Arithmetic curves on ball quotient surface
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)].
selfintersection number of an arithmetic curve
scientific article; zbMATH DE number 3861332
scientific article