Lectures on Elliptic Curves

✍ Scribed by J. W. S. Cassels

• Publisher: Cambridge University Press
• Year: 1991
• Tongue: English
• Leaves: 142
• Series: London Mathematical Society Student Texts
• Category: Library

✦ Synopsis

The study of special cases of elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centers of research in number theory. This book, addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Wei finite basis theorem, points of finite order (Nagell-Lutz), etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the "Riemann hypothesis for function fields") and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch. Many examples and exercises are included for the reader, and those new to elliptic curves, whether they are graduate students or specialists from other fields, will find this a valuable introduction.

✦ Table of Contents

Contents......Page 4
0 Introduction......Page 6
1 Curves of genus 0. Introduction......Page 8
2 p-adic numbers......Page 11
3 The local-global principle for conics......Page 18
4 Geometry of numbers......Page 22
5 Local-global principle. Conclusion of proof......Page 25
6 Cubic curves......Page 28
7 Non-singular cubics. The group law......Page 32
8 Elliptic curves. Canonical Form......Page 37
9 Degenerate laws......Page 44
10 Reduction......Page 47
11 The p-adic case......Page 51
12 Global torsion......Page 55
13 Finite Basis Theorem. Strategy and comments......Page 59
14 A 2-isogeny......Page 63
15 The weak finite basis theorem......Page 71
16 Remedial mathematics. Resultants......Page 80
17 Heights. Finite Basis Theorem......Page 83
18 Local-global for genus 1......Page 90
19 Elements of Galois cohomology......Page 94
20 Construction of the jacobian......Page 97
21 Some abstract nonsense......Page 103
22 Principal homogeneous spaces and Galois cohomology......Page 109
23 The Tate-Shafarevich group......Page 113
24 The endomorphism ring......Page 117
25 Points over finite fields......Page 123
26 Factorizing using elliptic curves......Page 129
Formulary......Page 135
Further Reading......Page 140
INDEX......Page 141