On absolute CM-periods, II
On absolute CM-periods, II
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在绝对 CM 周期上，II
DOI：
10.1353/ajm.1998.0053
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发表时间：
1998
影响因子：
1.7
通讯作者：
H. Yoshida
中科院分区：
文献类型：
--
作者：
H. Yoshida
对于CM-域K，Shimura通过分解具有复乘的交换变种的周期来定义周期符号pK。我们利用多重伽马函数的除值定义绝对周期符号gk，并猜想pk与gk重合，直到乘以代数数为止。考虑到Gal(q，q)的作用，我们给出了这个猜想的一个精化版本。我们证明了这些猜想是一致公式化的，并讨论了有力支持我们猜想的各种数值例子。在我们以前的文章(Y2)中，我们提出了一个猜想，它给出了Artin L-函数在S=0处的导数按CM-周期的表达式。然而，我们不能用这样的猜想来表示CM周期本身。(这一点将通过7中的一个例子清楚地说明。另见(Y2)，2开头的讨论。)在本文中，我们将给出一个用多重伽马函数在分割点的值来表示CM周期的猜想，并给出支持它的各种数值例子。让我们更准确地解释我们的想法和这篇论文的内容。设K是CM-域，JK是K到C的所有同构的集合，K是由JK生成的自由阿贝尔群。对于每个，Ik，Shimura定义(S2)，(S3)CM周期Pk(，)C，它是唯一确定的mod Q。我们将在1中复习周期符号Pk的基本性质。对于a，b C，让我们写a b if b=0A nda b Q.U SingpK，我们可以写Chowla-Selberg公式如下
For a CM-fieldK, Shimura defined the period symbolpK by factorizing periods of abelian varieties with complex multiplication. We define the absolute period symbolgK using division values of the multiple gamma function and conjecture that pK coincides with gK up to the multiplication by algebraic numbers. Taking the action of Gal(Q Q) into account, we present a refined version of this conjecture. We show that these conjectures are consistently formulated and discuss various numerical examples which support our conjectures strongly. In our previous paper (Y2), we formulated a conjecture which gives an expres- sion of the derivatives of Artin L-functions at s = 0 by CM-periods. However we could not express CM-periods themselves by such a conjecture. (This point will be shown explicitly by an example in 7. See also the discussion in the beginning of (Y2), 2.) In the present paper, we shall give a conjecture which expresses CM-periods by the values of the multiple gamma function at division points, and present various numerical examples which support it. Let us explain our ideas and the contents of this paper more precisely. Let K be a CM-field, JK be the set of all isomorphisms of K into C and IK be the free abelian group generated by JK. For every , IK, Shimura defined (S2), (S3) the CM-period pK( , ) C , which is uniquely determined mod Q.T he fundamental properties of the period symbol pK will be reviewed in 1. For a, b C, let us write a b if b =0a nda b Q.U singpK, we can write the Chowla-Selberg formula as