ell-adic topological Jacquet-Langlands duality

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 $E_{\infty}$-ring spectra on the \'{e}tale 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 "$\ell$-adic topological Jacquet-Langlands (TJL) dual" of $X$. We prove that there is a correspondence between: 1. certain irreducible representations of $Aut(\mathbb{G})$ occurring in $(\mathcal{K}(E(\mathbb{G}))^{\wedge}_{\ell})_*(X)$, and 2. certain supercuspidal irreducible representations of $GL_n$ occuring in the rational homotopy groups of the TJL dual of $X$. Here $E(\mathbb{G})$ is the Morava $E$-theory spectrum of a height $n$ formal group $\mathbb{G}$, and $\mathcal{K}(E(\mathbb{G}))^{\wedge}_{\ell}$ is its algebraic $K$-theory spectrum completed away from the characteristic of the ground field of $\mathbb{G}$. Finally, we prove that at height $1$, TJL duality preserves the $L$-factors. This means that the automorphic $L$-factor of the $GL_1$-representation associated to $(\mathcal{K}(E(\mathbb{G}))^{\wedge}_{\ell})_*(X)$ 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 $X$.