# Topology Seminar

## Analogs of Dirichlet $$L$$-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 $$J$$-homomorphism in the stable homotopy groups of spheres. The image of $$J$$ in $$\pi_{4k-1}^s(S^0)$$ is a cyclic group whose order is equal to the denominator of $$\zeta(1-2k)/2$$ (up to a factor of 2). The goal of this talk is to introduce a family of Dirichlet $$J$$-spectra that generalizes this connection.

We will start by reviewing Adams’s computation of the image of $$J$$. Using motivations from modular forms, we construct a family of Dirichlet $$J$$-spectra for each Dirichlet character. When conductor of the character is an odd prime $$p$$, the $$p$$-completion of the Dirichlet $$J$$-spectra splits as a wedge sum of $$K(1)$$-local invertible spectra. These summands are elements of finite orders in the $$K(1)$$-local Picard group.

We will then introduce a spectral sequence to compute homotopy groups of the Dirichlet $$J$$-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 $$L$$-functions in the homotopy groups of these Dirichlet $$J$$-spectra. Finally, we will establish a Brown-Comenetz duality for the Dirichlet $$J$$-spectra that resembles the functional equations of the corresponding Dirichlet $$L$$-functions. In this sense, the Dirichlet $$J$$-spectra we constructed are analogs of Dirichlet $$L$$-functions in chromatic homotopy theory.

Event contact: steven dot amelotte at rochester dot edu