The Chowla-Selberg Formula for CM Fields and the Colmez Conjecture

Abstract

In this thesis we start by giving a quick review of the classical Chowla-Selberg formula. We then recall a conjecture of Colmez which 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. It turns out that in the case in which the abelian variety is a CM elliptic curve, the conjecture of Colmez can be seen as a geometric reformulation of the classical Chowla-Selberg formula.

Then we will focus our attention on establishing a generalization of the classical Chowla- Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function Γ and an analogous function Γ2 at rational numbers.

Finally, we will study the above mentioned conjecture of Colmez. We will prove that if F is any fixed totally real number field of degree [F : Q] ≥ 3, then there are infinitely many CM extensions E ̸ F such that E ̸ Q is 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.