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 $\mathrm{taf}$.

Properties

Chromatic pattern

chromatic level $n=$	1	2	$\ge 3$
cohomology theory/spectrum $E=$	KO	TMF	TAF
algebraic group	${\mathrm{GL}}_1$	${\mathrm{GL}}_2$	$U(1,n-1)$
geometric object	multiplicative group	elliptic curve	Shimura variety
FQFT	superparticle	heterotic superstring	??

Related concepts

• topological modular forms
• automorphic form

References

Original articles

• Mark Behrens, Tyler Lawson, Topological automorphic forms, Mem. Amer. Math. Soc. (pdf)

• Tyler Lawson, An overview of abelian varieties in homotopy theory (pdf)

Lectures

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

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