Unitary $\operatorname{PSL}_2$ CM Fields and the Colmez Conjecture

TL;DR

This paper investigates specific unitary CM fields with Galois group structures involving $ ext{PSL}_2( ext{F}_q)$, computes relevant class functions, and proves Colmez's conjecture for these cases using recent refinements.

Contribution

It explicitly calculates class functions for these CM fields and proves Colmez's conjecture in this setting by applying recent average case refinements.

Findings

Colmez's conjecture is verified for these unitary CM fields.

Explicit calculations of class functions are provided.

Refinements of the average version of the conjecture are utilized.

Abstract

We study certain unitary CM fields whose Galois closure has Galois group $\operatorname{PSL}_2(\mathbb{F}_q) \times \mathbb{Z}/2\mathbb{Z}$. After investigating the CM types of these fields, we turn towards Colmez's conjectural formula on the Faltings heights of CM abelian varieties. We explicitly calculate the class functions appearing in the statement of the conjecture and then apply refinements of the recently proven average version to establish Colmez's conjecture in this case.