Elliptic curves, Fall 2004, Math 7280
http://www.math.lsu.edu/~verrill/teaching/math7280/index_part2.html

Week 9: Elliptic curves over F_p and Q_p.

• Lecture 22 (10/18/04): Chapter 11: The p-adic case: Passing from curves over Q to curves over Q_p.

  How do we think about points on a curve over Q_p? One way to find solutions in Z_p is to find solutions mod p, mod p², mod p³, and so on. There are various ways to think about this. Here's one picture:

  Note that this is an example, due to Selmer, where the curve is not an elliptic curve over Q, but it is an elliptic curve over Q_p for all primes p. We will discuss this later . Points in this picture are points (x,y) which satisfy the equation mod 5 and mod 25. This picture is reasonable as a graph of a Euclidean function. For the padics it doesn't make so much sense, since points which have coordinates which are equal mod 5 should be close (e.g., all the yellow points in the second picture, which will reduce mod 5 to the yellow point in the first picture, should be close to each other). One can devise ways to display the data to indicate this. For more ideas about how you might think about how to graph this, Jan Holly' paper Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields, The American Mathematical Monthly, 108 (2001) 721-728.

  This procedure only gives points in Z_p. To work in (1/p) Z_p, (1/p^2)Z_p, and so on, we can make a change of variables, e.g., write x1=x/p, y1=x/p, and so if (x,y) is on the curve 3x³ + 4y³ + 5=0, then (x1,y1) is on the curve 3x1³ + 4y1³ + 5p³=0. A solution (x1,y1) in Z_p to this equation, which can be constructed by taking solutions mod pⁿ for higher and higher powers of p, will give a (x,y) in (1/p)Z_p. This is pretty similar to what is actually done in this chapter, but we work with elliptic curves in the standard Weierstrass form (and get a filtration which is "in the opposite direction" to what is constucted via this method (increasing instead of decreasing).)

  In this chapter a p-adic filtration is defined on the group of rational points of the elliptic curve in Weiertrass form. Here's a picture; put your mouse over the image for its construction of the first step.

  Warning - the gif file is an animation, and is about 171K in size. Also, although it's slow, it might still be too fast to think through, but at least shows the order the things should be drawn into the diagram.

  

  Here is a pdf file with this picture and another for the next step of the filtration.

• Lecture 23 (10/20/04): Chapter 12: Global torsion
• Lecture 24 (10/22/04): Possibly Chapter 13: Idea of proof of finite basis theorem (Mordell-Weil theorem), or something relevant to this.

Week 10 Mordell-Weil finite basis theorem. The weak Mordell-Weil theorem.

Lectures this week will be given by Jean and Cristian. They will probably cover approximately chapters 13, 14, 15, setting up a 2 isogeny, and proving the week version of the theorem, that the quotient of C by 2C is finite.

• Lecture 25 (10/25/04): I finished off chapter 12, quickly talked about chapter 13.
• Lecture 26 (10/27/04): Chapter 14: Jean defined a map from E/2E.
• Lecture 27 (10/29/04): Chapter 15: Cristian on the finiteness of E/2E.

The proof of the Mordell theorem is somewhat algorithmic, and depends on "halving" points. The following picture shows a point R, and four other points P with 2P=R. Notice that the numbers used in writing down R are much bigger than for the other four points.

You can't necessarily "halve" a point and still end up with a rational point. To get a rational point you may need to first add a non trivial representative of E/2E. The first step of the proof of the Mordell theorem is to prove that this group is finite.

Week 11 Mordell-Weil finite basis theorem. Heights.

• Lecture 28 (11/1/04): j-invariants; refer to hand out from modular forms course last semester.
• Lecture 29 (11/3/04): Chapter 16, 17: Heights
• Lecture 30 (11/5/04): Chapter 17: Proof of the Mordell-Weil theorem for E(Q).

The second part of the proof of the Mordell theorem involves the concept of height of rational points on an elliptic curve. Here's a picture:

Week 12 Genus 1 curves over Q

• Lecture 31 (11/8/04): Chapter 18: Selmer's example showing the failure of local global principle for cubic curves. Here is a handout with more details.
• Lecture 32 (11/10/04): Chapter 19: Elements of Galois cohomology.
• Lecture 33 (11/12/04): Chapter 20: The Jacobian of an elliptic curve. Here is another handout with more details for the examples.

Week 13 Student talks

• Lecture 34 (11/15/04): Liqun talks on elliptic curves over finite feilds. ( Here's a copy of a couple of overheads used to accompany this talk )
• Lecture 35 (11/17/04): Matt and Richard talk on Elliptic curve cryptography.
• Lecture 36 (11/19/04): Karli and Liz talk on using elliptic curves in factorisation (Chapter 26). There are more details about this method in an article by Bjorn Poonen on elliptic curves (pages 12-15).

Week 14: No lectures

Week 15 Galois cohomology and the Tate Shaferavich group.

• Lecture 37 (11/29/04): Chapters 19 and 21, introduction to Galois Cohomology. Here is a handout with a summary of points from chapters 19, 21, 23, for lectures 37 and 39.
• Lecture 38 (12/01/04): Maiia and Matt talk on the endomorphism ring, chapter 24.
• Lecture 39 (12/03/04): Chapters 21 and 22, definition of the Tate Shaferevich group, and overview of the Birch-Swinnerton-Dyer conjecture.