On the Colmez conjecture for non-abelian CM fields View Full Text


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Article Info

DATE

2018-03

AUTHORS

Adrian Barquero-Sanchez, Riad Masri

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: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 Andreatta–Goren–Howard–Madapusi Pera and Yuan–Zhang. More... »

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References to SciGraph publications

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  • 1998-05. Sur la hauteur de Faltings des variétés abéliennes à multiplication complexe in COMPOSITIO MATHEMATICA
  • 2001. Classical Theory of Algebraic Numbers in NONE
  • 1999. Algebraic Number Theory in NONE
  • 1980. Arithmetic on Elliptic Curves with Complex Multiplication in NONE
  • 2009. Class Field Theory in NONE
  • 2013-06. On Colmez’s product formula for periods of CM-abelian varieties in MATHEMATISCHE ANNALEN
  • 1996. Field and Galois Theory in NONE
  • 2001. Periods in MATHEMATICS UNLIMITED — 2001 AND BEYOND
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  • 2018-05. CM fields of Dihedral type and the Colmez conjecture in MANUSCRIPTA MATHEMATICA
  • 1986. The Arithmetic of Elliptic Curves in NONE
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    http://scigraph.springernature.com/pub.10.1007/s40687-018-0119-3

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    44 schema:description 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 Andreatta–Goren–Howard–Madapusi Pera and Yuan–Zhang.
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