U. Haifa Topology & Geometry seminar: Sunday, May 22, 2022. Speaker: Andrew Salch (Wayne State). Title: “Stable homotopy groups and special values of L-functions”.
Time: 17:00 (Israel Time)
Location: On Zoom
In this talk, I will survey the known relationships between stable homotopy groups of Bousfield localizations of finite spectra and special values of L-functions. Given a p-local finite spectrum X, the “chromatic tower” of X is a tower of Bousfield localizations of X. The homotopy limit of this tower recovers X itself. A calculation of Ravenel and of Adams-Baird from the 1970s established that the homotopy groups of the first stage in this tower for the sphere spectrum, the E(1)-local sphere, are describable in terms of special values of the Riemann zeta-function. In the past 5 years, this theorem has been extended to a description, in terms of special values of L-functions arising in number theory, of the E(1)-local homotopy groups of many finite spectra other than the sphere spectrum. As a cute application, I give a topological proof of some cases of the Leopoldt conjecture, a well-known conjecture in algebraic number theory. Finally, time allowing, I explain what is known and what is conjectured about descriptions, in terms of special values of L-functions, of the orders of the homotopy groups of the higher layers in the chromatic tower (the E(n)-local layer for n>1) of a finite spectrum.
Meeting ID: 849 8759 1026
Markus Upmeier (Aberdeen) – May 29th
Denis-Charles Cisinski (Regensburg) – June 5th.