**
Lectures:**

**2008.06.27:** The final lectures will be by Gerhard Frey,
Univ. Duisburg-Essen, who gives a mini-course with the title `Duality in
Arithmetic, Brauer groups and Discrete Logarithms'. *Notice that this will be
in Aud. 10 (and not 9).*

*The first lecture by Frey will be on Thursday, 26 June, 13.15-14.00
in Aud. 8.*

I hope you have time and interest to come also to the first lecture. (Unfortunately, it was not possible to have Frey's first lecture on Tuesday 24 June).

**2008.06.24:** Because of a collision with an exam in the
course Analysis 3, I decided to cancel the exercises today and instead have the
lectures in the exercise hours, i.e., in room A110, 13.15-16.00.

I will give a little insight into contemporary research connected with the theory of elliptic curves, in particular the Taniyama-Shimura conjecture (now a theorem), the connection to Fermat's last theorem, and a little about Galois representations and the Langlands program.

I have written some notes about this - unfortunately only in Danish. They can be dowmnloaded here:

http://www.math.ku.dk/~kiming/papers/1999_fermat2/1999_fermat2_.pdf

We will certainly not have time to talk about all of this material, so this is just in case you are interested in reading something on your own.

**2008.06.20:** We finish the proof of the theorem of Billing
and Mahler.

**2008.06.17: **We finish the discussion of torsion on the
special families of elliptic curves and applications as in chap. 9 of the
notes.

After this we will look at a final application of algebraic number theory to the theory of elliptic curves, namely the theorem of Billing and Mahler that an elliptic curve over Q cannot have a rational point of order 11. Of course, this theorem is implied by Mazur's big theorem, but the point is that it is accessible also to us.

Perhaps we will not have time to present all details of the proof but only the overall structure + some details.

In any case, details are to be found in chap. 12 of the (updated) supplementary notes.

**2008.06.06:** We finish the discussion of Mordell's
theorem in the `irreducible case'.

After this 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. This is chap. 9 of the supplementary notes.

Today we started the proof of the theorem on torsion on elliptic curves of form $y^2=x^3+B$.

**2008.06.03:** We continue the discussion of Mordell's
theorem in the `irreducible case'.

Again, it is instructive to reflect on the sources of `finiteness' in Mordell's theorem. In the `irreducible' case, one could say that the sources are these: Every non-zero ideal in the ring of integers is a product of finitely many prime ideals, the class group is finite, and the unit group is finitely generated.

Today we got until and including the proof of the claim within the proof in section 10.3.

**2008.05.30:** We finish the discussion of 4 squares in
arithmetic progression.

After that we begin the proof of Mordell's theorem in the `irreducible case'. This is chap. 10 of the supplementary notes. (Note that section 10.2 which is a summary of basic results from algebraic number theory has been upgraded a bit since the beginning of the course).

Today we got until and including Lemma 4 in chap. 10.

**2008.05.27:** We finish III.5 with the
proof that the map alpha in the proposition pp. 85-86 is a
homomorphism.

There is no known *proven* algorithm for determining a set of
generators for the group of rational points on an elliptic curve over Q, or even
for just determining its rank.

One does know however, that the truth of the so-called Birch- and Swinnerton-Dyer conjecture gives such an algorithm. In general, one can of course work on the basis of this conjecture and get conditional results.

Today 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 in III.6.

After this, we begin to discuss an application of the method: There are no non-trivial examples of 4 squares of rational numbers in arithmetic progression:

**2008.05.23:** We discuss the proposition on page 79 in
III.4 as well as the proof of it.

After that, 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, in the present case the number of generators for the group of
rational points. 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 ... .

We discussed everything in III.5 today, except that the map alpha in the proposition pp. 85-86 is a homomorphism.

**2008.05.20:** We continue the proof of Mordell's theorem.
First we prove Lemma 3 in III.3, and then we begin proof of 'Weak
Mordell' for the case that the curve has a rational point of order 2.
This is in III.4.

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. I will say a little about this.

We discussed everything until, but not including, the proposition on page 79 in section III.4.

**2008.05.16:** We introduce the height concept and begin
discussing the strategy for proving that 'Weak Mordell implies Mordell': This is
the proof that '(Weak Mordell + height lemmas) implies Mordell'. This is in
Chap. III.1.

After that we begin proof of the height lemmas, Lemma 2 in III.2, and Lemma 3 in III.3.

Today we finished the proof of Lemma 2.

**2008.05.13:** We illustrate the theorem of Nagell-Lutz
with some simple examples.

Having proved the theorem of Nagell-Lutz, I find it natural to discuss elliptic curves over finite fields and the theory of reduction modulo p that has applications to the discussion of torsion points on elliptic curves over Q.

We first give some background on elliptic curves over finite fields. This is Chap. IV.1. We also mention the theorem of Hasse, and one of the big recent developments, namely the proof of the so-called Sato-Tate conjecture in 2006 by L. Clozel, M. Harris, and R. Taylor.

There is an excellent popular article in *Nature* on Sato-Tate
written by Barry Mazur. From within the institute you should have access to the
online version of the article:

http://www.nature.com/nature/journal/v443/n7107/full/443038a.html

After this we discuss reduction modulo p. This is Chap. IV.3. We will prove the `reduction modulo p theorem', p. 123, in the slightly more general form as given in exercise 4.12, p. 143. The proof is in chap. 7 of my notes. We illustrate with some numerical examples.

**2008.05.09:** II.4 continued + end of proof of
Nagell-Lutz.

Unfortunately, there's a slight problem with the argument on p. 52 of the book: The statement `If t_1=t_2 then P_1=-P_2, ...' is wrong. The argument is repaired in chap. 5 of my notes.

**2008.05.06:** We begin the discussion of the theorem of
Nagell-Lutz. This is II.3,4 in the book.

I will also give the strong form of
the Lemma in II.3, cf. exercise 2.11, p. 61 of the book. Cf. chap. 4 of my
notes.

We also mention Mazur's big theorem on torsion on elliptic curves defined over Q.

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.

**2008.05.02:** We start chapter II of the book: Points of
finite order, points of order 2 and 3, as well as a little about the complex
analytic picture.** **

**2008.04.29:** We finish the discussion of intersections
between lines and elliptic curves, and after that we introduce and discuss the
group law. This includes derivation of explicit formulas in the case
of simple Weierstrass equations.

**2008.04.25:** We continue with the material in chapter 1
of the supplementary notes:

We started discussing intersections between elliptic curves and lines (until and including p. 4 of the notes).

**2008.04.22:** 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 discussing things using projective coordinates all the way and also am avoiding Bezout's theorem in the general form.

Today we discussed motivations and perspectives, and also projective coordinates and projective curves, roughly corresponding to the material in the appendix of the book.