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 Artin L-functions at s = 0. In this paper, we prove that if F is any fixed totally real number field of degree [F : Q] ≥ 3, then there are infinitely many effective, "positive density" sets of 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 viewed as an explicit non-abelian Chowla-Selberg formula. Our results rely crucially on an averaged version of the Colmez conjecture which was recently proved independently by
In this paper we develop an effective procedure for expressing Stark units in real quadratic extensions of totally real fields as values of the Barnes multiple Gamma function at algebraic points. This procedure is used to explicitly generate non-abelian extensions of Q by special Gamma values. As a main component of our work, we develop an algorithm to compute Shintani sets in all dimensions. ' " " " " " " " " " " ( " " " " " " " " " " )
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