Topological Automorphic Forms II: examples, problems, and applications

I will survey some known computations of Topological Automorphic Forms. K-theory and TMF will be shown to be special cases to TAF. Certain TAF spectra have been identified with $BP\langle 2\rangle$ by Hill and Lawson, showing these spectra admit $E_{oo}$ ring structures. $K(n)$-local TAF gives instances of the higher real K-theories $EO_n$, one of which shows up in the solution of the Kervaire invariant one problem. Associated to the TAF spectra are certain approximations of the $K(n)$-local sphere, which are expected to see "Greek letter elements" in the same manner that TMF sees the divided beta family. Finally, I will discuss some partial results and questions concerning an automorphic forms valued genus which is supposed to generalize the Witten genus.

References

• Mark Behrens, Notes on the construction of TMF (2007).
• Mark Behrens and Tyler Lawson, Topological Automorphic Forms, Memoirs of the AMS 958 (2010).
• Paul Goerss, Topological modular forms (after Hopkins, Miller and Lurie), Séminaire Bourbaki, 2009.
• Mike Hopkins, Topological modular forms, the Witten genus and the Theorem of the cube, Proceedings of the 1994 ICM.
• Mike Hopkins, Algebraic Topology and Modular Forms, Proceedings of the 2002 ICM.
• Tyler Lawson, An overview of abelian varieties in homotopy theory (2008).

Doug Ravenel's web page for a seminar on topological automorphic forms contains a comprehensive list of references.