Topological Automorphic Forms.
- 1st ed.
- 1 online resource (167 pages)
- Memoirs of the American Mathematical Society ; v.204 .
- Memoirs of the American Mathematical Society .
Intro -- Contents -- Abstract -- Introduction -- 0.1. Background and motivation -- 0.2. Subject matter of this book -- 0.3. Organization of this book -- 0.4. Acknowledgments -- Chapter 1. p-divisible groups -- 1.1. Definitions -- 1.2. Classification -- Chapter 2. The Honda-Tate classification -- 2.1. Abelian varieties over finite fields -- 2.2. Abelian varieties over Fp -- Chapter 3. Tate modules and level structures -- 3.1. Tate modules of abelian varieties -- 3.2. Virtual subgroups and quasi-isogenies -- 3.3. Level structures -- 3.4. The Tate representation -- 3.5. Homomorphisms of abelian schemes -- Chapter 4. Polarizations -- 4.1. Polarizations -- 4.2. The Rosati involution -- 4.3. The Weil pairing -- 4.4. Polarizations of B-linear abelian varieties -- 4.5. Induced polarizations -- 4.6. Classification of weak polarizations -- Chapter 5. Forms and involutions -- 5.1. Hermitian forms -- 5.2. Unitary and similitude groups -- 5.3. Classification of forms -- Chapter 6. Shimura varieties of type U(1,n-1) -- 6.1. Motivation -- 6.2. Initial data -- 6.3. Statement of the moduli problem -- 6.4. Equivalence of the moduli problems -- 6.5. Moduli problems with level structure -- 6.6. Shimura stacks -- Chapter 7. Deformation theory -- 7.1. Deformations of p-divisible groups -- 7.2. Serre-Tate theory -- 7.3. Deformation theory of points of Sh -- Chapter 8. Topological automorphic forms -- 8.1. The generalized Hopkins-Miller theorem -- 8.2. The descent spectral sequence -- 8.3. Application to Shimura stacks -- Chapter 9. Relationship to automorphic forms -- 9.1. Alternate description of Sh(Kp) -- 9.2. Description of Sh(Kp)F -- 9.3. Description of Sh(Kp)C -- 9.4. Automorphic forms -- Chapter 10. Smooth G-spectra -- 10.1. Smooth G-sets -- 10.2. The category of simplicial smooth G-sets -- 10.3. The category of smooth G-spectra. 10.4. Smooth homotopy fixed points -- 10.5. Restriction, induction, and coinduction -- 10.6. Descent from compact open subgroups -- 10.7. Transfer maps and the Burnside category -- Chapter 11. Operations on TAF -- 11.1. The E-action of GU(Ap,) -- 11.2. Hecke operators -- Chapter 12. Buildings -- 12.1. Terminology -- 12.2. The buildings for GL and SL -- 12.3. The buildings for U and GU -- Chapter 13. Hypercohomology of adele groups -- 13.1. Definition of QGU and QU -- 13.2. The semi-cosimplicial resolution -- Chapter 14. K(n)-local theory -- 14.1. Endomorphisms of mod p points -- 14.2. Approximation results -- 14.3. The height n locus of Sh(Kp) -- 14.4. K(n)-local TAF -- 14.5. K(n)-local QU -- Chapter 15. Example: chromatic level 1 -- 15.1. Unit groups and the K(1)-local sphere -- 15.2. Topological automorphic forms in chromatic filtration 1 -- Bibliography -- Index.