The Hodge Laplacian operator on 1-forms on ℍ and 1-form E_\mathfraka¹

2023/07/23 by Romero, Otto
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2307.12209

Abstract

As is well known, we can average the eigenfunction y^s of the hyperbolic Laplacian on the hyperbolic plane by Γ a lattice in SL(2,ℝ) to obtain an automorphic form, the non-holomorphic Eisenstein series E_\mathfraka (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 Γ to define a 1-form E_\mathfraka¹ ( (z,v), s ). We see that E_\mathfraka¹ admits a Fourier expansion and calculates the corresponding coefficients. Also, we evaluate the integral ∫_γ E_\mathfraka¹ for when γ 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_\mathfraka¹.

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