Topology Seminar

Analogs of Dirichlet -functions in chromatic homotopy theory

Ningchuan Zhang, University of Pennsylvania

Friday, September 25th, 2020
2:00 PM - 3:00 PM
Zoom ID 677 596 7436

In the 1960’s, Adams computed the image of the -homomorphism in the stable homotopy groups of spheres. The image of in is a cyclic group whose order is equal to the denominator of (up to a factor of 2). The goal of this talk is to introduce a family of Dirichlet -spectra that generalizes this connection.

We will start by reviewing Adams’s computation of the image of . Using motivations from modular forms, we construct a family of Dirichlet -spectra for each Dirichlet character. When conductor of the character is an odd prime , the -completion of the Dirichlet -spectra splits as a wedge sum of -local invertible spectra. These summands are elements of finite orders in the -local Picard group.

We will then introduce a spectral sequence to compute homotopy groups of the Dirichlet -spectra. The 1-line in this spectral sequence is closely related to congruences of certain Eisenstein series. This explains appearance of special values of Dirichlet -functions in the homotopy groups of these Dirichlet -spectra. Finally, we will establish a Brown-Comenetz duality for the Dirichlet -spectra that resembles the functional equations of the corresponding Dirichlet -functions. In this sense, the Dirichlet -spectra we constructed are analogs of Dirichlet -functions in chromatic homotopy theory.

Event contact: steven dot amelotte at rochester dot edu