Period functions and the Selberg zeta function for the modular group.
Period functions and the Selberg zeta function for the modular group.
复制标题
模群的周期函数和 Selberg zeta 函数。
DOI：
--
复制
发表时间：
1997
期刊：
影响因子：
--
通讯作者：
D. Zagier
中科院分区：
文献类型：
--
作者：
John S. Lewis;D. Zagier
黎曼曲面 X 上的塞尔伯格迹公式将拉普拉斯算子的离散谱与表面的长度谱（即 X 上的闭合测地线的长度集合）联系起来。这种联系最引人注目地以塞尔伯格 zeta 函数的形式表达，该函数是复变量 s 的亚纯函数，根据长度谱定义为 <(s) > 1 并且在 s ∈ C 处都有零点，其中 s(1 − s) 是 L(X) 中拉普拉斯算子的特征值。我们感兴趣的是当 X 是上半平面 H 通过模群 Γ1 = SL(2,Z) 或扩展模群 Γ = GL(2,Z) 的商时的情况，其中 γ = ( a b c d ) ε Γ 通过 z 7→ (az + b)/(cz + d) 作用于 H，如果 det(γ) = +1 且 z 7→ (az̄ + b)/(cz̄ + d) 如果 det(γ) = −1。在这种情况下，X 的长度谱以实二次域中的类数和阶数单位给出，而拉普拉斯算子的本征函数是通常称为马斯波形的非全纯模函数。 （关于这个主题的很好的阐述可以在[6]和[7]中找到）。
The Selberg trace formula on a Riemann surface X connects the discrete spectrum of the Laplacian with the length spectrum of the surface, that is, the set of lengths of the closed geodesics of on X. The connection is most strikingly expressed in terms of the Selberg zeta function, which is a meromorphic function of a complex variable s that is defined for <(s) > 1 in terms of the length spectrum and that has zeros at all s ∈ C for which s(1 − s) is an eigenvalue of the Laplacian in L(X). We will be interested in the case when X is the quotient of the upper half-plane H by either the modular group Γ1 = SL(2,Z) or the extended modular group Γ = GL(2,Z), where γ = ( a b c d ) ∈ Γ acts on H by z 7→ (az + b)/(cz + d) if det(γ) = +1 and z 7→ (az̄ + b)/(cz̄ + d) if det(γ) = −1. In this case the length spectrum of X is given in terms of class numbers and units of orders in real quadratic fields, while the eigenfunctions of the Laplace operator are the non-holomorphic modular functions usually called Maass wave forms. (Good expositions of this subject can be found in [6] and [7]).