Picard Modular Surfaces Research Articles

Quelques relations entre les cohomologies des variétés de Shimura et celles de Griffiths–Schmid (cas du groupe SU(2,1))

Our aim is to study the cohomology groups of some coherent sheaves on a Griffiths–Schmid variety associated with an anisotropic Q-from of the unitary group SU(2,1). We define some transforms relating this cohomology to the coherent cohomology groups of some sheaves defined on certain threefolds, which are fibered in projective lines over Picard modular surfaces. In particular, we give a complete and explicit description, in terms of classical Picard modular forms, of the holomorphic (resp. anti-holomorphic) part of the 1-cohomology of the Griffiths-Schmid variety. From this, it results an explicit generating system for the part of the 2–cohomology which correspond to those automorphic representations whose archimedean component is a degenerate limit of discrete series.

Zeta dimension formula for picard modular cusp forms of neat natural congruence subgroups

Let $$\Gamma _K = \mathbb{U}((2,1),\mathfrak{D}_K )$$ be the full Picard modular group of the imaginary quadratic number fieldK. For all natural congruence subgroups Γk (m), m ≥ 3, acting freely on the two-dimensional complex unit ball, we prove an explicit polynomial formula for the dimensions of spaces of cusp forms of weightn ≥ 2. The coefficients of these polynomials in the natural variablesm, n are expressed by higher third and first Bernoulli numbers of the Dirichlet character χk of K and by values of Euler factors of the Riemann Zeta function and such factors of theL-series of χkat 2 or 3, respectively. The proof is based on detailed knowledge about classification of Picard modular surfaces. It combines algebraic geometric methods (Riemann-Roch, Vanishing- and Proportionality Theorem, curvature, structure of algebraic groups) with modern and classical number theoretic ones (representation densities, Tamagawa measure, strong approximation, functional equation forL-series).

Chern Numbers of Algebraic Surfaces — Hirzebruch's Examples are Picard Modular Surfaces

Mathematische NachrichtenVolume 126, Issue 1 p. 255-273 Article Chern Numbers of Algebraic Surfaces — Hirzebruch's Examples are Picard Modular Surfaces R.-P. Holzapfel, R.-P. Holzapfel Akademie der Wissenschaften der DDR Karl-Weierstraß-Institut für Mathematik DDR-1086 Berlin Mohrenstraße 39Search for more papers by this author R.-P. Holzapfel, R.-P. Holzapfel Akademie der Wissenschaften der DDR Karl-Weierstraß-Institut für Mathematik DDR-1086 Berlin Mohrenstraße 39Search for more papers by this author First published: 1986 https://doi.org/10.1002/mana.19861260117Citations: 8AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume126, Issue11986Pages 255-273 RelatedInformation

Arithmetic cycles on Picard modular surfaces and modular forms of Nebentypus.

Arithmetic cycles on Picard modular surfaces and modular forms of Nebentypus.

Symmetry Points and Chern Invariants of Picard Modular Surfaces

Symmetry Points and Chern Invariants of Picard Modular Surfaces