Zeta-determinant and torsion functions on Riemann surfaces of finite volume
type: Item
instance of: scholarly article
title: Zeta-determinant and torsion functions on Riemann surfaces of finite volume
zbMATH Open document ID: 0844.11037
DOI: 10.1007/BF02568004
author: Donggeng Gong
published in: Manuscripta Mathematica
publication date: 1996-09-01T00:00:00Z
full work available at URL: https://eudml.org/doc/156064
review text: Consider a finite volume Riemann surface \(M= H/ \Gamma\), where \(H\) is the Poincaré upper half plane and \(\Gamma\) is a cofinite discrete subgroup of \(\text{SL} (2, \mathbb{R})\). Consider the operator \(\Delta_{2k} f= y^2 ({{\partial^2 f} \over {\partial x^2}}+ {{\partial^2 f} \over {\partial y^2}} )- 2kiy {{\partial f} \over {\partial x}}\) on functions \(f\) which are square integrable with respect to \({{dx dy} \over y^2}\) and such that \(f(Az)= j_A (z,k) f(z)\) for all \(A\in \Gamma\). Here if \(Az= {{az+b} \over {cz+d}}\), \(j_A (z,k)= \exp (2ik \arg (cz+d))\). The spectral problem \(\Delta_{2k} f+ \lambda f=0\) determines a discrete as well as a continuous spectrum of \(\lambda= s(1- s)\), in general. The continuous spectrum is given by Eisenstein series. The spectral zeta function \(\zeta (s, \mu)\) is a sum over \(s= s_n\) corresponding to the discrete spectrum of terms \((s_n (1- s_n)- \mu(1- \mu))^{-s}\) plus an integral over the continuous spectrum of the same function of \(s\). The \(\zeta\)-determinant function for \(\text{Re } \lambda>1\) is \(\text{det} (\Delta_{2k}- \lambda (1- \lambda))= e^{- (\partial/ \partial s) (\zeta (s, \lambda))_{s=0}}\). This is valid provided \(\zeta (s, \mu)\) extends to an analytic function in a neighborhood of \(s=0\). Since \(\zeta (s, \mu)\) is a Mellin transform of a theta function \(\Theta_\Gamma (t)\), one need only estimate \(\Theta_\Gamma (t)\) as \(t\) goes to 0. This is done using the Selberg trace formula, which expresses the sum over the discrete spectrum of \(h(r_n)\), for \(\lambda_n= {1\over 4}+ r^2_n\) as a sum of 10 terms coming from the 4 types of conjugacy classes (central, hyperbolic, elliptic, and parabolic) in \(\Gamma\). The result for \(h(r)= \exp(- ({1\over 4}+ z^2) t)\) is given in Theorem 2. Some references for the Selberg trace formula are \textit{J. Fischer} [Lect. Notes Math. 1253 (1987; Zbl 0618.10029)], \textit{D. Hejhal} [Lect. Notes Math. 1001 (1983; Zbl 0543.10020)], and the revie…
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 30F15
Mathematics Subject Classification ID: 58J52
zbMATH DE Number: 784051
zbMATH Keywords: finite volume Riemann surface
zbMATH Keywords: Mellin transform
zbMATH Keywords: Selberg trace formula
zbMATH Keywords: discrete spectrum
zbMATH Keywords: resolvent trace formula
zbMATH Keywords: zeta-determinant function
zbMATH Keywords: conjugacy class zeta functions
zbMATH Keywords: \(L^ 2\)-analytic torsion function
reviewed by: Audrey A. Terras
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784051