Super-zeta functions and regularized determinants associated to cofinite Fuchsian groups with finite-dimensional unitary representations

TL;DR

This paper introduces super-zeta functions to regularize determinants of hyperbolic Laplacians on non-compact Riemann surfaces, revealing symmetries linked to spectral data and Selberg zeta functions.

Contribution

It develops a novel super-zeta approach for regularized determinants, capturing spectral symmetries not seen in previous methods.

Findings

Defined super-zeta functions encoding spectral data

Derived a formula relating determinants to the Selberg zeta function

Revealed symmetry $z\leftrightarrow 1-z$ in the determinant expression

Abstract

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $\chi$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $\Delta$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over $M$ associated to $\chi$. From the spectral theory of $\Delta$, 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 $\Delta$ in such a way that they can be used to define the regularized determinant of $\Delta-z(1-z)I$. The resulting formula for the regularized determinant of $\Delta-z(1-z)I$ 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.