On the appearance of Eisenstein series through degeneration

Abstract

Let Γ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane H, and let M = Γ\H be the associated finite volume hyperbolic Riemann surface. If γ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If γ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If γ ∈ Γ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.

Applications of Kronecker’s limit formula for elliptic Eisenstein series

“…Since the hyperbolic Eisenstein series are in L 2 (M ), the expression (1) admits a spectral expansion which involves the parabolic Eisenstein series; see [JKvP10] and [KM79]. If one considers a degenerating sequence of Riemann surfaces obtained by pinching a geodesic, then the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface; see [Fa07] and [GJM08]. If one studies a family of elliptically degenerating surfaces obtained by re-uniformizing at a point with increasing order, then the corresponding elliptic Eisenstein series converge to parabolic Eisenstein series on the limit surface; see [GvP09].…”

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

“…Referring to [Fa07], [GJM08], or [Ri04], e.g., where detailed proofs are provided, we recall that the series (4) converges absolutely and locally uniformly for any z ∈ H and s ∈ C with Re(s) > 1, and that the series is invariant with respect to Γ. A straightforward computation shows that the hyperbolic Eisenstein series satisfies the differential equation…”

“…Using elementary considerations involving counting functions, one has that the series in (28) converges absolutely and locally uniformly for z, w ∈ M and s ∈ C with Re(s) > 1; see [GJM08] for details. Therefore, we get…”

On the spectral expansion of hyperbolic Eisenstein series

“…The method of proof in [2] involves a detailed analysis of the differential equation satisfied by the hyperbolic Eisenstein series. The main results in [2] are reproved in [3] using counting function arguments and Stieltjes integral representations of various Eisenstein series.…”

“…Referring to [2,3,11], or [10], where detailed proofs are provided, we recall that the series (6) converges absolutely and locally uniformly for any z ∈ H and s ∈ C with Re(s) > 1, and that it is invariant with respect to Γ . A straightforward computation shows that the series (6) satisfies the differential equation…”