nLab topological automorphic form

Idea

Topological automorphic forms are a generalization of topological modular forms: where the latter come with the moduli space of elliptic curves, topological automorphic forms are associated to a given Shimura variety. Moreover, just as topological modular forms refine to the tmf-spectrum representing the corresponding cohomology theory, so every Shimura variety induces a cohomology theory $taf$.

Properties

Chromatic pattern

chromatic level $n =$	1	2	$\geq 3$
cohomology theory/spectrum $E =$	KO	TMF	TAF
algebraic group	$GL_1$	$GL_2$	$U(1,n-1)$
geometric object	multiplicative group	elliptic curve	Shimura variety
FQFT	superparticle	heterotic superstring	??

Related concepts

• topological modular forms
• automorphic form

References

The definition is due to

• Mark Behrens, Tyler Lawson, Topological automorphic forms, Mem. Amer. Math. Soc., 204 (2010), no. 958 (arXiv:math/0702719)

An introductory survey is in

• Tyler Lawson, An overview of abelian varieties in homotopy theory (arXiv:0810.0507)

Lecture notes include

• Mark Behrens, Topological Automorphic Forms, Lecture series (2011) (lecture notes)

• Doug Ravenel, Seminar on topological automorphic forms (2009-11) (web)

• Dylan Wilson and Paul VanKoughnett, TAF reading course, 2013 (web)

• Chermaine Sia, Topological automorphic forms, talk at Talbot Workshop 2013 (pdf)