Zeta-determinant and torsion functions on Riemann surfaces of finite volume
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 reviewer [Harmonic analysis on symmetric spaces and applications. I. (Springer, 1985; Zbl 0574.10029)]. Next the author plugs the function \(h(r)= (r^2+ (s- {1\over 2})^2 )^{-1} -(r^2+ (a- {1\over 2})^2 )^{-1}\) for \(\text{Re} (a)>1\) and \(\text{Re} (s)>1\) into Selberg's trace formula to get the resolvent trace formula. The result is Theorem 3 expressing the zeta-determinant function \(\text{det} (\Delta_{2k}- \lambda(1- \lambda))\) in terms of zeta functions related to the four types of conjugacy classes in \(\Gamma\)-hyperbolic, elliptic, parabolic, and central. There are also two constants to be determined. One is \(\widetilde {c}\), the torsion factor. The asymptotics of the zeta-determinant and the conjugacy class zeta functions as \(\lambda\) goes to infinity determine the constants. The formula (44) for \(\widetilde {c}\) is rather complicated. It is remarked that the torsion factor deserves further investigation. In the last section the \(L^2\)-analytic torsion function of certain differential operators on \(M\) is expressed in terms of the zeta functions and the torsion factor. Previous papers of \textit{P. Sarnak} [Commun. Math. Phys. 110, 113-120 (1987; Zbl 0618.10023)], \textit{I. Efrat} [ibid. 119, 443-451 (1988; Zbl 0661.10038); ibid. 138, 607 (1991; Zbl 0722.11029)]\ and \textit{S. Koyama} [Proc. Am. Math. Soc. 113, 303-311 (1991; Zbl 0744.11031)]\ had considered special cases such as \(M\) compact or the weight \(k=0\). Related papers are \textit{A. Voros} [Commun. Math. Phys. 110, 439-465 (1987; Zbl 0631.10025)]\ and \textit{F. Williams} [Manuscr. Math. 77, 17-39 (1992; Zbl 0789.22024)].
scientific article; zbMATH DE number 784051