Virtual characters on L-functions

TL;DR

This paper explores the development and application of Heilbronn and arithmetic Heilbronn characters, along with supercharacter theory, to analyze the holomorphy of quotients of Artin L-functions, including those related to CM elliptic curves.

Contribution

It introduces new applications of Heilbronn and supercharacter theories to study the holomorphy of Artin L-function quotients, expanding existing methods in the field.

Findings

Heilbronn characters aid in understanding L-function holomorphy.

Arithmetic Heilbronn characters extend analysis to CM elliptic curves.

Supercharacter theory provides new insights into Artin L-functions.

Abstract

In this expository note we show the inception and development of the Heilbronn characters and their application to the holomorphy of quotients of Artin L-functions. Further we use arithmetic Heilbronn characters introduced by Wong, to deal with holomorphy of quotients of certain L-functions, e.g,, L-functions associated to CM elliptic curves. Furthermore we use the supercharacter theory introduced by Diaconis and Isaacs to study Artin L-functions associated to such characters. We conclude the note surveying about various other unconditional approaches taken based on character theory of finite groups.