Lectures on elliptic curves

✍ Scribed by Cassels J.W.S.

• Publisher: CUP
• Year: 1991
• Tongue: English
• Leaves: 144
• Series: London Mathematical Society Student Texts 24
• 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

Cover......Page 1
Title......Page 2
Contents......Page 6
0 Introduction......Page 8
1 Curves of genus 0. Introduction......Page 10
2 p-adic numbers......Page 13
3 The local-global principle for conics......Page 20
4 Geometry of numbers......Page 24
5 Local-global principle. Conclusion of proof......Page 27
6 Cubic curves......Page 30
7 Non-singular cubics. The group law......Page 34
8 Elliptic curves. Canonical Form......Page 39
9 Degenerate laws......Page 46
10 Reduction......Page 49
11 The p-adic case......Page 53
12 Global torsion......Page 57
13 Finite Basis Theorem. Strategy and comments......Page 61
14 A 2-isogeny......Page 65
15 The weak finite basis theorem......Page 73
16 Remedial mathematics. Resultants......Page 82
17 Heights. Finite Basis Theorem......Page 85
18 Local-global for genus 1......Page 92
19 Elements of Galois cohomology......Page 96
20 Construction of the jacobian......Page 99
21 Some abstract nonsense......Page 105
22 Principal homogeneous spaces and Galois cohomology......Page 111
23 The Tate-Shafarevich group......Page 115
24 The endomorphism ring......Page 119
25 Points over finite fields......Page 125
26 Factorizing using elliptic curves......Page 131
Formulary......Page 137
Further Reading......Page 142
INDEX......Page 143