Formal vector spaces over a local field of positive characteristic

Abstract: Let $O$ be the ring of power series in one variable over a finite field, with $K$ its fraction field. We introduce the notion of a "formal $K$-vector space"; this is a certain kind of $K$-vector space object in the category of formal schemes. This concept runs parallel to the established notion of a formal $O$-module, but in many ways formal $K$-vector spaces are much simpler objects. Our main result concerns the Lubin-Tate tower, which plays a vital role in the local Langlands correspondence for $GL_n(K)$. Let $A_m$ be the complete local ring parametrizing deformations of a fixed formal $O$-module over the residue field, together with Drinfeld level $m$ structure. We show that the completion of the union of the $A_m$ has a surprisingly simple description in terms of formal $K$-vector spaces. This description shows that the generic fiber of the Lubin-Tate tower at infinite level carries the structure of a perfectoid space. As an application, we find a family of open neighborhoods of this perfectoid space whose special fibers are certain remarkable varieties over a finite field which we are able to make completely explicit. It is shown in joint work with Mitya Boyarchenko that the $\ell$-adic cohomology of these varieties realizes the local Langlands correspondence for a certain class of supercuspidal representations of $GL_n(K)$.