On the wave representation of hyperbolic, elliptic, and parabolic Eisenstein series

TL;DR

This paper presents a unified wave-based framework for constructing hyperbolic, elliptic, and parabolic Eisenstein series on finite volume hyperbolic Riemann surfaces, linking them through integral transforms of wave kernels.

Contribution

It introduces a novel unified approach using wave operator integral transforms to derive Eisenstein series, connecting heat, Poisson, and wave kernels.

Findings

Unified wave-based construction of Eisenstein series

Explicit integral transform expressions for hyperbolic and elliptic series

Representation of parabolic Eisenstein series via wave operator transforms

Abstract

We develop a unified approach to the construction of the hyperbolic and elliptic Eisenstein series on a finite volume hyperbolic Riemann surface. Specifically, we derive expressions for the hyperbolic and elliptic Eisenstein series as integral transforms of the kernel of a wave operator. Established results in the literature relate the wave kernel to the heat kernel, which admits explicit construction from various points of view. Therefore, we obtain a sequence of integral transforms which begins with the heat kernel, obtains a Poisson and wave kernel, and then yields the hyperbolic and elliptic Eisenstein series. In the case of a non-compact finite volume hyperbolic Riemann surface, we finally show how to express the parabolic Eisenstein series in terms of the integral transform of a wave operator.