# Elliptic Curves

Princeton University Press, 1992 - Mathematics - 427 pages

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

### 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.