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.