Ebook: Variations on a Theme of Euler: Quadratic Forms, Elliptic Curves, and Hopf Maps

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.

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Introduction. Quadratic Forms. Algebraic Varieties. Plane Algebraic Curves. Space Elliptic Curves. Quadratic Spherical Maps. Hurwitz Problem. Arithmetic of Quadratic Maps. Answers and Hints to Selected Exercises. Appendixes. Index.

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Introduction. Quadratic Forms. Algebraic Varieties. Plane Algebraic Curves. Space Elliptic Curves. Quadratic Spherical Maps. Hurwitz Problem. Arithmetic of Quadratic Maps. Answers and Hints to Selected Exercises. Appendixes. Index.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-14
Quadratic Forms....Pages 15-38
Algebraic Varieties....Pages 39-82
Plane Algebraic Curves....Pages 83-121
Space Elliptic Curves....Pages 123-164
Quadratic Spherical Maps....Pages 165-198
Hurwitz Problem....Pages 199-256
Arithmetic of Quadratic Maps....Pages 257-303
Answers and Hints to Selected Exercises....Pages 305-320
Back Matter....Pages 321-347

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Introduction. Quadratic Forms. Algebraic Varieties. Plane Algebraic Curves. Space Elliptic Curves. Quadratic Spherical Maps. Hurwitz Problem. Arithmetic of Quadratic Maps. Answers and Hints to Selected Exercises. Appendixes. Index.
Content:
Front Matter....Pages i-xi
Introduction....Pages 1-14
Quadratic Forms....Pages 15-38
Algebraic Varieties....Pages 39-82
Plane Algebraic Curves....Pages 83-121
Space Elliptic Curves....Pages 123-164
Quadratic Spherical Maps....Pages 165-198
Hurwitz Problem....Pages 199-256
Arithmetic of Quadratic Maps....Pages 257-303
Answers and Hints to Selected Exercises....Pages 305-320
Back Matter....Pages 321-347
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