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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)
Published electronically: November 23, 2009
MathSciNet review: 2640996
Keywords:Homotopy groups, cohomology theories, automorphic forms, Shimura varieties
MSC: Primary 55N34; Secondary 11F23, 14G35, 55N22, 55P43

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Table of Contents


  • Introduction
  • Chapter 1. $p$-divisible groups
  • Chapter 2. The Honda-Tate classification
  • Chapter 3. Tate modules and level structures
  • Chapter 4. Polarizations
  • Chapter 5. Forms and involutions
  • Chapter 6. Shimura varieties of type $U(1,n-1)$
  • Chapter 7. Deformation theory
  • Chapter 8. Topological automorphic forms
  • Chapter 9. Relationship to automorphic forms
  • Chapter 10. Smooth $G$-spectra
  • Chapter 11. Operations on TAF
  • Chapter 12. Buildings
  • Chapter 13. Hypercohomology of adele groups
  • Chapter 14. $K(n)$-local theory
  • Chapter 15. Example: chromatic level 1


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 ( $\mathit{TAF}$) are related to Shimura varieties in the same way that $\mathit{TMF}$ 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 $\mathit{TAF}$, 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.

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