### Geometry-Topology Seminar

Thursday, September 17, 2020 - 4:30pm

#### Ningchuan Zhang

UPenn

Location

University of Pennsylvania

via Zoom

This is the first Geometry-Topology Seminar of the fall semester. We will meet weekly at this same time throughout the semester. The Zoom link is: https://upenn.zoom.us/j/91890239234 This same Zoom link will apply for future talks as well. It is set to open at 4 PM so that speakers can come on early and check out their technology setups. The talks will begin at 4:30 PM. We will also stay afterwards, say from 5:30 - 6 PM to chat with one another, as an online substitute for going out to dinner with our speakers. We encourage everyone to have a nice bottle of wine at hand for that social half hour. For further information about the seminar, please contact Mona Merling (mmerling@math.upenn.edu), Davi Maximo (dmaxim@math.upenn.edu) or Herman Gluck (gluck@math.upenn.edu).

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. Then we will introduce a spectral sequence to compute their homotopy groups. The $1$-line in this spectral sequence is closely related to congruences of Eisenstein series. This explains appearance of special values of Dirichlet $L$-functions in the homotopy groups of these Dirichlet $J$-spectra.

Finally, we 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.