Abstract

A fundamental result in local class field theory is the 1965 paper of Lubin and Tate, which classifies the abelian extensions of a nonarchimedean local field in terms of an algebraic structure known as a one-dimensional formal module. We’ll review this result, and show how the question of constructing nonabelian extensions leads to the study of the Lubin-Tate tower, which can be viewed as an infinitesimal version of the classical tower of modular curves X(pⁿ).

By results of Harris-Taylor and Boyer, the cohomology of the Lubin-Tate tower encodes precise information about non-abelian extensions of the local field (namely, it realizes the local Langlands correspondence). The Lubin-Tate tower has a horribly singular special fiber, which hinders any direct study of its cohomology, but we will show that after blowing up a singularity there is a model for the tower whose reduction contains a very curious nonsingular hypersurface defined over a finite field—curious because it seems to have the maximum number of rational points relative to its topology. We will write down the equation for this hypersurface and formulate a conjecture (alas, still unproved) regarding its zeta function.