The Hodge Laplacian operator on 1-forms on $\mathbb{H}$ and 1-form $E_\mathfrak{a}^1$

TL;DR

This paper explores the properties of a specific 1-form eigenfunction on the hyperbolic plane, averaging it over a lattice to define a new form, and analyzes its Fourier expansion, integrals, and an analog of the Rankin-Selberg method.

Contribution

It introduces a novel 1-form eigenfunction on the hyperbolic plane, constructs its automorphic form, and develops an analog of the Rankin-Selberg method for it.

Findings

Fourier expansion of the 1-form automorphic form obtained.

Explicit calculation of Fourier coefficients.

Evaluation of integrals over horocycles and geodesics.

Abstract

As is well known, we can average the eigenfunction $y^s$ of the hyperbolic Laplacian on the hyperbolic plane by $\Gamma$ a lattice in $\mathbf{SL}(2,\mathbb{R})$ to obtain an automorphic form, the non-holomorphic Eisenstein series $E_\mathfrak{a} (z,s)$. In this note, we choose a particular eigenfunction $y^s dx$ of the Hodge-Laplace operator for 1-forms on the hyperbolic plane. Then, we average by $\Gamma$ to define a 1-form $E_\mathfrak{a}^1 \big( (z,v), s \big)$. We see that $E_\mathfrak{a}^1$ admits a Fourier expansion and calculates the corresponding coefficients. Also, we evaluate the integral $\int_{\gamma} E_\mathfrak{a}^1$ for when $\gamma$ is a lifting of horocycles and closed geodesics in the unit tangent bundle. Finally, we will obtain an analog to the Rankin-Selberg method for $E_\mathfrak{a}^1$.