2023/11/16 by Andrew Salch, Salch, Andrew, Matthias Strauch +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2311.10225
openalex publication_date 2023/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We embed the Lubin-Tate tower into a larger tower of formal schemes, the "degenerating Lubin-Tate tower." We construct a topological realization of the degenerating Lubin-Tate tower, i.e., a compatible family of presheaves of E∞-ring spectra on the étale site of each formal scheme in the degenerating Lubin-Tate tower, which agrees on the base of the tower with the Goerss-Hopkins presheaf on Lubin-Tate space. We define and prove basic properties of nearby cycle and vanishing cycle presheaves of spectra on formal schemes. We apply these constructions to our spectrally-enriched degenerating Lubin-Tate tower to produce, for every spectrum X, an "ℓ-adic topological Jacquet-Langlands (TJL) dual" of X. We prove that there is a correspondence between: 1. certain irreducible representations of Aut(\mathbbG) occurring in (K(E(\mathbbG))\wedgeℓ)_*(X), and 2. certain supercuspidal irreducible representations of GLn occuring in the rational homotopy groups of the TJL dual of X. Here E(\mathbbG) is the Morava E-theory spectrum of a height n formal group \mathbbG, and K(E(\mathbbG))\wedgeℓ is its algebraic K-theory spectrum completed away from the characteristic of the ground field of \mathbbG. Finally, we prove that at height 1, TJL duality preserves the L-factors. This means that the automorphic L-factor of the GL1-representation associated to (K(E(\mathbbG))\wedgeℓ)_*(X) by TJL duality is precisely the p-local Euler factor in a meromorphic L-function whose special values in the left half-plane recover the orders of the KU-local stable homotopy groups of X.