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Elliptic Curves. (MN-40), Volume 40 : Mathematical Notes - Anthony W. Knapp

Elliptic Curves. (MN-40), Volume 40

Mathematical Notes

Paperback Published: 25th October 1992
ISBN: 9780691085593
Number Of Pages: 448

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An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects--elliptic curves and modular forms--come together in Eichler-Shimura theory, which constructs elliptic curves out of modular forms of a special kind. The converse, that all rational elliptic curves arise this way, is called the Taniyama-Weil Conjecture and is known to imply Fermat's Last Theorem.

Elliptic curves and the modeular forms in the Eichler- Shimura theory both have associated L functions, and it is a consequence of the theory that the two kinds of L functions match. The theory covered by Anthony Knapp in this book is, therefore, a window into a broad expanse of mathematics--including class field theory, arithmetic algebraic geometry, and group representations--in which the concidence of L functions relates analysis and algebra in the most fundamental ways.

Developing, with many examples, the elementary theory of elliptic curves, the book goes on to the subject of modular forms and the first connections with elliptic curves. The last two chapters concern Eichler-Shimura theory, which establishes a much deeper relationship between the two subjects. No other book in print treats the basic theory of elliptic curves with only undergraduate mathematics, and no other explains Eichler-Shimura theory in such an accessible manner.

List of Figures
List of Tables
Preface
Standard Notation
Overviewp. 3
Curves in Projective Space
Projective Spacep. 19
Curves and Tangentsp. 24
Flexesp. 32
Application to Cubicsp. 40
Bezout's Theorem and Resultantsp. 44
Cubic Curves in Weierstrass Form
Examplesp. 50
Weierstrass Form, Discriminant, j-invariantp. 56
Group Lawp. 67
Computations with the Group Lawp. 74
Singular Pointsp. 77
Mordell's Theorem
Descentp. 80
Condition for Divisibility by 2p. 85
E(Q)/2E(Q), Special Casep. 88
E(Q)/2E(Q), General Casep. 92
Height and Mordell's Theoremp. 95
Geometric Formula for Rankp. 102
Upper Bound on the Rankp. 107
Construction of Points in E(Q)p. 115
Appendix on Algebraic Number Theoryp. 122
Torsion Subgroup of E(Q)
Overviewp. 130
Reduction Modulo pp. 134
p-adic Filtrationp. 137
Lutz-Nagell Theoremp. 144
Construction of Curves with Prescribed Torsionp. 145
Torsion Groups for Special Curvesp. 148
Complex Points
Overviewp. 151
Elliptic Functionsp. 152
Weierstrass p Functionp. 153
Effect on Additionp. 162
Overview of Inversion Problemp. 165
Analytic Continuationp. 166
Riemann Surface of the Integrandp. 169
An Elliptic Integralp. 174
Computability of the Correspondencep. 183
Dirichlet's Theorem
Motivationp. 189
Dirichlet Series and Euler Productsp. 192
Fourier Analysis on Finite Abelian Groupsp. 199
Proof of Dirichlet's Theoremp. 201
Analytic Properties of Dirichlet L Functionsp. 207
Modular Forms for SL(2,Z)
Overviewp. 221
Definitions and Examplesp. 222
Geometry of the q Expansionp. 227
Dimensions of Spaces of Modular Formsp. 231
L Function of a Cusp Formp. 238
Petersson Inner Productp. 241
Hecke Operatorsp. 242
Interaction with Petersson Inner Productp. 250
Modular Forms for Hecke Subgroups
Hecke Subgroupsp. 256
Modular and Cusp Formsp. 261
Examples of Modular Formsp. 265
L Function of a Cusp Formp. 267
Dimensions of Spaces of Cusp Formsp. 271
Hecke Operatorsp. 273
Oldforms and Newformsp. 283
L Function of an Elliptic Curve
Global Minimal Weierstrass Equationsp. 290
Zeta Functions and L Functionsp. 294
Hasse's Theoremp. 296
Eichler-Shimura Theory
Overviewp. 302
Riemann surface X[subscript 0](N)p. 311
Meromorphic Differentialsp. 312
Properties of Compact Riemann Surfacesp. 316
Hecke Operators on Integral Homologyp. 320
Modular Function j([tau])p. 333
Varieties and Curvesp. 341
Canonical Model of X[subscript 0](N)p. 349
Abstract Elliptic Curves and Isogeniesp. 359
Abelian Varieties and Jacobian Varietyp. 367
Elliptic Curves Constructed from S[subscript 2]([Gamma](N))p. 374
Match of L Functionsp. 383
Taniyama-Weil Conjecture
Relationships among Conjecturesp. 386
Strong Weil Curves and Twistsp. 392
Computations of Equations of Weil Curvesp. 394
Connection with Fermat's Last Theoremp. 397
Notesp. 401
Referencesp. 409
Index of Notationp. 419
Indexp. 423
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691085593
ISBN-10: 0691085595
Series: Mathematical Notes
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 448
Published: 25th October 1992
Country of Publication: US
Dimensions (cm): 22.99 x 15.29  x 2.77
Weight (kg): 0.73