Eta invariants for flat manifolds

Using H. Donnelly result from the article "Eta Invariants for G-Spaces" we calculate the eta invariants of the signature operator for almost all 7-dimensional flat manifolds with cyclic holonomy group. In all cases this eta invariants are an integer numbers. The article was motivated by D. D. Long and A. Reid article "On the geometric boundaries of hyperbolic 4-manifolds, Geom. Topology 4, 2000, 171-178Comment: 18 pages, a new version with referees comment

Regularity of the eta function on manifolds with cusps

On a spin manifold with conformal cusps, we prove under an invertibility condition at infinity that the eta function of the twisted Dirac operator has at most simple poles and is regular at the origin. For hyperbolic manifolds of finite volume, the eta function of the Dirac operator twisted by any homogeneous vector bundle is shown to be entire.Comment: 22 pages, 2 figure

Resolvent and lattice points on symmetric spaces of strictly negative curvature

We study the asymptotics of the lattice point counting function N(x,y;r)=#{γ∈Γ:d(x,γy)} for a Riemannian symmetric space X obtained from a semisimple Lie group of real rank one and a discontinuous group Γ of motions in X, such that Γ∖X has finite volume. We show that as r→∞ , for each ε>0 . The constant 2ρ corresponds to the sum of the positive roots of the Lie group associated to X, and n = dimX. The sum in the main term runs over a system of orthonormal eigenfunctions φj∈L2(Γ∖X) of the Laplacian, such that the eigenvalues ρ2−ν2j are less than 4nρ2/(n+1)2

Sums of Kloosterman sums for real quadratic number fields

We estimate sums of Kloosterman sums for a real quadratic number field F of the type where c runs through the integers of F that satisfy C⩽|N(c)|0. An estimate not taking cancellation between Kloosterman sums into account would yield . The exponent is less sharp than occurs in the bound , obtained in our paper in J. reine angew. Math. 535 (2001) 103–164 for sums of Kloosterman sums where c runs over integers satisfying , . The proof is based on the Kloosterman-spectral sum formula for the corresponding Hilbert modular group. The Bessel transform in this formula has a product structure corresponding to the infinite places of F. This does not fit well to the bounds depending on N(c) and c/c′. Nevertheless, we do obtain non-trivial bounds for S