Volume 23, Issue 5

W. Luo and P. Sarnak have proved the quantum unique ergodicity property for Eisenstein series on PSL(2, ℤ)\ℍ. Their result is quantitative in the sense that they find the precise asymptotics of the measure considered. We extend their result to Eisenstein series on , where 𝒪 is the ring of integers in a totally real field of degree n over ℚ with narrow class number one, using the Eisenstein series considered by I. Efrat. We also give an expository treatment of the theory of Hecke operators on non-holomorphic Hilbert modular forms.

We study the relation between certain local and global numerical invariants of a projective surface given by the Stückrad–Vogel self-intersection cycle. In the smooth case we find a relation between the degrees of the tangent, the secant and the first polar variety which generalizes classical relations between the so-called elementary invariants of surfaces. For singular surfaces this relation involves also the contributions of singularities to the self-intersection cycle. In order to make the formula really effective for singular surfaces, we need to describe the movable part of the self-intersection cycle on the singular locus. This is done explicitly for a particular class of non-normal del Pezzo surfaces.

Wolff potentials are used to study the measures μ for which □ w = μ where □ is the d'Alembertian.

We study one-level and two-level densities for low-lying zeros of symmetric power L -functions in the level aspect. This allows us to completely determine the symmetry types of some families of symmetric power L -functions with prescribed sign of functional equation. We also compute the moments of one-level density and exhibit mock-Gaussian behavior discovered by Hughes & Rudnick.

This paper classifies irreducible Harish-Chandra modules over the loop-Virasoro algebra, which turn out to be highest weight modules, lowest weight modules and evaluation modules of the intermediate series (all wight spaces are 1-dimensional). As a by-product, we obtain a classification of irreducible Harish-Chandra modules over truncated Virasoro algebras. We also establish an irreducibility criterion for Verma modules over the loop Virasoro algebra and determine necessary and sufficient conditions for the simple quotient of a Verma module to be a Harish-Chandra module.

In the first part of the paper we present a topological classification of fake lens spaces of dimension ≥ 5 whose fundamental group is the cyclic group of order any N ≥ 2. The classification is stated in terms of the simple structure sets in the sense of surgery theory. The results use and extend the results of Wall and others in the cases N = 2 and N odd and the results of the authors of the present paper in the case N = 2 K . In the second part we study the suspension map between the simple structure sets of lens spaces of different dimensions. As an application we obtain an inductive geometric description of the torsion invariants of fake lens spaces.

Let X be a (connected and reduced) complex space. A q-collar of X is a bounded domain whose boundary is a union of a strongly q -pseudoconvex, a strongly q -pseudoconcave and two flat (i.e. locally zero sets of pluriharmonic functions) hypersurfaces. Finiteness and vanishing cohomology theorems obtained in [Saracco and Tomassini, Math. Z. 256: 737–748, 2007, Bull. Sci. Math. 132: 232–245, 2008] for semi q -coronae are generalized in this context and lead to results on extension problems and removable sets for sections of coherent sheaves and analytic subsets.