Klein, David Christian (2024)
Hyperbolic and elliptic Eisenstein series in n-dimensional hyperbolic space.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00027466
Dissertation, Erstveröffentlichung, Verlagsversion

Kurzbeschreibung (Abstract)

The classical non-holomorphic Eisenstein series E^par_p(z,s) on the upper half-plane ℍ is associated to a parabolic fixed point p of a Fuchsian subgroup Γ ⊆ PSL_2(ℝ) of the first kind. Hyperbolic and elliptic analogues of E^par_p(z,s) were also studied, namely non-holomorphic Eisenstein series which are associated to a pair of hyperbolic fixed points of Γ or a point in the upper half-plane, respectively. In particular, von Pippich derived Kronecker limit type formulas for elliptic Eisenstein series on the upper half-plane.

In the present thesis we consider hyperbolic and elliptic Eisenstein series in the n-dimensional hyperbolic upper half-space ℍ^n for a discrete group Γ of orientation-preserving isometries of ℍ^n which has finite hyperbolic volume. Here we realize these isometries as certain matrices with entries in the Clifford numbers. We define the hyperbolic Eisenstein series E^hyp_(Q_1,Q_2)(P,s) associated to a pair (Q_1,Q_2) of hyperbolic fixed points of Γ and the elliptic Eisenstein series E^ell_Q(P,s) associated to a point Q ∈ ℍ^n. First we prove the absolute and locally uniform convergence of these series for s ∈ ℂ with Re(s)>n-1. Then we derive some other basic properties of E^hyp_(Q_1,Q_2)(P,s) and E^ell_Q(P,s) like Γ-invariance, smoothness and certain differential equations that are satisfied by these Eisenstein series.

We establish the meromorphic continuations of the hyperbolic Eisenstein series E^hyp_(Q_1,Q_2)(P,s) and the elliptic Eisenstein series E^ell_Q(P,s) in s to the whole complex plane. For that we employ the relations between these Eisenstein series and the so-called hyperbolic kernel function K^hyp(P,Q,s), which is meromorphically continued to all s ∈ ℂ by means of its spectral expansion. In this way we also establish the meromorphic continuation of E^hyp_(Q_1,Q_2)(P,s) via its spectral expansion, and further obtain the meromorphic continuation of E^ell_Q(P,s) by expressing it in terms of K^hyp(P,Q,s). Moreover, we determine the possible poles of E^hyp_(Q_1,Q_2)(P,s) and E^ell_Q(P,s).

Using the aforementioned meromorphic continuations, we investigate the behaviour of the hyperbolic Eisenstein series E^hyp_(Q_1,Q_2)(P,s) and the elliptic Eisenstein series E^ell_Q(P,s) at the point s=0 via their Laurent expansions. We determine the first two terms in the Laurent expansions of E^hyp_(Q_1,Q_2)(P,s) and E^ell_Q(P,s) at s=0 for arbitrary n and Γ. Eventually, we refine the Laurent expansion of E^hyp_(Q_1,Q_2)(P,s) for n=2, Γ=PSL_2(ℤ) and n=3, Γ=PSL_2(ℤ[i]), as well as the Laurent expansion of E^ell_Q(P,s) for n=3, Γ=PSL_2(ℤ[i]), and obtain Kronecker limit type formulas in these specific cases.

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