2005.04.13: Perspectives ...
I will give some indications - mostly at the intuitive/descriptive level - of why elliptic curves are enormously important in contemporary arithmetic. This is because they are important test cases of the so-called Langlands program, a vast system of conjectures generalizing (for instance) such things as the quadratic reciprocity law. The Langlands program is the most complicated system of conjectures ever seen in the history of mathematics (ok, so this is 'just' my claim, but ... it's true
The connection to Langlands also is what ties elliptic curves up with Fermat's last theorem, and I may get a little into that as well.
I have written a survey note on some of the issues arising in connection with the proof of Fermat's last theorem (unfortunately only in Danish). They can be downloaded here: http://www.math.ku.dk/~kiming/papers/surv/1999_fermat2/1999_fermat2_.pdf 2005.04.11:
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 note: http://www.math.ku.dk/~kiming/lecture_notes/2002-2003-elliptic_curves/torsion_on_special_curves.pdf 2005.04.06:
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. And when it works, it's great ... 2005.04.04:
Some concluding remarks about how - using maple - one can actually fill in the details of the proof of the fact that $\phi$ in the proposition on page 79 is a homomorphism.
After that we proceed 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 tick is the fact that any integer can be written as plus-minus a finite
number of primes ... 2005.03.30:
Lectures cancelled today due to lecturer's illness. 2005.03.23 + 2005.03.28:
Easter vacation. 2005.03.21:
We continue the proof of weak Mordell for the case that the elliptic curve has a rational point of order 2. 2005.03.16:
End of the height story and begin proof of weak Mordell in case the curve has a rational point of order 2. This is III.4.
Notice that in the book certain details are left as an exercise to the reader, - cf. the remarks at the end of page 81. I will explain how to fill in these details. This actually requires quite a lot of computation, but fortunately one can use maple for that. 2005.03.14:
Proof that '(Weak Mordell + height lemmas) implies Mordell'.
After that we prove Lemma 2 in III.2, and Lemma 3 in III.3. I will ask you to read most of the proof of Lemma 3' on your own (it's quite horrible to write this argument on the blackboard ...). 2005.03.09:
We begin discussion of heights, Chap. III, 1-3.
Heights is one of the tools that are used in the proof of Mordell's theorem.
Today we talked about definition of heights and the strategy for proving that 'Weak Mordell implies Mordell'. 2005.03.07:
We finish the proof of the `reduction modulo p theorem'. 2005.03.02:
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. 2005.02.28:
We finish the proof of the lemma in http://www.math.ku.dk/~kiming/lecture_notes/2002-2003-elliptic_curves/nagell_lutz2.pdf
as well as the proof of the Nagell-Lutz theorem. 2005.02.23:
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: http://www.math.ku.dk/~kiming/lecture_notes/2002-2003-elliptic_curves/nagell_lutz2.pdf 2005.02.21:
A little about elliptic curves over the complex numbers, and then we continue the discussion of Nagell-Lutz.
We will see a slightly stronger form of the Lemma on p. 48. We will use this note: http://www.math.ku.dk/~kiming/lecture_notes/2002-2003-elliptic_curves/nagell_lutz1.pdf 2005.02.16:
II.1,3: Points of low order, begin the Theorem of Nagell-Lutz: Seminar by Peter Petersen. 2005.02.14:
The group law.
If you took 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.
Here is a note about intersections between a line and a projective curve: http://www.math.ku.dk/~kiming/lecture_notes/2002-2003-elliptic_curves/points_of_intersection.pdf
We discussed the group law and the explicit formulas for addition of points in a slightly more precise way than the book (I will use projective coordinates all the way). You can take notes of this at the lecture.
If you want to see the addition formulas for the case of a completely general Weierstrass equation, they are here: http://www.math.ku.dk/~kiming/lecture_notes/2002-2003-elliptic_curves/add.pdf
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. 2005.02.09:
Simple Weierstrass equations, irreducibility of elliptic curves, intersection between projective lines and cubic curves.
We will need this note: http://www.math.ku.dk/~kiming/lecture_notes/2002-2003-elliptic_curves/irred.pdf 2005.02.07:
Diophantine equations, projective coordinates and curves, smooth curves, elliptic curves, Weierstrass equations.