Notes on the arithmetic of Hecke L-functions

TL;DR

This paper provides an expository overview of algebraic Hecke characters, their foundational properties, and explores new insights into the ratios of critical values of associated L-functions, highlighting a previously unnoticed signature variation.

Contribution

It systematically develops the foundations of algebraic Hecke characters and introduces novel observations on the ratios of critical L-values, including a new signature aspect.

Findings

New insights into the ratios of successive critical values of Hecke L-functions.

Identification of a delicate signature variation in the ratios.

Enhanced understanding of algebraic Hecke characters and their arithmetic properties.

Abstract

This is an expository article that concerns the various related notions of algebraic idele class characters, the Groessencharaktere of Hecke, and cohomological automorphic representations of 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 Guenter Harder, especially drawing attention to a delicate signature that apparently has not been noticed before.