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

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 $E_{\mathfrak{a}}^1((z,v),s)$. 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 $E_{\mathfrak{a}}^1$.

14 pages