On the rationalization of $\mathrm{K}(n)$-local determinant sphere

In their recent breakthroughs, Barthel--Schlank--Stapleton--Weinstein used $p$-adic arithmetic geometry to compute both the rationalization of the $\mathrm{K}(n)$-local sphere and the algebraic Picard group of the $\mathrm{K}(n)$-local category. The latter is topologically generated by the $\mathrm{K}(n)$-local ordinary and determinant spheres. 

In this project, we study the rationalization of the $\mathrm{K}(n)$-local determinant sphere. Assuming the strong chromatic splitting conjecture, Gross--Hopkins duality gives a prediction for its rational Poincaré series $D_n(T)$. Independently, we compute the rational cohomology of the Steinberg representation for $\mathrm{GL}_n(\mathbb{Z}_p)$ and observe that its Poincaré series agrees up to a shift with $D_n(T)$. This numerical coincidence suggests a deep connection between the two seemingly unrelated calculations. We are currently working towards proving this connection through $p$-adic arithmetic geometry.

This is work in progress with Tobias Barthel, Jacob Lerma, Guchuan Li, and Wei Yang.