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Seminar on topological automorphic forms, Fall 2009.

This is my attempt to understand Topological automorphic forms by Mark Behrens and Tyler Lawson. This 159 page memoir describes a general construction which includes that of the theory of topological modular forms (tmf) as a special case. It includes a lot of number theory and algebraic geometry not familiar to most topologists.

Before attempting to read it one should look at Lawson's introduction, An overview of abelian varieties in homotopy theory, 37 pages. It describes some of the backgroup and motivation for the constructions and definitions in TAF.

Another useful introduction is Topological algebraic geometry by Paul Goerss. This consists of over 100 slides for a series of lectures given in Copenhagen in June, 2008. Here you can find quick discussions of schemes, sheaves, stacks, topoi and other relevant notions from abstract algebraic geometry. For a record fast description of the moduli stack of elliptic curves over the complex numbers, see What is ... a stack?

Goerss' writeup of his March, 2009 Bourbaki talk Topological Modular Forms [after Hopkins, Miller and Lurie], 33 pages, is the best single account I have seen of the construction of tmf. ArXiv link.   For a very concise introduction to the relevant facts about elliptic curves and modular forms, see Appendix B (starting on page 53) of Ando-Hopkins-Strickland.


Seminar meetings
A collection of related papers

Papers and preprints of Mark Behrens
Papers and preprints of Tyler Lawson
Elliptic spectra, the Witten genus, and the theorem of the cube  M. Ando, M. J. Hopkins,  and N. P. Strickland
Hecke operators as operations in elliptic cohomology Andrew J. Baker
Computation of the homotopy of the spectrum tmf Tilman Bauer
ON THE HOMOTOPY THEORY OF K-LOCAL SPECTRA AT AN ODD PRIME A. K. BOUSFIELD
Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups Ethan S. Devinatz and Michael J. Hopkins
Nilpotence and stable homotopy theory I Ethan S. Devinatz, Michael J. Hopkins and Jeffrey H. Smith
UNIQUENESS THEOREMS FOR CERTAIN TRIANGULATED CATEGORIES POSSESSING AN ADAMS SPECTRAL SEQUENCE JENS FRANKE
Papers and preprints of Paul Goerss
Quasi-coherent sheaves on the Moduli Stack of Formal Groups Paul Goerss 
A resolution of the K(2)-local sphere at the prime 3 Paul Goerss, Hans-Werner Henn, Mark Mahowald and Charles Rezk 
Moduli Spaces of Commutative Ring Spectra Paul Goerss and Mike Hopkins
Formal completion of the Jacobian of plane curves and higher real K-theories  V. Gorbounov and M. Mahowald
The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory
B. H. Gross and M. J. Hopkins
Automorphic forms and cohomology theories on Shimura curves of small discriminant  Michael Hill and Tyler Lawson
 
Complex oriented cohomology theories and the language of stacks  Notes on course given by Mike Hopkins 
From elliptic curves to homotopy theory  Mike Hopkins and Mark Mahowald 
A survey of elliptic cohomology  Jacob Lurie 
The stack of formal groups in stable homotopy theory  Niko Naumann 

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Created September 14, 2009; last modified November 18, 2009.