Super-zeta functions and regularized determinants associated to cofinite Fuchsian groups with finite-dimensional unitary representations
Abstract
Let M be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let denote a finite dimensional unitary representation of the fundamental group of M. Let denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over M associated to . From the spectral theory of , there are three distinct sequences of numbers: The first coming from the eigenvalues of L2 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, -(s,z) and +(s,z) that encode the spectrum of in such a way that they can be used to define the regularized determinant of -z(1-z)I. The resulting formula for the regularized determinant of -z(1-z)I in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry z 1-z, which could not be seen in previous works, due to a different definition of the regularized determinant.