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.