THE HODGE LAPLACIAN OPERATOR ON 1-FORMS ON IHI AND 1-FORM Ea1

As is well known, we can average the eigenfunction y s of the hyperbolic Laplacian on the hyperbolic plane by Gamma a lattice in SL(2, R) to obtain an automorphic form: the nonholomorphic Eisenstein series Ea(z, s ) . In this note, we choose a particular eigenfunction ysdx of the Hodge-Laplace operator for 1-forms on the hyperbolic plane IHI. We can average ysdx by Gamma to define a 1-form E 1 ( ( z, v ) , s ) . a We shall prove that E 1 efficients. Also, we evaluate the integral f a admits a Fourier expansion and calculates the corresponding co gamma Ea1 for when gamma is a lifting of horo cycles and closed geodesics in the unit tangent bundle. Finally, we will obtain an analog to the Rankin-Selberg method for E a 1 via horo cycles.