[Paper Review] Zeta functions and regularized determinants related to the Selberg trace formula

Arash Momeni, Alexei B. Venkov|arXiv (Cornell University)|Aug 29, 2011

Advanced NMR Techniques and Applications32 references3 citations

TL;DR

This paper establishes novel connections between zeta functions, regularized determinants, and the Selberg trace formula for Fuchsian groups with unitary representations. It derives determinant representations of zeta functions for identity, hyperbolic, elliptic, and parabolic contributions using spectral theory, and proves determinant identities linking automorphic Laplacians on different Riemann surfaces via the Jacquet-Langlands correspondence.

ABSTRACT

For a general Fuchsian group of the first kind with an arbitrary unitary representation we define zeta functions related to the contributions of the identity, hyperbolic, elliptic and parabolic conjugacy classes in Selberg's trace formula. We present Selberg's zeta function in terms of a regularized determinant of the automorphic Laplacian. We also present the zeta function for the identity contribution in terms of a regularized determinant of the Laplacian on the two dimensional sphere. We express the zeta functions for the elliptic and parabolic contributions in terms of certain regularized determinants of one dimensional Schroedinger operator for harmonic oscillator. We decompose the determinant of the automorphic Laplacian into a product of the determinants where each factor is a determinant representation of a zeta function related to Selberg's trace formula. Then we derive an identity connecting the determinants of the automorphic Laplacians on different Riemannian surfaces related to the arithmetical groups. Finally, by using the Jacquet-Langlands correspondence we connect the determinant of the automorphic Laplacian for the unit group of quaternions to the product of the determinants of the automorphic Laplacians for certain cogruence subgroups.

Motivation & Objective

• To define zeta functions associated with each conjugacy class type (identity, hyperbolic, elliptic, parabolic) in the Selberg trace formula for Fuchsian groups of the first kind with unitary representations.
• To express Selberg's zeta function and contributions from each conjugacy class type as regularized determinants of differential operators (Laplacian, Schrödinger operators).
• To decompose the regularized determinant of the automorphic Laplacian into a product of zeta function-related determinants.
• To derive determinant identities for commensurable Fuchsian groups and establish a link via the Jacquet-Langlands correspondence between quaternionic unit groups and congruence subgroups.
• To extend the framework of regularized determinants to non-compact, finite-volume Riemann surfaces with continuous spectrum and resonances.

Proposed method

• Define zeta functions for each conjugacy class type (I, H, E, P) via trace formula contributions, using spectral zeta regularization and analytic continuation.
• Express the identity contribution zeta function as a regularized determinant of the Laplacian on the 2-sphere.
• Represent elliptic and parabolic contributions via regularized determinants of one-dimensional Schrödinger operators for the harmonic oscillator.
• Use spectral zeta functions and analytic continuation to define regularized determinants of the automorphic Laplacian, incorporating discrete eigenvalues and continuous spectrum.
• Apply the Selberg trace formula with test functions in the strip |Im(r)| < 1/2 + ε, ensuring absolute convergence of integrals and series.
• Leverage the Jacquet-Langlands correspondence to relate the determinant of the automorphic Laplacian on the unit group of a quaternion algebra to products of determinants on congruence subgroups.

Experimental results

Research questions

• RQ1 How can zeta functions associated with each conjugacy class type in the Selberg trace formula be expressed as regularized determinants of differential operators?
• RQ2 What is the relationship between the regularized determinant of the automorphic Laplacian and the decomposition into contributions from identity, hyperbolic, elliptic, and parabolic conjugacy classes?
• RQ3 How do determinant identities for commensurable Fuchsian groups emerge from the spectral theory of automorphic forms?
• RQ4 In what way does the Jacquet-Langlands correspondence relate the determinant of the automorphic Laplacian on the unit group of a quaternion algebra to those on congruence subgroups?
• RQ5 How can regularized determinants be consistently defined for non-compact, finite-volume Riemann surfaces with continuous spectrum and resonances?

Key findings

• The Selberg zeta function is expressed as a regularized determinant of the automorphic Laplacian via spectral zeta regularization.
• The identity contribution zeta function is represented as the regularized determinant of the Laplacian on the 2-sphere.
• The elliptic and parabolic contributions are expressed as regularized determinants of one-dimensional Schrödinger operators for the harmonic oscillator.
• The determinant of the automorphic Laplacian decomposes into a product of determinants, each corresponding to a zeta function associated with a conjugacy class type.
• For cocompact groups, the trace formula reduces to ∑h(λₖ) = I + H + E, omitting the continuous spectrum and parabolic terms.
• The Jacquet-Langlands correspondence yields a determinant identity linking the automorphic Laplacian on the unit group of a quaternion algebra to a product of determinants on specific congruence subgroups.