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On the Colmez conjecture for non-abelian CM fields

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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 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.

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Notes

  1. The shaded parallelogram in Fig. 2 is the subset of the Shintani cone \(\mathcal {C}(\varepsilon )\) determined by the inequalities \(0 < t_1 \le 1\) and \(0 \le t_2 < 1\), which correspond to the inequalities appearing in the definition of \(\mathcal {R}\left( \varepsilon , \mathfrak {D}_{E/F}^{-1}\right) \).

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Acknowlegements

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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  1. Escuela de Matemática, Universidad de Costa Rica, San José, 11501, Costa Rica

    Adrian Barquero-Sanchez

  2. Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX, 77843-3368, USA

    Riad Masri

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Barquero-Sanchez, A., Masri, R. On the Colmez conjecture for non-abelian CM fields. Res Math Sci 5, 10 (2018). https://doi.org/10.1007/s40687-018-0119-3

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