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Variations on a Theme of Euler

Quadratic Forms, Elliptic Curves, and Hopf Maps

  • Book
  • © 1994

Overview

Authors:
  1. Takashi Ono

    1. The Johns Hopkins University, Baltimore, USA

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Part of the book series: University Series in Mathematics (USMA)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xi
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  2. Introduction

    • Takashi Ono
    Pages 1-14

  3. Quadratic Forms

    • Takashi Ono
    Pages 15-38

  4. Algebraic Varieties

    • Takashi Ono
    Pages 39-82

  5. Plane Algebraic Curves

    • Takashi Ono
    Pages 83-121

  6. Space Elliptic Curves

    • Takashi Ono
    Pages 123-164

  7. Quadratic Spherical Maps

    • Takashi Ono
    Pages 165-198

  8. Hurwitz Problem

    • Takashi Ono
    Pages 199-256

  9. Arithmetic of Quadratic Maps

    • Takashi Ono
    Pages 257-303

  10. Answers and Hints to Selected Exercises

    • Takashi Ono
    Pages 305-320

  11. Back Matter

    Pages 321-347
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Keywords

About this book

The first six chapters and Appendix 1 of this book appeared in Japanese in a book of the same title 15years aga (Jikkyo, Tokyo, 1980).At the request of some people who do not wish to learn Japanese, I decided to rewrite my old work in English. This time, I added a chapter on the arithmetic of quadratic maps (Chapter 7) and Appendix 2, A Short Survey of Subsequent Research on Congruent Numbers, by M. Kida. Some 20 years ago, while rifling through the pages of Selecta Heinz Hopj (Springer, 1964), I noticed a system of three quadratic forms in four variables with coefficientsin Z that yields the map of the 3-sphere to the 2-sphere with the Hopf invariant r =1 (cf. Selecta, p. 52). Immediately I feit that one aspect of classical and modern number theory, including quadratic forms (Pythagoras, Fermat, Euler, and Gauss) and space elliptic curves as intersection of quadratic surfaces (Fibonacci, Fermat, and Euler), could be considered as the number theory of quadratic maps-especially of those maps sending the n-sphere to the m-sphere, i.e., the generalized Hopf maps. Having these in mind, I deliveredseverallectures at The Johns Hopkins University (Topics in Number Theory, 1973-1974, 1975-1976, 1978-1979, and 1979-1980). These lectures necessarily contained the following three basic areas of mathematics: v vi Preface Theta Simple Functions Aigebras Elliptic Curves Number Theory Figure P.l.

Reviews

From a review of the Japanese-language edition:
`A beautifully written book...The statement of the problem is very clear-that is, [the author] claims that one aspect of classical and modern number theory can be considered as the number theory of Hopf maps-and then he solves this problem....skillfully and perspectively organized...This book will be a good introductory textbook...There has never been a textbook similar to this....I highly recommend this book.'
Michio Kuga, Professor, late of State University of New York at Stony Brook

Authors and Affiliations

  • The Johns Hopkins University, Baltimore, USA

    Takashi Ono

Bibliographic Information

  • Book Title: Variations on a Theme of Euler

  • Book Subtitle: Quadratic Forms, Elliptic Curves, and Hopf Maps

  • Authors: Takashi Ono

  • Series Title: University Series in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4757-2326-7

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Takashi Ono 1994

  • Hardcover ISBN: 978-0-306-44789-1Published: 30 November 1994

  • Softcover ISBN: 978-1-4419-3241-9Published: 06 December 2010

  • eBook ISBN: 978-1-4757-2326-7Published: 09 March 2013

  • Edition Number: 1

  • Number of Pages: XI, 347

  • Topics: Functional Analysis, Operator Theory

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