Introduction to elliptic curves. (Introduktion til elliptiske kurver).

Book: J. H. Silverman, J. Tate: Rational Points on Elliptic Curves.

Undergraduate Texts in Mathematics, Springer 1992.

Download a list of Errata here.

 

Course plan: Chapters I-IV + App. A.1,2 + misc. notes. Extra: Chapter VI.

 

23.05:  A little about Fermat's last theorem.

I've witten some simple notes about this. Unfortunately, they're only in Danish, but at least they contain references to other literature.

The notes are there as a possible way to start with this topic. In the lectures we'll only discuss some of the 'big' ideas.

Download here: pdf

20.05:  VI.4: Complex multiplication (continued).

VI.5: Abelian extensions of Q(i).

 

16.05:  Holiday (Store bededag).

13.05:  VI.3: Galois representations.

VI.4: Complex multiplication.

 

09.05:  VI.1,2: Abelian extensions of number fields. Algebraic points on elliptic curves.

06.05:  IV. 4: Lenstra factorization (continued).

Begin Chap. VI: Complex multiplication.

 

02.05:  A little about cryptography + IV. 4: Lenstra factorization.

29.04:  IV.1,3:  Elliptic curves over finite fields, and reduction modulo p (continued).

 

25.04:  Mordell's theorem: The 'irreducible case': End of the argument.

IV.1,3:  Elliptic curves over finite fields, and reduction modulo p.

 

18.04 + 22.04:  Easter vacation.

15.04:  Mordell's theorem: The 'irreducible case' (continued).

 

11.04:  III.6: Examples (continued).

Mordell's theorem: The 'irreducible case'. Notes:  pdf

08.04: III.6: Examples (continued).

 

04.04:  Mordell's theorem: III.5 (continued). III.6: Examples.

01.04:  Mordell's theorem: III.4 (continued). III.5.

 

28.03:  Mordell's theorem: III.4.

25.03:  Mordell's theorem: III.2,3.

 

21.03:  Begin of proof of Mordell's theorem: III.1,2.

18.03:  Torsion on some special classes of elliptic curves (continued).

 

14.03:  II.5: Additional remarks concerning torsion points. Examples of torsion groups:  pdf

Torsion on some special classes of elliptic curves. Notes:  pdf

11.03: II.4: Torsion points have integer coefficients (continued):  pdf

Note: There's an error on p. 52 of the book: The statement 'If t_1=t_2 then P_1= -P_2, ...' is wrong. This whole argument needs to be rewritten.

That's why we're using a slightly different argument (see the notes).

 

07.03: Exercise 2.11, p. 61: Stronger form of lemma in II.3:  pdf

II.4: Torsion points have integer coefficients.

04.03: II.1: Points of finite order (continued).

II.2: Elliptic curves over the complex numbers.

II.3: The discriminant and begin of proof of Nagell-Lutz.

 

28.02: Degenerate group laws (continued).

II.1: Points of finite order.

25.02: I.4: The group law on elliptic curves. Explicit formulas.

III.7 + Exercises 3.14, 3.15 (p.106): Degenerate group laws. We will use the following notes:

Degenerate group laws:  pdf

 

21.02: I.2: The group law: General discussion. (Continued).

I.4: The group law on elliptic curves. Explicit formulas.

If you want to have a closer look at the use of Bezout's theorem, you can have a look at these notes by Helena Verrill.

If you're interested: Here are addition formulas for elliptic curves with general Weierstrass equations:  pdf

18.02: Exercise 1.13(a), p. 35.

I.2: The group law: General discussion.

 

14.02: App. A.: non-singular points.

I.3: Weierstrass equations and non-singular points, definition of elliptic curves. Rational maps.

An elliptic curve does not contain a line (pdf).

11.02: App. A: Proj. coordinates and curves.

 

07.02: I.1: Rational points on conics.

04.02: Introduction: Background and motivation. Overview of first part of the course.