Lectures in Mathematics. ETH Zürich

The Ball and Some Hilbert Problems

Authors: Holzapfel, Rolf-Peter

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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.

Table of contents (8 chapters)

  • Elliptic Curves, the Finiteness Theorem of Shafarevič

    Holzapfel, Rolf-Peter

    Pages 1-9

  • Picard Curves

    Holzapfel, Rolf-Peter

    Pages 11-37

  • Uniformizations and Differential Equations of Euler-Picard Type

    Holzapfel, Rolf-Peter

    Pages 39-63

  • Algebraic Values of Picard Modular Theta Functions

    Holzapfel, Rolf-Peter

    Pages 65-100

  • Transcendental Values of Picard Modular Theta Constants

    Holzapfel, Rolf-Peter

    Pages 101-118

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-3-0348-9051-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $99.00
price for USA
  • ISBN 978-3-7643-2835-1
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
The Ball and Some Hilbert Problems
Authors
Series Title
Lectures in Mathematics. ETH Zürich
Copyright
1995
Publisher
Birkhäuser Basel
Copyright Holder
Birkhäuser Verlag
eBook ISBN
978-3-0348-9051-9
DOI
10.1007/978-3-0348-9051-9
Softcover ISBN
978-3-7643-2835-1
Edition Number
1
Number of Pages
160
Number of Illustrations and Tables
3 b/w illustrations
Topics