Arithmetic curves on ball quotient surface
type: Item
instance of: scholarly article
title: Arithmetic curves on ball quotient surface
zbMATH Open document ID: 0542.14027
DOI: 10.1007/BF02330006
author: Rolf-Peter Holzapfel
published in: Annals of Global Analysis and Geometry
publication date: 1983-01-01T00:00:00Z
review text: Let \(\Gamma\) be an arithmetic group acting on the two-dimensional complex unit ball B. Special subdiscs D of B, called rational discs, project to algebraic curves on B/\(\Gamma\). These curves are called arithmetic curves. In the article there is proved a formula for the selfintersection number of an arithmetic curve on a certain smooth model of the Baily- Borel compactification B/\(\Gamma\), which is called the filled singularity resolution of B/\(\Gamma\) along D/\(\Gamma\). The proof of the selfintersection formula is a combination of Hirzebruch-Mumford's proportionality theory and a fine analysis of a stepwise resolution procedure for quotient singularities of surfaces. - The intersection formula is applied to the classification of Picard modular surfaces of Gauß and Eisenstein numbers. For this purpose the author uses a fine classification criterion for rational surfaces by means of Chern numbers and curve configurations. In the meantime the results have been effectively applied by the author to the study of automorphic forms, the monodromy group of an Euler partial differential equation (system) and of the integrals on Riemann surfaces of equation type \(y^ 3=x(x-1)(x-u)(x- v)\) and to related problems. The corresponding results can be found in the author's monograph ''Geometry and arithmetic around Euler partial differential equations'' (Mathematics and its applications, Dordrecht (1985)].
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 14D22
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 3861332
zbMATH Keywords: rational discs
zbMATH Keywords: selfintersection number of an arithmetic curve
zbMATH Keywords: filled singularity resolution
zbMATH Keywords: Picard modular surfaces
zbMATH Keywords: automorphic forms
zbMATH Keywords: monodromy group
zbMATH Keywords: integrals on Riemann surfaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3861332