On the Colmez conjecture for non-abelian CM fields
On the Colmez conjecture for non-abelian CM fields
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关于非阿贝尔 CM 域的 Colmez 猜想
DOI：
10.1007/s40687-018-0119-3
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发表时间：
2016
影响因子：
1.2
通讯作者：
R. Masri
中科院分区：
文献类型：
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作者：
Adrian Barquero;R. Masri
科尔梅兹猜想将阿贝尔簇与CM域E的整数环的复数乘法的法尔丁高度与Artin L-函数在$$s= 0 $$s =0的对数导数联系起来。本文证明，如果F是任何固定的全真实的数域，其度为$$[F：\mathbb {Q}] \ge 3$$[F：Q]≥3，则存在无限多个有效的CM扩张E / F的“正密度”集，使得$$E/\mathbb {Q}$$E/Q是非阿贝尔的，并且Colmez猜想对E成立。此外，这些CM扩张被明确地构造为在F的素理想的任意指定集合上分歧。我们还证明了Colmez猜想是真实的一个通用类的非阿贝尔CM领域称为Weyl CM领域，并使用此开发一个算术统计方法的Colmez猜想计数CM领域的固定程度和有界判别式的基础上。我们说明了这些结果通过评估Faltings高度的Jacobian的亏格2超椭圆曲线与复杂的乘法由一个非交换的四次CM字段的巴恩斯双伽玛函数在代数参数。这可以看作是一个显式的非阿贝尔Chowla-Selberg公式。我们的结果依赖于最近由Andreatta-Goren-Howard-Madapusi Pera和张元独立证明的Colmez猜想的平均版本。
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 $$s=0$$s=0. In this paper, we prove that if F is any fixed totally real number field of degree $$[F:\mathbb {Q}] \ge 3$$[F:Q]≥3, then there are infinitely many effective, “positive density” sets of CM extensions E / F such that $$E/\mathbb {Q}$$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.
在反射场上定义的二属 CM 曲线示例
影响因子：
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作者：
Bouyer F
通讯作者：
Bouyer F