论文标题
超级Zeta功能和与有限维统一表示的富富奇偶群相关的正则决定因素
Super-zeta functions and regularized determinants associated to cofinite Fuchsian groups with finite-dimensional unitary representations
论文作者
论文摘要
令$ m $为有限的量，非紧缩双曲线riemann表面，可能具有椭圆固定点，让$χ$表示$ m $的基本组的有限维度单一表示。令$δ$表示双曲线拉普拉斯式的柔软的laplacian，该拉普拉斯人在平滑的套件上，超过$ m $，与$χ$相关。从$δ$的光谱理论中，有三个不同的数字序列：第一个来自$ l^{2} $ eigenfunctions的特征值，第二个来自与连续光谱相关的共振，第三个是负整数的集合。使用这些光谱数据序列，我们采用了超级Zeta方法进行正则化，并引入了两个超级Zeta功能，$ \ z _-（s，z）$和$ \ z _+（s，z）$（s，z）$（S，Z）$，以$δ$的频谱的方式以某种方式可以用来定义正规化的确定性$δ-Z（1- Z）（1- Z）I。 $δ-Z（1-Z）i $正规化决定因素的最终公式在Selberg Zeta功能方面，请参见Theorem 5.3，编码对称性$ z \ leftrightArrow 1-Z $，由于以前的工作，由于对正则确定性的不同定义，这在以前的工作中无法看到。
Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $χ$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $Δ$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$ associated to $χ$. From the spectral theory of $Δ$, there are three distinct sequences of numbers: The first coming from the eigenvalues of $L^{2}$ eigenfunctions, the second coming from resonances associated to the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, $\Z_-(s,z)$ and $\Z_+(s,z)$ that encode the spectrum of $Δ$ in such a way that they can be used to define the regularized determinant of $Δ-z(1-z)I$. The resulting formula for the regularized determinant of $Δ-z(1-z)I$ in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry $z\leftrightarrow 1-z$, which could not be seen in previous works, due to a different definition of the regularized determinant.
