Speaker: Andrew Salch (Wayne State University)
Title: ell-adic topological Jacquet-Langlands duality
Abstract: I will describe the construction of an “ell-adic topological Jacquet-Langlands (TJL) dual” of each finite CW-complex. One effect of the TJL dual is that the representations of the Morava stabilizer group, which arise in the height n Morava E-theory of a finite spectrum X, get turned into continuous actions of GL_n(Z_p) on the ell-adic TJL dual of X. On the level of homotopy groups, these actions of GL_n(Z_p) correspond, via the classical ell-adic Jacquet-Langlands correspondence, to the representa- tions of the Morava stabilizer group on the Morava E-theory groups of X. When n=1, these representa- tions of GL_1 have associated automorphic L-factors. I will explain how the product of those L-factors, i.e., the “automorphic L-function of a finite CW-complex X,” analytically continues to a meromorphic function on the complex plane whose special values in the left half-plane recover the orders of the KU-local stable homotopy groups of X. Part of the construction of the TJL dual is a solution to an old problem on construction of E_\infty-ring realizations of the Lubin-Tate tower. I will describe that solution, which is joint work with Matthias Strauch.
URL associated with Seminar: https://www.benszczesny.com/osuhomotopyseminar/
