................... 27
5.1 Hermitian forms ........................................... 27
5.2 Unitary and similitude groups ............................. 30
5.3 Classification of forms ................................... 31
Chapter 6 Shimura varieties of type U(1, n — 1) .............. 35
6.1 Motivation ................................................ 35
6.2 Initial data .............................................. 36
6.3 Statement of the moduli problem ........................... 37
6.4 Equivalence of the moduli problems ........................ 39
6.5 Moduli problems with level structure ...................... 41
6.6 Shimura stacks ............................................ 42
Chapter 7 Deformation theory .................................. 45
7.1 Deformations of p-divisible groups ........................ 45
7.2 Serre-Tate theory ......................................... 47
7.3 Deformation theory of points of Sh ........................ 47
Chapter 8 Topological automorphic forms ....................... 51
8.1 The generalized Hopkins-Miller theorem .................... 51
8.2 The descent spectral sequence ............................. 54
8.3 Application to Shimura stacks ............................. 56
Chapter 9 Relationship to automorphic forms ................... 57
9.1 Alternate description of Sh(Kp) ........................... 57
9.2 Description of Sh(Kp)F .................................... 59
9.3 Description of Sh(Kp)C .................................... 59
9.4 Automorphic forms ......................................... 62
Chapter 10 Smooth G-spectra ................................... 65
10.1 Smooth G-sets ............................................. 65
10.2 The category of simplicial smooth G-sets .................. 67
10.3 The category of smooth G-spectra .......................... 69
10.4 Smooth homotopy fixed points .............................. 70
10.5 Restriction, induction, and coinduction ................... 71
10.6 Descent from compact open subgroups ....................... 72
10.7 Transfer maps and the Burnside category ................... 74
Chapter 11 Operations on TAF .................................. 79
11.1 The E∞-action of GU( P,∞) ................................. 79
11.2 Hecke operators ........................................... 80
Chapter 12 Buildings .......................................... 87
12.1 Terminology ............................................... 87
12.2 The buildings for GL and SL ............................... 88
12.3 The buildings for U and GU ................................ 89
Chapter 13 Hypercohomology of adele groups .................... 95
13.1 Definition of QGU and QU ................................... 95
13.2 The semi-cosimplicial resolution .......................... 95
Chapter 14 K(n)-local theory ................................. 103
14.1 Endomorphisms of mod p points ............................ 103
14.2 Approximation results .................................... 105
14.3 The height n locus of Sh(Kp) ............................. 108
14.4 K(n)-local TAF ........................................... 113
14.5 K(n)-local QU ............................................ 118
Chapter 15 Example: chromatic level 1 ........................ 121
15.1 Unit groups and the K(l)-local sphere .................... 121
15.2 Topological automorphic forms in chromatic
filtration 1 ............................................. 123
Bibliography .................................................. 129
Index ......................................................... 133
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