Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces

TL;DR

This paper introduces the first Selberg type zeta functions for noncompact higher rank locally symmetric spaces, specifically for Hilbert modular surfaces, and explores their meromorphic extensions, functional equations, and applications to class number asymptotics.

Contribution

It presents novel Selberg type zeta functions for Hilbert modular surfaces, extending the theory to higher rank noncompact spaces and deriving new functional equations and applications.

Findings

Zeta functions have meromorphic extensions to the entire complex plane.

They satisfy functional equations.

Application to asymptotic class number formulas.

Abstract

We present the first example of the Selberg type zeta function for noncompact higher rank locally symmetric spaces. We study certain Selberg type zeta functions and Ruelle type zeta functions attached to the Hilbert modular group of a real quadratic field. We show that they have meromorphic extensions to the whole complex plane and satisfy functional equations. The method is based on considering the differences among several Selberg trace formulas with different weights for the Hilbert modular group. Besides as an application of the differences of the Selberg trace formula, we also obtain an asymptotic average of the class numbers of indefinite binary quadratic forms over the real quadratic integer ring.