Signature $(n-2,2)$ CM Types and the Unitary Colmez Conjecture

TL;DR

This paper investigates the Colmez conjecture for unitary CM fields, reducing its complexity and proving additional cases by analyzing class functions and Galois actions.

Contribution

It advances the understanding of the Colmez conjecture specifically for unitary CM fields by reducing the problem and proving more cases using Galois symmetry.

Findings

Reduced the Colmez conjecture to a special case for unitary CM fields

Proved additional cases of the Colmez conjecture using Galois actions

Analyzed class functions arising in the context of CM types

Abstract

Colmez conjectured a formula relating the Faltings height of CM abelian varieties to a certain linear combination of log derivatives of $L$-functions. In this paper, we study the case of unitary CM fields and by studying the class functions that arise, we reduce the conjecture to a special case. Using the Galois action, we prove more cases of the Colmez Conjecture.