Topological Automorphic Forms
Lecture 1: Modular forms and topology
In this survey talk I will describe how modular forms give invariants
of manifolds, and how these invariants detect elements of the homotopy
groups of spheres. These invariants pass through a cohomology theory
of Topological Modular Forms (TMF). I will review the role that
K-theory plays in detecting periodic families of elements in the
homotopy groups of spheres (the image of the J homomorphism) in terms
of denominators of Bernoulli numbers. I will then describe how
certain higher families of elements (the divided beta family) are
detected by certain congruences between q-expansions of modular forms.
Slides
used in the first lecture.
Lecture 2: Topological Automorphic Forms I: definition.
I will review the definition of certain moduli spaces of abelian
varieties (Shimura varieties) which generalize the role that the
moduli space of elliptic curves plays in number theory. Associated to
these Shimura varieties are cohomology theories of Topological
Automorphic Forms (TAF) which generalize the manner in which
Topological Modular Forms are associated to the moduli space of
elliptic curves. These cohomology theories arise as a result of a
theorem of Jacob Lurie.
Mark's notes for the second lecture.
Lecture 3: 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<2> 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.
Mark's notes for the third lecture.
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.