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.

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.