On the ratios of Barnes' multiple gamma functions to the p-adic analogues
On the ratios of Barnes' multiple gamma functions to the p-adic analogues
复制标题
关于 Barnes 的多重伽玛函数与 p 进类似物的比率
DOI：
10.1016/j.jnt.2018.11.022
复制
发表时间：
2019
影响因子：
0.7
通讯作者：
Kashio Tomokazu
中科院分区：
文献类型：
--
作者：
Komori Yasushi;Matsumoto Kohji;Tsumura Hirofumi;Tomokazu Kashio;Shin-ya Koyama;Kashio Tomokazu
我们考虑与全真实的域F的窄射线类c相联系的偏zeta函数f（s，c）。我们有一个典型分解′（0，c）=∑ i X（c，i），其中i覆盖F的所有真实的嵌入。符号X（c，i）表示Hiroyuki Yoshida的类不变量，其被定义为巴恩斯的多个伽马函数的对数和一些校正项的有限和。吉田研究了exp（X（c，ι））、斯塔克单位和志村周期符号之间的关系。Yoshida和作者还定义并研究了p-adic类比Xp（c，i），得到了“比”[exp <$（X（c，i））：exp <$（Xp（c，i））]与Gross-Stark单位之间的显式关系。在前一篇文章中，作者证明了exp（X（c，i））的某些乘积的代数性。在本文中，我们证明了它的p-adic模拟。作为应用，我们得到了相同的“比率”和斯塔克单位之间的显式关系。这样，我们就可以统一讨论秩为1的阿贝尔Stark猜想关于真实的空间和Gross的p-adic类比.
We consider the partial zeta function ζ (s, c) associated with a narrow ray class c of a totally real field F. We have a canonical decomposition ζ′(0, c)=∑ ι X (c, ι) where ι runs over all real embeddings of F. The symbol X (c, ι) denotes Hiroyuki Yoshida's class invariant defined as a finite sum of log of Barnes' multiple gamma functions and some correction terms. Yoshida studied the relation between the values of exp(X (c, ι)), Stark units, and Shimura's period symbol. Yoshida and the author also defined and studied the p-adic analogue X p (c, ι), and obtained an explicit relation between the “ratios”[exp(X (c, ι)): exp p(X p (c, ι))] and Gross–Stark units. In a previous paper, the author proved the algebraicity of some products of exp(X (c, ι))'s. In this paper, we prove its p-adic analogue. As an application, we obtain an explicit relation between the same “ratios” and Stark units. As a result, we can discuss the rank 1 abelian Stark conjecture with respect to real places and Gross'p-adic analogue uniformly.