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Tables of Contents for Elliptic Curves
Chapter/Section Title
Page #
Page Count
List of Figures
x
List of Tables
x
Preface
xi
Standard Notation
xv
Overview
3
398
Curves in Projective Space
Projective Space
19
5
Curves and Tangents
24
8
Flexes
32
8
Application to Cubics
40
4
Bezout's Theorem and Resultants
44
6
Cubic Curves in Weierstrass Form
Examples
50
6
Weierstrass Form, Discriminant, j-invariant
56
11
Group Law
67
7
Computations with the Group Law
74
3
Singular Points
77
3
Mordell's Theorem
Descent
80
5
Condition for Divisibility by 2
85
3
E(Q)/2E(Q), Special Case
88
4
E(Q)/2E(Q), General Case
92
3
Height and Mordell's Theorem
95
7
Geometric Formula for Rank
102
5
Upper Bound on the Rank
107
8
Construction of Points in E(Q)
115
7
Appendix on Algebraic Number Theory
122
8
Torsion Subgroup of E(Q)
Overview
130
4
Reduction Modulo p
134
3
p-adic Filtration
137
7
Lutz-Nagell Theorem
144
1
Construction of Curves with Prescribed Torsion
145
3
Torsion Groups for Special Curves
148
3
Complex Points
Overview
151
1
Elliptic Functions
152
1
Weierstrass & Function
153
9
Effect on Addition
162
3
Overview of Inversion Problem
165
1
Analytic Continuation
166
3
Riemann Surface of the Integrand
169
5
An Elliptic Integral
174
9
Computability of the Correspondence
183
6
Dirichlet's Theorem
Motivation
189
3
Dirichlet Series and Euler Products
192
7
Fourier Analysis on Finite Abelian Groups
199
2
Proof of Dirichlet's Theorem
201
6
Analytic Properties of Dirichlet L Functions
207
14
Modular Forms for SL(2, Z)
Overview
221
1
Definitions and Examples
222
5
Geometry of the q Expansion
227
4
Dimensions of Spaces of Modular Forms
231
7
L Function of a Cusp Form
238
3
Petersson Inner Product
241
1
Hecke Operators
242
8
Interaction with Petersson Inner Product
250
6
Modular Forms for Hecke Subgroups
Hecke Subgroups
256
5
Modular and Cusp Forms
261
4
Examples of Modular Forms
265
2
L Function of a Cusp Form
267
4
Dimensions of Spaces of Cusp Forms
271
2
Hecke Operators
273
10
Oldforms and Newforms
283
7
L Function of an Elliptic Curve
Global Minimal Weierstrass Equations
290
4
Zeta Functions and L Functions
294
2
Hasse's Theorem
296
6
Eichler-Shimura Theory
Overview
302
9
Riemann surface X0(N)
311
1
Meromorphic Differentials
312
4
Properties of Compact Riemann Surfaces
316
4
Hecke Operators on Integral Homology
320
13
Modular Functions j(τ)
333
8
Varieties and Curves
341
8
Canonical Model of X0(N)
349
10
Abstract Elliptic Curves and Isogenies
359
8
Abelian Varieties and Jacobian Variety
367
7
Elliptic Curves Constructed from S2(Γ0(N))
374
9
Match of L Functions
383
3
Taniyama-Weil Conjecture
Relationships among conjectures
386
6
Strong Weil Curves and Twists
392
2
Computations of Equations of Weil Curves
394
3
Connection with Fermat's Last Theorem
397
4
Notes
401
8
References
409
10
Index of Notation
419
4
Index
423