Ball and Surface Arithmetics - Rolf-Peter Holzapfel

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Résumé :
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Biographie:
Dr. Rolf-Peter Holzapfel ist Mitglied der Arbeitsgruppe Allgebraische Geometrie und Zahlentheorie am Max-Planck-Institut zur F?rderung der Wissenschaften, Berlin....

Sommaire:
1 Abelian Points.- 1.1 Cyclic Points.- 1.2 Graphs of Abelian Points.- 1.3 Geometric Interpretation.- 1.4 Derived Representations.- 1.5 The Differential Relation.- 1.6 Stepwise Resolutions of Cyclic Points.- 1.7 Continued Fractions and Selfintersection Numbers.- 1.8 Reciprocity Law for Geometric Sums.- 1.9 Explicit Dedekind Sums.- 1.10 Eisenstein Sums.- 1.11 Hirzebruch's Sum.- 1.12 Geometric Interpretation.- 1.13 Quotients and Coverings of Modifications.- 1.14 Selfintersections of Quotient Curves.- 1.15 The Bridge Algorithm.- 1.16 First Orbital Properties.- 1.17 Local Orbital Euler Numbers.- 1.18 Absorptive Numbers.- 2 Orbital Curves.- 2.1 Point Arrangements on Curves.- 2.2 Euler Heights of Orbital Curves.- 2.3 The Geometric Local-Global Principle.- 2.4 Signature Heights of Orbital Curves.- 3 Orbital Surfaces.- 3.1 Regular Arrangements on Surfaces.- 3.2 Basic Invariants and Fixed Point Theorem.- 3.3 Euler Heights.- 3.4 Signature Heights.- 3.5 Quasi-homogeneous Points, Quotient Points and Cusp Points.- 3.6 Quasi-smooth Orbital Surfaces.- 3.7 Open Orbital Surfaces.- 3.8 Orbital Decompositions.- 4 Ball Quotient Surfaces.- 4.1 Ball Lattices.- 4.2 Neat Ball Cusp Lattices.- 4.3 Invariants of Neat Ball Quotient Surfaces.- 4.4 ?-Rational Discs.- 4.5 Cusp Singularities, Reflections and Elliptic Points.- 4.6 Orbital Ball Quotient Surfaces and Molecular.- 4.7 Invariants of Disc Quotient Curves.- 4.8 Invariants of Ball Quotient Surfaces.- 4.9 Global Proportionality.- 4.10 Orbital Decompositions and the Finiteness Theorem.- 4.11 Leading Examples.- 4.12 Towards the Count of Ball Metrics on Non-Compact Surfaces.- 5 Picard Modular Surfaces.- 5.1 Classification Diagram.- 5.2 Picard Modular Surface of the Field of Eisenstein Numbers.- 5.3 Picard Modular Surface of the Field of Gauss-Numbers.- 5.4 Kodaira Classification of Picard Modular Surfaces.- 5.5 Special Results and Examples.- 5A Volumes of Fundamental Domains of Picard Modular Groups.- 5A.1 The Order of Finite Unitary Groups.- 5A.2 Index of Congruence Subgroups.- 5A.3 Local Volumina.- 5A.4 The Global Volume.- 6 ?-Orbital Surfaces.- 6.1 Introduction.- 6.2 Arrangements with Rational Coefficients.- 6.3 Finite Morphisms of ?-Orbital Surfaces.- 6.4 Functorial Properties for Rational Invariants.- 6.5 Euler and Signature Heights.- 6.6 Reduction of Galois-Finite Morphisms.- 6.7 Local Base Changes.- 6.8 Global Base Changes.- 6.9 Explicit Hurwitz Formulas for Finite Surface Coverings.- 6.10 Finite Coverings of Ruled Surfaces and the Inequality c12 ? 2c2.