Abstract

Chromatic homotopy theory decomposes the category of spectra at a prime p into a collection of categories according to certain periodicities. There is one of these categories for each natural number n and it is called the K(n)-local category. Unfortunately, when n > 1, even the K(n)-local categories are often quite difficult to understand computationally. ¹ When n = 2, significant progress has been made but above n = 2 many basic computational questions are open. Even the most well-behaved ring spectra in the K(n)-local category, the height n Morava E-theories, still hold plenty of mysteries. In order to understand E-cohomology, Hopkins, Kuhn, and Ravenel constructed a character map for each E-theory landing in a form of rational cohomology. They proved that the codomain of their character map serves as an approximation to E-cohomology in a precise sense. It turns out that many of the deep formal properties of the K(n)-local category can be expressed in terms of simple formulas or relations satisfied by simple formulas on the codomain of the character map. This is intriguing for several reasons: The codomain of the character map is not K(n)-local, it is a rational approximation to E-cohomology so it removes all of the torsion from E-cohomology. The codomain of the character map for the E-cohomology of a finite group is a simple generalization of the ring of class functions on the group. Thus, in this case, these deep properties of the K(n)-local category are reflected in combinatorial and group theoretic properties of certain types of conjugacy classes in the group. Finally, a certain Q -algebra, known as the Drinfeld ring of infinite level structures, arises from topological considerations as the coefficients of the codomain of the character map. This Q -algebra plays an important role in the local Langlands program and the group actions that feed into the local Langlands correspondence are closely related to fundamental properties of the Morava E-theories.