UID:
almahu_9947367851302882
Format:
1 online resource (363 p.)
ISBN:
1-281-79736-7
,
9786611797362
,
0-08-087165-8
Series Statement:
North-Holland mathematics studies ; 54
Content:
Real Elliptic Curves
Note:
Includes index.
,
Front Cover; Real Elliptic Curves; Copyright Page; Contents; Preface; Chapter 0. Introduction; 0.10 Research; 0.20 Historical and bibliographic notes; 0.30 Prerequisites and exposition; 0.40 Indexing; 0.50 Acknowledgments; PART I: ELLIPTIC INTEGRALS; Chapter 1. Examples of elliptic integrals; 1.10 Some integrals associated with an ellipse; 1.20 The simple pendulum; 1.30 The lemniscate integral; Chapter 2. Some addition theorems; 2.10 Examples of addition theorems; 2.20 The arcsine integral; 2.30 Fagnano's theorem; 2.40 Euler's addition theorem; 2.50 Other addition theorems
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Chapter 3. Development of some discoveries made prior to 18273.10 Linear fractional substitutions; 3.20 Generalized Legendre form; 3.30 Some of Legendre's work; 3.40 Gauss's arithmetic - geometric mean; PART II: ELLIPTIC FUNCTIONS; Chapter 4. Inverting the integral; 4.10 Abel's Recherches; 4.20 Jacobi's Fundamenta Nova; 4.30 Gauss's work on elliptic functions; 4.40 The question of priority; Chapter 5. Theta functions; 5.10 Origins; 5.20 Definitions; 5.30 Properties of theta functions; Chapter 6. The introduction of analytic function theory; 6.10 Early history; 6.20 Lattices in C
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6.30 Fields of elliptic functions6.40 Some applications of Cauchy's and Liouville's work; 6.50 Theta functions treated with analytic function theory; Chapter 7. Weierstrass's work on elliptic functions; 7.10 Introduction; 7.20 Weierstrass's Vorlesungen; 7.30 Weierstrass's theory reordered; 7.40 Representation of doubly periodic functions; 7.50 An addition theory for ß; 7.60 A relation between Weierstrass's s function and 01; Chapter 8. Riemann surfaces; 8.10 Introduction; 8.20 Definitions; 8.30 Some properties of the Riemann sphere; 8.40 Some properties of C/L; 8.50 Surfaces of genus one
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8.60 The divisor class groupChapter 9. The elliptic modular function; 9.10 Introduction; 9.20 Definition and elementary properties; 9.30 Reflection of J across ? c1CD; 9.40 Modular functions; 9.50 An inversion problem; Chapter 10. Algebraic function fields; 10.10 Definitions and introduction; 10.20 Extensions; 10.30 The Riemann surface of complex algebraic function field; 10.40 A theorem of coequivalence; 10.50 The Riemann-Roch theorem; PART III: REAL ELLIPTIC CURVES; Chapter 11. Real algebraic function fields and compact Klein surfaces; 11.10 Real algebraic function fields
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11.20 Klein surfaces11.30 Symmetric Riemann surfaces; 11.40 A theorem of coequivalence; Chapter 12. The species and geometric moduli of a real elliptic curve; 12.10 The extended modular; 12.20 Species; 12.30 Geometric moduli; 12.40 Real lattices; Chapter 13. Automorphisms of real elliptic curves; 13.10 The automorphism group of ? s,t; 13.20 The orbit subspaces of ? s,t; 13.30 Orthogonal trajectories; Chapter 14. From species and geometric moduli to defining equations; 14.10 Introduction; 14.20 Species 2 and 1; 14.30 Species 0; 14.40 Other quartic defining equations
,
Chapter 15. The divisor class group of y s, t
,
English
Additional Edition:
ISBN 0-444-86233-1
Language:
English
