We will discuss Section 3 (pages 25-29) proving associativity of elliptic curve addition.
As time permits, we will discuss the statement of the Riemann Roch theorem.
The final week of meetings will go over introductory material on elliptic curves from Milne’s book
Read the introduction about some of the uses of elliptic curves (including sphere packings and cryptography!)
pages 5-6 should be review.
- Read the definition of a nonsingular curve.
- Work example 1.5 to get a test for a Weierstrass polynomial to define a nonsingular curve.
- Read Prop 1.8 on intersection numbers.
- Read up through page 11 completing the section on intersection numbers
- Review projective plane curves, translating what Milne does back into the language of Proj(S.).
- Learn the statement of Bezout’s theorem.
- We will discuss a later result (8.6) from Humphreys: let G be an affine algebraic group over an algebraically closed field k, then G is isomorphic to a closed subgroup of GL(n) (over k) for some n.
- 7.3. Let G be an affine algebraic group. Show that G^0 is a normal subgroup of finite index.
- 7.4. Show that the Zariski closure of a subgroup is a subgroup.
- 7.4.B. Show that the kernel and images are closed subgroups, and that the image of the connected component of the identity is the connected component of the image.
- 8.1. State what an action is in the category of k-varieties.
- Show that GLn acts on P^n, on the flag variety.
We will start on our discussion of elliptic curves as time permits.
The meeting will continue with Humphrey’s book.
- Work through the details of the construction of the Grassmannian as a k-scheme.
- Work the the construction of the flag variety
- Read 6.4. (automorphisms of the projective line) and prepare the proof that that the automorphism group is PGL(2,k).
- Read the section 6 on complete varieties (pages 45-46).