文档介绍：Elliptic curve handbook
Ian Connell
Course Notes
McGill University
August 1996
Foreword
The first version of this handbook was a set of notes of about 100 pages handed out to the class of an intro-
ductory course on elliptic curves given in the 1990 fall semester at McGill University in Montreal. Since then
I have added to the notes, holding to the principle: If I look up a certain topic a year from now I want all
the details right at hand, not in an “exercise”, so if I’ve forgotten something I won’t waste time. Thus there
is much that an ordinary text would either condense, or relegate to an exercise. But at the same time I have
maintained a solid mathematical style with the thought of sharing the handbook.
Montreal,
August, 1996.
Contents
1 Introduction to Elliptic Curves. 1
j
The a
b
c ’s and ..................................... 1
Quartic to Weierstrass ....................................... 4
Projective coordinates. ...................................... 9
Cubic to Weierstrass: Nagell’s algorithm . . . . . . . . . . . . . . . . . . . .......... 12
Example 1: Selmer curves . . . . . . . . . . . . . . . . . . . . . . . . . . ...... 13
Example 2: Desboves curves . . . . ........................... 16
Example 3: Intersection of quadric surfaces . . . . . . . . . . . . . . . . . . . . . . . 17
Singular points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Example: No E has or ........................... 23
Z
Affine coord. ring, function field, generic points . . . . . . . . . . . . . . . . . . . . . . . . . 24
The group law: nonsingular case . . . . . . . . . . . . . . . . . . . . . . . .......... 25
Halving points ...................................... 30
The division polynomials . ................................ 34
Remarks on the group of division points . . . . . . . . . . . . . . . .