Analogs of Dirichlet $L$-functions in chromatic homotopy theory

The relation between Eisenstein series and the $J$-homomorphism is an important topic in chromatic homotopy theory at height $1$. Both sides are related to the special values of the Riemann $\zeta$-function. Number theorists have studied the twistings of the Riemann $\zeta$-functions and Eisenstein series by Dirichlet characters. Motivated by the Dirichlet equivariance of these Eisenstein series, we introduce the Dirichlet $J$-spectra in this paper. The homotopy groups of the Dirichlet $J$-spectra are related to the special values of the Dirichlet $L$-functions. Moreover, we find Brown-Comenetz duals of the Dirichlet $J$-spectra, whose formulas resemble 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.