CM fields of Dyhedral type and the Colmez conjecture

TL;DR

This paper explores CM fields of dihedral type, computes associated Artin L-functions, and links the Colmez conjecture to Hecke characters, proving it for certain unitary CM types and on average for others.

Contribution

It computes Artin L-functions for dihedral CM fields and establishes the Colmez conjecture for specific unitary CM types and on average for others.

Findings

Colmez conjecture holds for unitary CM types of signature (n-1,1)

Colmez conjecture holds on average for fixed CM fields with signature (n-r,r)

Relation between Colmez conjecture and log derivatives of Hecke characters

Abstract

In this paper, we consider some CM fields which we call of dihedral type and compute the Artin $L$-functions associated to all CM types of these CM fields. As a consequence of this calculation, we see that the Colmez conjecture in this case is very closely related to understanding the log derivatives of certain Hecke characters of real quadratic fields. Recall that the `abelian case' of the Colmez conjecture, proved by Colmez himself, amounts to understanding the log derivatives of Hecke characters of $\Q$ (cyclotomic characters). In this paper, we also prove that the Colmez conjecture holds for `unitary CM types of signature $(n-1, 1)$' and holds on average for `unitary CM types of a fixed CM number field of signature $(n-r, r)$'.