  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 $X$.
