Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces
Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces
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希尔伯特模曲面的Selberg迹公式与Selberg型zeta函数的差异
DOI：
10.1016/j.jnt.2014.07.019
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发表时间：
2015
影响因子：
0.7
通讯作者：
Yasuro Gon
中科院分区：
文献类型：
--
作者：
Yoshiaki Okazaki;Yasuro Gon and Takayuki Oda;Yasuro Gon;Aoi Honda and Yoshiaki Okazaki;Yasuro Gon
给出了非紧高阶局部对称空间的Selberg型Zeta函数的一个例子。这是Selberg未发表的工作[26]在非紧致情况下的推广。研究了实二次域的Hilbert模群上的某些Selberg型Zeta函数和Ruelle型Zeta函数。我们证明了它们具有到整个复平面的亚纯扩张，并且满足函数方程。该方法考虑了不同权重的Selberg迹公式对于Hilbert模群的不同。此外，作为Selberg迹公式的一个应用，我们还得到了实二次整数环上不定二次形式的类数的渐近平均。
We present an example of the Selberg type zeta function for non-compact higher rank locally symmetric spaces. This is a generalization of Selberg's unpublished work [26] to non-compact cases. We study certain Selberg type zeta functions and Ruelle type zeta functions attached to the Hilbert modular group of a real quadratic field. We show that they have meromorphic extensions to the whole complex plane and satisfy functional equations. The method is based on considering the differences among several Selberg trace formulas with different weights for the Hilbert modular group. Besides as an application of the differences of the Selberg trace formula, we also obtain an asymptotic average of the class numbers of indefinite binary quadratic forms over the real quadratic integer ring.