Remote access

How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2294  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax

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: http://dx.doi.org/10.1090/S0065-9266-09-00573-0
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

View full volume PDF

View other years and numbers:

Table of Contents


Chapters

  • 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

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 ( $\mathit{TAF}$) 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.

References [Enhancements On Off] (What's this?)


Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia