Number Theory Seminar

The $\mathrm{GL}_{2n}$ Rapoport-Zink space is a moduli space of supersingular $p$-divisible groups of dimension $n$ and height $2n$, with a quasi-isogeny to a fixed base point. After the $\mathrm{GL}_2$ Rapoport-Zink space, which is zero-dimensional, the $\mathrm{GL}_4$ Rapoport-Zink space has the most fundamental moduli description, yet relatively little of its specific geometry has been explored. In this talk, I will give a full description of the geometry of the $\mathrm{GL}_4$ Rapoport-Zink space, including the connected components, irreducible components, and intersection behavior of the irreducible components. As an application of the main result, I will also give a description of the supersingular locus of the Shimura variety for the group $\mathrm{GU}(2,2)$ over a prime split in the relevant imaginary quadratic field.