On the Colmez conjecture for non-abelian CM fields

TL;DR

This paper proves the Colmez conjecture for infinitely many non-abelian CM fields constructed over fixed totally real fields, including explicit examples and a statistical approach, extending the conjecture's validity beyond abelian cases.

Contribution

It establishes the Colmez conjecture for infinitely many non-abelian CM fields, including explicit constructions and a statistical framework for broader classes.

Findings

Proved Colmez conjecture for infinitely many non-abelian CM fields.

Explicitly constructed ramified CM extensions satisfying the conjecture.

Evaluated Faltings height of Jacobians with non-abelian CM in terms of Barnes double Gamma function.

Abstract

The Colmez conjecture relates the Faltings height of an abelian variety with complex multiplication by the ring of integers of a CM field $E$ to logarithmic derivatives of certain Artin $L$--functions at $s=0$. In this paper, we prove that if $F$ is any fixed totally real number field of degree $[F:\mathbb{Q}] \geq 3$, then there are infinitely many CM extensions $E/F$ such that $E/\mathbb{Q}$ is $\textit{non-abelian}$ and the Colmez conjecture is true for $E$. Moreover, these CM extensions are explicitly constructed to be ramified at "arbitrary" prescribed sets of prime ideals of $F$. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be seen as an explicit non-abelian Chowla-Selberg formula. A crucial input to the proofs is an averaged version of the Colmez conjecture which was recently proved independently by Andreatta-Goren-Howard-Madapusi Pera and Yuan-Zhang.