On the Colmez conjecture for non-abelian CM fields

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  1. Modular Forms are Everywhere: Celebration of Don Zagier’s 65th Birthday


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 \([F:\mathbb {Q}] \ge 3\), then there are infinitely many effective, “positive density” sets of CM extensions E / F such that \(E/\mathbb {Q}\) 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 Andreatta–Goren–Howard–Madapusi Pera and Yuan–Zhang.


Author's contributions


The authors were partially supported by the NSF grants DMS-1162535 and DMS-1460766 during the preparation of this work. A. B-S. was also partially supported by the University of Costa Rica.

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Authors and Affiliations

  1. 1.Escuela de MatemáticaUniversidad de Costa RicaSan JoséCosta Rica
  2. 2.Department of Mathematics, Mailstop 3368Texas A&M UniversityCollege StationUSA

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