Analogs of Dirichlet L-functions in chromatic homotopy theory

Abstract

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 ζ-function. Number theorists have studied the twistings of the Riemann ζ-function 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.