2006.04.07: I will give an introductory lecture about more advanced topics in the theory of elliptic curves, specifically about the Taniyama-Shimura conjecture and its ties to the Langlands program which is the big contemporary challenge in algebraic number theory.
Some notes about this are these (in Danish):
We can obviously only give the briefest hints about this and only talk about a smaller subset of the topics dealt with in the notes.
2006.04.04: End of proof of the theorem of Billing and Mahler.
2006.03.31: We finish the discussion of Mordell's theorem in the `irreducible case'.
After that I will start explaining the proof of the theorem og Billing and
Mahler (from 1940) that states that no elliptic curve over Q has
a rational point of order 11. We will use this note:
2006.03.28: We will discuss Mordell's theorem in the
`irreducible case', i.e., the case where the curve is given by a Weierstrass
equation $y^2=f(x)$ with $f$ irreducible. For this case we will need some
algebraic number theory which I will review. We use this note:
2006.03.24: We will discuss torsion on elliptic curves of
the special forms $y^2=x^3+Ax$ or $y^2=x^3+B$, as well as applications of this
analysis to the question of representing numbers as sums of 2 cubes. We use this
2006.03.21: We will discuss how to attempt computing the rank of an elliptic curve with a point of order 2. The method is not guaranteed to work but does so sometimes. This is III.6.
2006.03.17: We continue with the final part of the
proof of weak Mordell (for the case that the curve has a rational point of order
2). This is III.5.
Mordell's theorem is a finiteness theorem (i.e., a statement that something is finite). This finiteness actually comes from 2 sources. One source is the theory of heights, specifically Lemma 1, p. 64. The other source is of an arithmetic nature: Analyzing the arguments of III.5 you will be able to see that the second point that really makes things work is the fact that any integer can be written as plus-minus a finite number of primes ... .
2006.03.14: We continue the proof of weak Mordell for the case that the elliptic curve has a rational point of order 2. Some remarks about the construction of the `2-isogeny' that plays a prominent role in the proof: One way of constructing this is to look at the function field attached to the elliptic curve.
2006.03.10: End of proofs of height lemmas. Begin proof of 'Weak Mordell' in case the curve has a rational point of order 2. This is III.4.
2006.03.07: Today we will first discuss the strategy for proving that 'Weak Mordell implies Mordell': This is the proof that '(Weak Mordell + height lemmas) implies Mordell'.
After that we begin proof of the height lemmas, Lemma 2 in III.2, and Lemma 3 in III.3.
2006.03.03: After proving the theorem of Nagell-Lutz, I find it natural to discuss reduction mod p and what happens with torsion points under reduction mod p. This is Chap. IV, 3. I will prove the `reduction modulo p theorem', p. 123, in the slightly more general form as given in exercise 4.12, p. 143.
The content of the lecture is in the following note:
After discussing reduction modulo p we begin the study of heights in Chap. III, 1-3.
2006.02.28: We finish the proof of the Nagell-Lutz theorem and give some info on subsequent developments.
2006.02.24: We continue with II.4 which is the hard
part of the theorem of Nagell-Lutz, i.e., the statement that rational affine
torsion points on curves of form y^2 = x^3 + ax^2 + bx + c with a,b,c integers
have integer coordinates.
Unfortunately, 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/repaired. This is done in the following note:
2006.02.21: We begin the discussion of the theorem of
Nagell-Lutz. This is II.3,4.
I will also give the strong form of the Lemma in II.3, cf. exercise 2.11, p. 61 of the book. For this we need the following note:
2006.02.17: Some background info on the group law, and a little about elliptic curves over the complex numbers (cf. chap. II.2). Then we discussed point of order 2 and 3 on elliptic curves over Q (or the algebraic closure of Q, or C); cf. chap. II.1.
2006.02.14: The group law. Explicit formulas.
If you want to see the addition formulas for the case of a completely general Weierstrass equation, they are here:
It's a good exercise to verify these more general formulas. If you do this exercise, you can model the computations on what I'll show you at the lectures.
2006.02.10: Simple Weierstrass equations, irreducibility of elliptic curves, intersection between projective lines and cubic curves.
2006.02.07: Diophantine equations, projective coordinates and curves, smooth curves, elliptic curves, Weierstrass equations.
If you take notes at the first 3 lectures, you don't need to read the book yet, - except for the discussion of conics in Chap. 1: What I'm doing corresponds to Chap. 1 + some of the beginning of Appendix A. The point is that I'm avoiding Bezout's theorem and discussing things using projective coordinates all the way.
We will need these notes: