Variations on a Theme of Euler

Quadratic Forms, Elliptic Curves, and Hopf Maps

Takashi Ono (Autor)

Buch | Softcover

347 Seiten

2010 | Softcover reprint of hardcover 1st ed. 1995
Springer-Verlag New York Inc.
978-1-4419-3241-9 (ISBN)

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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.

I. Introduction.- 1. Quadratic Forms.- 2. Algebraic Varieties.- 3. Plane Algebraic Curves.- 4. Space Elliptic Curves.- 5. Quadratic Spherical Maps.- 6. Hurwitz Problem.- 7. Arithmetic of Quadratic Maps.- Answers and Hints to Selected Exercises.- A1.4. Chronological Table and Record.- References.- Appendix 2. A Short Survey of Subsequent Research on Congruent Numbers.- Masanari Kida.- References.

Erscheint lt. Verlag	6.12.2010
Reihe/Serie	University Series in Mathematics
Zusatzinfo	XI, 347 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	152 x 229 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
ISBN-10	1-4419-3241-0 / 1441932410
ISBN-13	978-1-4419-3241-9 / 9781441932419
Zustand	Neuware