Notes on the arithmetic of Hecke L-functions

Abstract

This is an expository article that concerns the various related notions of algebraic idèle-class characters, the Größencharaktere of Hecke, and cohomological automorphic representations of \(\text {GL}(1)\), all under the general title of algebraic Hecke characters. The first part of the article systematically lays the foundations of algebraic Hecke characters. The only pre-requisites are: basic algebraic number theory, familiarity with the adelic language, and basic sheaf theory. Observations that play a crucial role in the arithmetic of automorphic L-functions are also discussed. The second part of the article, on the ratios of successive critical values of the Hecke L-function attached to an algebraic Hecke character, concerns certain variations on a theorem of Harder [6], especially drawing attention to a delicate signature that apparently has not been noticed before.