Elliptic Curves

Front Cover
Princeton University Press, 1992 - Mathematics - 427 pages
0 Reviews

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.

 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Overview
3
Curves in Projective Space
19
Cubic Curves in Weierstrass Form
50
Mordells Theorem
80
Torsion Subgroup of EQ
130
Complex Points 1 Overview
151
Elliptic Functions
152
Weierstrass p Function
153
Geometry of the q Expansion
227
Dimensions of Spaces of Modular Forms
231
Function of a Cusp Form
238
Petersson Inner Product
241
Hecke Operators
242
Interaction with Petersson Inner Product
250
Modular Forms for Hecke Subgroups 1 Hecke Subgroups
256
Modular and Cusp Forms
261

Effect on Addition
162
Overview of Inversion Problem
165
Analytic Continuation
166
Riemann Surface of the Integrand
169
An Elliptic Integral
174
Computability of the Correspondence
183
Dirichlets Theorem 1 Motivation
189
Dirichlet Series and Euler Products
192
Fourier Analysis on Finite Abelian Groups
199
Proof of Dirichlets Theorem
201
Analytic Properties of Dirichlet L Functions
207
Modular Forms for 512 Z 1 Overview
221
Definitions and Examples
222
Examples of Modular Forms
265
Function of a Cusp Form
267
Dimensions of Spaces of Cusp Forms
271
Hecke Operators
273
Oldforms and Newforms
283
Function of an Elliptic Curve 1 Global Minimal Weierstrass Equations
290
Zeta Functions and L Functions
294
Hasses Theorem
296
TaniyamaWeil Conjecture
386
Notes
401
References
409
Index of Notation
419
Copyright

Common terms and phrases

Popular passages

Page 409 - On thé trace formula for Hecke operators, Acta. Math. 132 (1974), 245-281. [87] M. Eichler, The basis problem for modular forms and the traces of the Hecke operators, Modular functions of one variable. I, Lecture Notes in Math. 320 (1973), 75-151. [88] D. Mumford, A remark on Mahler's compactness theorem, Proc. Amer. Math. Soc. 28 (1971), 289-294. MR 43 #2157. [89] S. Lang, Elliptic functions, Addison-Wesley, Reading, Mass., 1973. MR 53 # 13117.
Page 416 - J. Velu, Courbes elliptiques sur Q ayant bonne réduction en dehors de (11} , CR Acad.

References to this book

All Book Search results »

Bibliographic information