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Topological automorphic forms
About this Title
Mark Behrens and Tyler Lawson
Publication: Memoirs of the American Mathematical Society
Publication Year:
2010; Volume 204, Number 958
ISBNs: 978-0-8218-4539-4 (print); 978-1-4704-0572-4 (online)
DOI: https://doi.org/10.1090/S0065-9266-09-00573-0
Published electronically: November 23, 2009
Keywords: Homotopy groups,
cohomology theories,
automorphic forms,
Shimura varieties
MSC: Primary 55N35; Secondary 55Q51, 55Q45, 11G15
Table of Contents
Chapters
- Introduction
- 1. $p$-divisible groups
- 2. The Honda-Tate classification
- 3. Tate modules and level structures
- 4. Polarizations
- 5. Forms and involutions
- 6. Shimura varieties of type $U(1,n-1)$
- 7. Deformation theory
- 8. Topological automorphic forms
- 9. Relationship to automorphic forms
- 10. Smooth $G$-spectra
- 11. Operations on $TAF$
- 12. Buildings
- 13. Hypercohomology of adele groups
- 14. $K(n)$-local theory
- 15. Example: chromatic level $1$
Abstract
We apply a theorem of J. Lurie to produce cohomology theories associated to certain Shimura varieties of type $U(1,n-1)$. These cohomology theories of topological automorphic forms ($TAF$) are related to Shimura varieties in the same way that $TMF$ is related to the moduli space of elliptic curves. We study the cohomology operations on these theories, and relate them to certain Hecke algebras. We compute the $K(n)$-local homotopy types of these cohomology theories, and determine that $K(n)$-locally these spectra are given by finite products of homotopy fixed point spectra of the Morava E-theory $E_n$ by finite subgroups of the Morava stabilizer group. We construct spectra $Q_U(K)$ for compact open subgroups $K$ of certain adele groups, generalizing the spectra $Q(\ell )$ studied by the first author in the modular case. We show that the spectra $Q_U(K)$ admit finite resolutions by the spectra $TAF$, arising from the theory of buildings. We prove that the $K(n)$-localizations of the spectra $Q_U(K)$ are finite products of homotopy fixed point spectra of $E_n$ with respect to certain arithmetic subgroups of the Morava stabilizer groups, which N. Naumann has shown (in certain cases) to be dense. Thus the spectra $Q_U(K)$ approximate the $K(n)$-local sphere to the same degree that the spectra $Q(\ell )$ approximate the $K(2)$-local sphere.- J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8
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