Ball and Surface Arithmetics

ISBN-13: 9783322901712 / Angielski / Miękka / 2012 / 414 str.

Rolf-Peter Holzapfel

Opis

This monograph is based on the work of the author on surface theory con nected with ball uniformizations and arithmetic ball lattices during several years appearing in a lot of special articles. The first four chapters present the heart of this work in a self-contained manner (up to well-known ba sic facts) increased by the new functorial concept of orbital heights living on orbital surfaces. It is extended in chapter 6 to an explicit HURWITZ theory for CHERN numbers of complex algebraic surfaces with the mildest singularities, which are necessary for general application and proofs. The chapter 5 is dedicated to the application of results in earlier chapters to rough and fine classifications of PICARD modular surfaces. For this part we need additionally the arithmetic work of FEUSTEL whose final results are presented without proofs but with complete references. We had help ful connections with Russian mathematicians around VENKOV, VINBERG, MANIN, SHAFAREVICH and the nice guide line of investigations of HILBERT modular surfaces started by HIRZEBRUCH in Bonn. More recently, we can refer to the independent (until now) study of Zeta functions of PICARD modular surfaces in the book L-R] edited by LANGLANDS and RAMAKR ISHN AN. The basic idea of introducing arrangements on surfaces comes from the monograph BHH], (BARTHEL, HOFER, HIRZEBRUCH) where linear ar rangements on the complex projective plane ]p2 play the main role."

Spis treści

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.

O autorze

Dr. Rolf-Peter Holzapfel ist Mitglied der Arbeitsgruppe "Allgebraische Geometrie und Zahlentheorie" am Max-Planck-Institut zur Förderung der Wissenschaften, Berlin.