# How do topological automorphic forms fit into homotopy theory and what makes them interesting?

Topological automorphic forms (TAF) were introduced by Mark Behrens and Tyler Lawson in 2007, being to Shimura varieties what topological modular forms (TMF) is to the moduli stack of elliptic curves.

While a previous question asked about TMF, there does not seem to be a similar question for TAF (and indeed not many questions about it at MathOverflow at all).

Question: How do TAF fit into the current research programs in homotopy theory, and what are the connections between them and other concepts, such as TMF?

For instance, the nLab page on TAF has the following comparison between $$\mathrm{KO}$$, $$\mathrm{TMF}$$, and $$\mathrm{TAF}$$: (Modified LaTeX reproduction. The scare quotes on “$$\geq3$$” are to account for the fact that while $$\mathrm{TAF}$$ is a kind of analogue of $$\mathrm{TMF}$$ at higher heights, it also specializes to $$n=1,\ 2$$. See Charles Rezk's comment for great pointers in this direction.)

• A nice survey on this topic recently appeared in the arXiv: arxiv.org/abs/1901.07990 – Denis Nardin Feb 11 at 19:13
• Also this: arxiv.org/pdf/0810.0507.pdf – Drew Heard Feb 11 at 19:14
• The TAF spectra provide examples of E_oo-rings of higher heights (i.e., which are L_n-local for n>2). They're "concrete" applications of Lurie's realization theorem, where I've put "concrete" in scare quotes because (afaik) it's very hard to actually compute much about the homotopy groups of these spectra, let alone the extra data of power operations that comes from the E_oo-ring structure. I see it as part of the more general program to find algebra(o-geometr)ically approachable L_n-local E_oo-rings which, like TMF, can hopefully ... – skd Feb 12 at 4:07
• ... detect elements in the homotopy groups of spheres and "complete" (i.e., K(n)-localize) to homotopy fixed points of Morava E-theories. Behrens and Hopkins discuss this latter point for TAF spectra in arxiv.org/abs/0910.0617. Like TMF, these TAF spectra might also be the target of orientations like the Euler characteristic, Atiyah-Bott-Shapiro, and Ando-Hopkins-Rezk orientations/genera, but afaik this is not known yet (although Mark Behrens has some notes on his webpage). – skd Feb 12 at 4:08
• The chart isn't entirely accurate: strictly speaking, TAFs exist at heights 1 and 2 as well. There are even papers (arxiv.org/abs/0902.2603, arxiv.org/abs/1301.3233) which carry out explicit calculations of some examples of height 2 TAFs. These examples tend to look sort-of-like TMF-with-level-structures: sometimes they are exactly the same as TMFs (for reasons unclear), and sometimes they are a little different from any TMF. – Charles Rezk Feb 13 at 17:53