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The Ball and Some Hilbert Problems

  • Book
  • © 1995

Overview

Authors:
  1. Rolf-Peter Holzapfel

    1. Fachbereich Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

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Part of the book series: Lectures in Mathematics. ETH Zürich (LM)

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-vii
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  2. Elliptic Curves, the Finiteness Theorem of Shafarevič

    • Rolf-Peter Holzapfel
    Pages 1-9

  3. Picard Curves

    • Rolf-Peter Holzapfel
    Pages 11-37

  4. Uniformizations and Differential Equations of Euler-Picard Type

    • Rolf-Peter Holzapfel
    Pages 39-63

  5. Algebraic Values of Picard Modular Theta Functions

    • Rolf-Peter Holzapfel
    Pages 65-100

  6. Transcendental Values of Picard Modular Theta Constants

    • Rolf-Peter Holzapfel
    Pages 101-118

  7. Arithmetic Surfaces of Kodaira-Picard Type and some Diophantine Equations

    • Rolf-Peter Holzapfel
    Pages 119-131

  8. Appendix I A Finiteness Theorem for Picard Curves With Good Reduction (by J. Estrada-Sarlabous)

    • J. Estrada-Sarlabous
    Pages 133-143

  9. Appendix II The Hilbert Problems 7, 12,21 and 22

    • Rolf-Peter Holzapfel
    Pages 145-148

  10. Back Matter

    Pages 149-160
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Keywords

About this book

As an interesting object of arithmetic, algebraic and analytic geometry the complex ball was born in a paper of the French Mathematician E. PICARD in 1883. In recent developments the ball finds great interest again in the framework of SHIMURA varieties but also in the theory of diophantine equations (asymptotic FERMAT Problem, see ch. VI). At first glance the original ideas and the advanced theories seem to be rather disconnected. With these lectures I try to build a bridge from the analytic origins to the actual research on effective problems of arithmetic algebraic geometry. The best motivation is HILBERT'S far-reaching program consisting of 23 prob­ lems (Paris 1900) " . . . one should succeed in finding and discussing those functions which play the part for any algebraic number field corresponding to that of the exponential function in the field of rational numbers and of the elliptic modular functions in the imaginary quadratic number field". This message can be found in the 12-th problem "Extension of KRONECKER'S Theorem on Abelian Fields to Any Algebraic Realm of Rationality" standing in the middle of HILBERTS'S pro­ gram. It is dedicated to the construction of number fields by means of special value of transcendental functions of several variables. The close connection with three other HILBERT problems will be explained together with corresponding advanced theories, which are necessary to find special effective solutions, namely: 7. Irrationality and Transcendence of Certain Numbers; 21.

Authors and Affiliations

  • Fachbereich Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Rolf-Peter Holzapfel

Bibliographic Information

  • Book Title: The Ball and Some Hilbert Problems

  • Authors: Rolf-Peter Holzapfel

  • Series Title: Lectures in Mathematics. ETH Zürich

  • DOI: https://doi.org/10.1007/978-3-0348-9051-9

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Birkhäuser Verlag 1995

  • Softcover ISBN: 978-3-7643-2835-1Published: 01 December 1994

  • eBook ISBN: 978-3-0348-9051-9Published: 06 December 2012

  • Edition Number: 1

  • Number of Pages: 160

  • Number of Illustrations: 3 b/w illustrations

  • Topics: Geometry, Number Theory, Algebraic Geometry, Analysis

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