2012/07/26 by Jared Weinstein, Weinstein, Jared
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1207.6424
openalex publication_date 2012/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 GLn(K). Let Am 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 Am 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 ℓ-adic cohomology of these varieties realizes the local Langlands correspondence for a certain class of supercuspidal representations of GLn(K).