On the spectral expansion of hyperbolic Eisenstein series

Abstract

In this article, we determine the spectral expansion, meromorphic continuation, and location of poles with identifiable singularities for the scalar-valued hyperbolic Eisenstein series. Similar to the form-valued hyperbolic Eisenstein series studied in Kudla and Millson (Invent Math 54:193-211, 1979), the scalar-valued hyperbolic Eisenstein series is defined for each primitive, hyperbolic conjugacy class within the uniformizing group associated to any finite volume hyperbolic Riemann surface. Going beyond the results in Kudla and Millson (Invent Math 54:193-211, 1979) and Risager (Int Math Res Not 41:2125-2146, 2004, we establish a precise spectral expansion for the hyperbolic Eisenstein series for any finite volume hyperbolic Riemann surface by first proving that the hyperbolic Eisenstein series is in L 2 . Our other results, such as meromorphic continuation and determination of singularities, are derived from the spectral expansion.

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

“…, n − 1). To obtain the meromorphic continuation across the lines Re(s) = 1/2 − 2ℓ, we mimic the method used, e.g., in the proof of Theorem 2 of [JKvP10] or in [vP10]. Namely, for ℓ = 0, we move the line of integration to Re(s) = 1/2 + ε for some ε > 0 sufficiently small such that E par p k (z, s) has no poles in the strip 1/2 − ε < Re(s) < 1/2 + ε.…”

“…A scalar-valued, non-holomorphic, hyperbolic Eisenstein series was defined in [JKvP10], and the authors proved that the series admits a meromorphic continuation to all s ∈ C. Let γ ∈ Γ be a primitive, hyperbolic element with stabilizer Γ γ . Let L γ be the geodesic in H which is invariant by the action of γ on H. Then, for z ∈ M and s ∈ C with Re(s) > 1, the hyperbolic Eisenstein series is defined by E hyp γ (z, s) =…”

Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces

“…Following Kudla and Millson's point of view, the scalar-valued analogue of the hyperbolic Eisenstein series is defined in [3,4], and [10]. It is defined as follows.…”

“…Furthermore, J. Jorgenson, J. Kramer and A.-M. v. Pippich [10], in 2010, proved that the hyperbolic Eisenstein series is a square integrable function on Γ\H 2 and obtained the spectral expansion associated to the hyperbolic Laplace-Beltrami operator −∆ precisely. It is given as follows.…”

“…In our previous paper [5], we defined the hyperbolic Eisenstein series for a loxodromic element of the cofinite Kleinian groups acting on 3-dimensional hyperbolic space and proved the results analogous to [10]. We also in [7] consider the asymptotic behavior of the hyperbolic Eisenstein series for the degeneration of 3-dimensional hyperbolic manifolds and obtain the results corresponding to [2,3].…”