CM points and weight 3/2 modular forms.

The theta correspondence has been an important tool in the theory of automorphic forms with plentiful applications to arithmetic questions. In this paper, we consider a specific theta lift for an isotropic quadratic space V over Q of signature (1, 2). The theta kernel we employ associated to the lift has been constructed by Kudla-Millson (e.g., [29, 30]) in much greater generality for O(p, q) (U(p, q)) to realize generating series of cohomo-logical intersection numbers of certain, ’special ’ cycles in locally symmetric spaces of orthogonal (unitary) type as holomorphic Siegel (Hermitian) mod-ular forms. In our case for O(1, 2), the underlying locally symmetric space M is a modular curve, and the special cycles, parametrized by positive in-tegers N, are the classical CM points Z(N); i.e., quadratic irrationalities of discriminant −N in the upper half plane. We survey the results of [16] and of our joint work with Bruinier [12] on using this particular theta kernel to define lifts of various kinds of functions F on the underlying modular curve M. The theta lift is given b

SDSS DR7 superclusters. Morphology

We study the morphology of a set of superclusters drawn from the SDSS DR7. We calculate the luminosity density field to determine superclusters from a flux- limited sample of galaxies from SDSS DR7, and select superclusters with 300 and more galaxies for our study. The morphology of superclusters is described with the fourth Minkowski functional V3, the morphological signature (the curve in the shapefinder's K1-K2 plane) and the shape parameter (the ratio of the shapefinders K1/K2). We investigate the supercluster sample using multidimensional normal mixture modelling, and use Abell clusters to identify our superclusters with known superclusters and to study the large-scale distribution of superclusters. The superclusters in our sample form three chains of superclusters; one of them is the Sloan Great Wall. Most superclusters have filament-like overall shapes. Superclusters can be divided into two sets; more elongated superclusters are more luminous, richer, have larger diameters, and a more complex fine structure than less elongated superclusters. The fine structure of superclusters can be divided into four main morphological types: spiders, multispiders, filaments, and multibranching filaments. We present the 2D and 3D distribution of galaxies and rich groups, the fourth Minkowski functional, and the morphological signature for all superclusters. Widely different morphologies of superclusters show that their evolution has been dissimilar. A study of a larger sample of superclusters from observations and simulations is needed to understand the morphological variety of superclusters and the possible connection between the morphology of superclusters and their large-scale environment.Comment: Comments: 20 pages, 18 figures, accepted for publication in Astronomy and Astrophysic