Second moments of incomplete Eisenstein series and applications

Abstract

We prove a second moment formula for incomplete Eisenstein series on the homogeneous space Γ\G with G the orientation preserving isometry group of the real (n + 1)-dimensional hyperbolic space and Γ⊂ G a non-uniform lattice. This result generalizes the classical Rogers' second moment formula for Siegel transform on the space of unimodular lattices. We give two applications of this moment formula. In Chapter 5 we prove a logarithm law for unipotent flows making cusp excursions in a non-compact finite-volume hyperbolic manifold. In Chapter 6 we study the counting problem counting the number of orbits of Γ-translates in an increasing family of generalized sectors in the light cone, and prove a power saving estimate for the error term for a generic Γ-translate with the exponent determined by the largest exceptional pole of corresponding Eisenstein series. When Γ is taken to be the lattice of integral points, we give applications to the primitive lattice points counting problem on the light cone for a generic unimodular lattice coming from SO₀(n+1,1)(ℤ\SO₀(n+1,1).