Problems from the E-theory seminar

This is currently a placeholder for questions raised by the E-theory seminar going on at MIT. The short version of this would be to simply go through and write down everything that has a bold-block “Question” or “Conjecture” next to it, but I think that there are enough little things that it’s worth going through in more detail.

(This is incomplete.)

- When is the map from the $K(n)$-local Picard group to the algebraic Picard group surjective or injective? What can one say about finiteness, or finite generation, or completeness, of the source, target, and kernel?
- Do the lower Bousfield-Kuhn functors ${\Phi}_{t}$, applied to the total unstable power operation for ${E}_{n}$, act nontrivially? Does this give some filtration of the total power operation on ${E}_{n}$?
- Does the “inertia groupoid” functor ${\Lambda}_{G}$ respect ${E}_{\mathrm{\infty}}$ structures?
- Can Marsh’s computations of the $E$-theory of linear algebraic groups over finite fields be expressed algebro-geometrically? (They appear to be connected to symmetric powers of formal schemes.)
- Ando has a program for studying ${E}_{\mathrm{\infty}}$ orientations of $E(2)$-local $\mathrm{TMF}$ by finding a presentation of $\mathrm{MU}$ in this category, and then finding the space of ${E}_{\mathrm{\infty}}$ maps $\mathrm{MU}\to \mathrm{TMF}$. (This may already or imminently be being considered by a graduate student, and this problem has been considered by Jan-David Möllers in his thesis.)
- (H. Miller) How does the Morava stabilizer group ${\mathbb{S}}_{n}$ act on the formal group ${C}_{t}\otimes \mathbb{G}$ appearing in Nathaniel Stapleton’s character theory? (It then “acts” on ${\mathbb{Q}}_{p}/{\mathbb{Z}}_{p}$^{n-t}, but also acts on the base scheme.)
- (N. Stapleton) Does going down in height by one give the determinant representation of ${\mathbb{S}}_{n}$?
- (S. Glasman) What happens when $n=1$, $t=0$? In this case the ring ${C}_{t}$ is a maximal ramified extension of ${\mathbb{Q}}_{p}$.
- (H. Miller) Is this character theory the unit of an adjunction between “height $n$ stuff” and “height $t$ stuff”, for some definition of “stuff”?
- Is there a nice category of spaces over $\pi $-finite spaces with fibers finite CW-complexes? In particular, can one make sense of the intertia groupoid in this category?
- What spectra can be decomposed $K(n)$-locally using attaching maps along exotic elements of the $K(n)$-local Picard group?
- Is there an analogous notion of “vector bundle” which is classified by analogues of the spaces $\mathrm{BU}(m)$, which come equipped with associated bundles which are fibered in determinantal spheres?
- What is the topological Hochschild homology of the Lubin-Tate spectrum ${E}_{n}$? The $K(n)$-localization is ${E}_{n}$ itself (G. Horel) and $\mathrm{THH}(\mathrm{KU})\simeq \mathrm{KU}\wedge \Sigma {\mathrm{KU}}_{\mathbb{Q}}$ (McClure-Staffeldt).
- Does $\mathrm{THH}({E}_{n})$ classify some sort of deformations of the Honda formal group law along with an automorphism? (It is claimed that ${E}_{n}$ doesn’t have a moduli-theoretic interpretation in the terms of derived algebraic geometry.)
- Is the Dennis trace map from the algebraic $K$-theory of the group algebra ${E}_{n}[G]$ to ${E}_{n}\wedge {\Sigma}_{+}^{\mathrm{\infty}}\mathrm{LBG}$ related to chromatic redshift?
- Is there a surjection from the exotic elements of the $K(p-1)$-local Picard group onto $\mathbb{Z}/p$, as at $n=1$ and $n=2$? Does this split?
- (Hughes, Lau, Peterson) The quotient of the mod-2 cohomology of $\mathrm{BU}\u27e82k\u27e9$ by images of the classes in odd degree by the elements in the Steenrod algebra has an algebro-geometric interpretation, in terms of a symmetric power of a scheme of divisors.
- How does the dual Steenrod algebra act on the Cartier dual scheme to the symmetric power from the previous question?
- Does $\mathrm{BU}\u27e82k\u27e9$ refine to a space $X(k)$ whose $E$-cohomology realizes this symmetric power?