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

What is an elliptic curve?

Why elliptic curves?

Useful links:

• The number theory web includes links on elliptic curves.
• Eric Weisstein Math World web pages on elliptic curves.
• Franz Lemmermeyer's list of books and other references on elliptic curves.
• William Stein's elliptic curves course.
• J. S. Milne's Elliptic curves course
• Fermingier's list of links on elliptic curves.
• Certicom 's Online Elliptic Curve Cryptography Tutorial. See especially their interactive java program which adds points on elliptic curves. (Click on their link to "Geometric Elliptic Curve Model".)
• UCL Microelectronics Laboratory links on elliptic curves.

Pictures

	1) Here's an example of an elliptic curve, with some construction lines for the group law which we will study. Some points on this curve, such as (0,0) and (-1,-2) are easy to find, but others such as (-5248681/4020025,16718705378/8060150125) would be difficult to find without using the group law.
2) Here's a picture of a family of the "standard" elliptic curves in Weierstrass form, drawn using maple, with the command with(plots): contourplot(x^3 - 3*x - y^2, x=-3..3, y=-3..3, contours=[-7,-5,-4,-3,-2,-1,0,1,1.95,2,10], axes=NONE,filled=true, numpoints=2000); 12 curves are plotted, all of the form y2 = x3 - 3x + b where b varies. The spaces between the curves are coloured. Question: In this picture, what's special about the curves where b=2 and b=-2?
3) Here's another elliptic curve, but not in the standard form, this is in a family (x+y+1)(x+y+xy) = bxy, and we've taken b=1.01 here. This was drawn in Maple with: contourplot((x+y+1)*(x*y+y+x)-1.01*x*y,x=-3..3,y=-3..3, contours=[0], axes=NONE, numpoints=6000); Question: What happens when b=1? Any other interesting values of b? Try sketching more members of this family.

Notes on lectures

Week 1: Conics and projective space.

• Lecture 1 (8/23/04): Discussed lines, conics and cubics, and how conics can be parametrised. (please read chapter 0 in text book).
• Lecture 2 (8/25/04): Defined projective space. We will often go from affine curves to their projectivisation. A definition of the real projective plane can be found on pages 11-12 of Miles Reid's book undergraduate algebraic geometry. There is an animated version of this on Mr Shiner's cool physics web page. Miles Reid wrote knitting instructions for making a Boy's surface (appeared in an old issue of "2-manifold", I believe; see also "The Knitting of Surfaces", Eureka - The Journal of the Archimedeans (Cambridge University Mathematical Society) 34, October 1971, pp21-26.); Claire Irving also describes how to knit the projective plane . Joe Fields gives instructions for how to make one out of cardboard.
• Lecture 3 (8/27/04): Chapter 1. More on parametrisation, and statement of Bezout's Theorem (we only need a special case). Take care that in Theorem 1, you must assume that the conic is not the union of two lines, e.g., the conic given by xy=0.

  The set of real points of the singular cubic which Cassels writes about in chapter looks something like the following, near the origin:

  This was produced in maple with
  with(plots):
  plotsetup(`cps`,plotoutput=`singularcubic.eps`,plotoptions=`portrait,height=100pt,width=100pt`);
  contourplot(x^2 - y^2 -(x-2*y)*(x^2+y^2),x=-1..1,y=-1..1,contours=[0],axes=BOXED, style=point,thickness=3, numpoints=10000);

  The above version is drawn as from the implicit equation for the curve. But as Cassels explains, this curve can be parametrized. Here is the curve drawn using a parametrization (a simple change of variables from the version in the book, so that infinity maps to infinity)

  This was produced in maple with
  with(plots):
  F := proc(u) u*(1 + 4*u + 3*u^2)/(1 + 4*u + 5*u^2) end proc
  plot([F(s)*(1/s+2),F(s),s=-3..1]);

Week 2: p-adic numbers

• Lecture 4 (8/30/04): First 2 pages of Chapter 2 on p-adic numbers. To illustrate how spaces with the ultrametric inequality differ geometrically from spaces with valuations where we can have |x+y| = |x|+|y|, we showed that any point in a unit ball center 0 can be considered the center of the ball. More generally, for any y in a ball radius e, center x, B_e(x), we have B_e(y)=B_e(x). So usual intuition about distance does not hold for the p-adics. Jan Holly discusses other ways to think about p-adics geometrically in her paper Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields, The American Mathematical Monthly, 108 (2001) 721-728.
• Lecture 5 (9/1/04): Second 2 pages of Chapter 2 on p-adic numbers.
• Lecture 6 (9/3/04): Completed Chapter 2. To get familiar with p-adic numbers, try doing some computations with them with computer algebra packages, many of which can compute with p-adic numbers. E.g., here is an example in pari, and you can also work through Mark van Hoeij's maple worksheet for p-adics.

Week 3: Hasse's local – global principle for conics

• Lecture 7 (9/8/04):
  • (1): A few more examples about p-adic numbers. As an unusual application of 2-adic numbers, we will discuss Paul Monsky's proof that you can't dissect a square into an odd number of equal area triangles, which appeared in On dividing a square into triangles, Amer. Math. Monthly, Vol. 77 No. 2, Feb. 1970, pp. 161-164. You can learn about the p-adic methods involved in an article on the Construction of C_p and Extension of p-adic valuations to C by Mike Hamburg. I gave out a handout containing an outline of Monsky's proof, and several explanatory pictures. (See note below for correction). The following is an example of the coloring of points. Drawing your own version, following Monsky's instructions, is good practice for working with p-adics:
    
  
  Note: Ronald Wotzlaw points out to me there are a couple of errors in the picture of the red/blue/green coloring. Points at (1/6, 1/8), and (1/6, 7/8) should be green, and the colored points have coordinates a/b with 0 less than or equal to a less than or equal to b less than or equal to 9 and b not 0.
  I intend to correct these at some point in the future, in the meantime, please be aware of these points when reading the file.
  • (2): Start chapter 3, which is on Hasse's local – global principle for conics. (Note that Z_p is an example of a local ring.)
    Details of the local-global principle are also written by Johnathan Bloom and are programmed into computer pacakges such as magma.
• Lecture 8 (9/10/04): Finished chapter 3. We covered most of chapter 4, on the " geometry of numbers". (We won't need much on the geometry of numbers; for more, you might refer to the MAA book, The geometry of numbers, by Olds, Lax and Davidoff. This field is particularly concerned with lattices, and has applications in coding theory. Dave Rusin's Mathematical Atlas lists subtopics in this area. )

Week 4: Hasse's local-global principle for conics

• Lecture 9 (9/13/04): We finished chapter 4 and chapter 5 on the local global principle for conics.
• Lecture 10 (9/17/04): Review of material covered so far. For more details about p-adic numbers, see Cremona's notes on p-adics. Alan Baker has a much more extensive set of notes on p-adic analysis. On conics, you might also find Franz Lemmermeyer's article on Conics - a poor man's elliptic curves interesting (though it's a bit more advanced than what we've done so far). The math department computer run magma, which also knows about conics and the local global principle .

  Here is a handout (pdf version) (or ps version) with some pictures to explain in more details lemma 1, page 17.

  Here is a little picture illustrating the result about sets of area larger than k containing at least k+1 points lattice distances apart from each other (in the case k=4 in dimension 2). Put your mouse over the image for an animation.

  

Week 5: Cubic curves

• Lecture 11 (9/20/04): Cubic curves, chapter 6 (start - the singular case).
• Lecture 12 (9/22/04): Cubic curves, chapter 6 (middle - chord tangent process - exceptional points).
  The curves we looked at as an example are in a family, x³ + y³ =a, where a is a cube free integer. A few examples are in the following picture: This was produced in maple with:
  
  contourplot(x^3 + y^3,x=-9..9,y=-9..9,contours=[-200,-99,-1,1,2,3,4,5,30,101],axes=BOXED,filled=true,thickness=1, numpoints=10000); The exceptional point is way off at infinity in the direction of the line x+y=0. Any other point gives infinitely many points via the chord tangent process.
  Here's a picture of the process Cassels talks about for the curve x³ + y³ = 9, starting with the point (1,2). Put your mouse over the image and the animation will start. It will show the generation of 6 more points. (If you want to run it again you'll have to reload the page). The first point obtained after (1,2) is (-17/7,20/7); then we get (188479/90391, -36520/90391). As explained in Cassels' book, we can carry on going forever, and the denominator of the coordinates will always increase. (This picture was created using Maple and gimp.)
• Lecture 13 (9/24/04): Cubic curves, chapter 6 (end). Start chapter 7 on the group law on elliptic curves.

Week 6: The group law on elliptic curves

• Lecture 14 (9/27/04): chapter 7 on the group law on elliptic curves; proof of associativity. The following picture corresponds to the proof of associativity, though the diagram is nice due to the special choice of the curve. (this was created in Maple with
  k:=1.1; contourplot({x*(x-1)*(x+1)-k*y*(y+1)*(y-1),x,y,x-1,x+1,y-1,y+1,y-(x+1)/2},x=-3..3,y=-3..3,contours=[0],axes=BOXED,thickness=3, numpoints=10000);
  and edited with gimp, to add labels; try this with different values of k; actually, I first saw it drawn like this in a picture of the proof of associativity of the group law on Duco van Straaten's door, in Mainz University.)
  We want to show that it makes sense to write A+B+C in this diagram, i.e., (A+B) + C = A + (B+C). In the picture c1, c2, c3, c4 are intermediate construction points. We have to show c3=c4, i.e., the lines really do meet the curve where they cross, and the picture does not look like: This can be shown using classical algebraic geometry techniques, in particular a special case of Bezout's theorem.
• Lecture 15 (9/29/04): chapter 7 on the group law on elliptic curves. Discussion of the Riemann Roch theorem for curves.
• Lecture 16 (10/1/04): chapter 7: The Riemann Roch theorem and the group law on elliptic curves. We sketched an idea of how the group law we've defined is isomorphic to the divisor class group of the curve. This is covered in more detail in Silverman's book on the arithmetic of elliptic curves. Although this seems a more complicated way to do things, it has advantages that its easy to prove some results, like associativity, and also this generalizes to curves of higher genus.

Week 7: The Weierstrass form of elliptic curves

• Lecture 17 (10/4/04): Chapter 8. Elliptic curves in Weierstrass form.
• Lecture 18 (10/6/04): Finish chapter 8. How to put elliptic curves in the projective place into Weierstrass form.

Week 8: The group law on singular elliptic curves

• Lecture 19 (10/11/04): Chapter 9: Degenerate group laws - what happens when the curve is singular. We need to know about this because a non-singular curve may become singular when we reduce modulo p, which we will do in future, just as we understood conics through looking at them modulo p and over the p-adics.

  In the cuspidal case, the "obvious" parametrization (in picture below) also gives rize to an isomorphism of groups, where on the line, we just have the usual addition group law.

  
  For your own pictures, try variants of the following maple commands:
  L:=proc(a,b,x) ((b^2 + a*b + a^2)*x - b^2*a^2)/(a+b) end proc;
  cuspidal_picture := proc(a,b)
  plot([x^(3/2), -x^(3/2),1, x/a, x/b, -x/(a+b), L(1/a,1/b,x)] ,x=min(a,b,-a-b)..max(1/a^2,1/b^2,1/(a+b)^2)+1, colour=[red,red,blue,green,green,green,black]);
  end proc;
  cuspidal_picture(1 , 1/2);
  cuspidal_picture(1.2 , 0.62);
• Lecture 20 (10/13/04): Chapter 9: Case where cubic has a double point (also called a node).
• Lecture 21 (10/15/04): Chapter 10: Reduction. Passing from Q_p to F_p and back.