Math 702 schedule
Tuesday |
Thursday |
Jan. 9
In Lecture:
| Jan. 11
In Lecture:
- Overview of the course
- The cryptography of sharing secrets
- The Weil pairing
Associated Reading:
- Silverberg and Boneh on multilinear maps
- The part of Silverman's elliptic curves book about the Weil pairing.
- Miller on the Weil Pairing
Video of class (downloadable)
|
Jan. 16
In Lecture:
- Review of the theory of curves
Associated Reading:
- Hartshorne, Chapter 1 on constructing curves from valuations on their function fields
- The part of Silverman's elliptic curves book about curves
Video of class (downloadable)
| Jan. 18
In Lecture:
- Review of elliptic curves
- The Weil pairing
Associated Reading:
- The part of Silverman's elliptic curves book about the Weil pairing.
- Miller on the Weil Pairing
Video of class (downloadable)
|
Jan. 23
In Lecture:
- Proof of Weil reciprocity
Associated Reading:
- Hartshorne, Chapter 1 on constructing curves from valuations on their function fields
- The part of Silverman's elliptic curves book about curves
Video of class (downloadable)
| Jan. 25
In Lecture: (Meeting at 9:00 AM in 4N30)
- Miller's algorithm for computing the Weil pairing
Associated Reading:
- The part of Silverman's elliptic curves book about the Weil pairing.
- Miller on the Weil Pairing
Video of class (downloadable)
|
Jan. 30
In Lecture:
- Miller's algorithm, continued
Associated Reading:
- Miller on the Weil Pairing
Video of class (downloadable)
| Feb. 1
In Lecture: (Meeting at 9:00 AM in 4N30)
- End of the discussion of Miller's algorithm
- Covers of curves
Associated Reading:
- The part of Silverman's elliptic curves book covers of curves
- Miller on the Weil Pairing
Video of class (downloadable)
|
Feb. 6
No class!
|
Feb. 13
In Lecture: (Meeting at 9:00 AM in 4N30)
- Covers of curves, Hurwitz Theorem
- Galois theory over function fields of elliptic curves
- Division polynomials and the multiplication by N map
Associated Reading:
- The part of Silverman's elliptic curves book covers of curves
Video of class (downloadable)
| Feb. 15
In Lecture: (Meeting at 9:00 AM in 4N30)
- Division polynomials, continued
- A pairing from torsion points times roots of unity to torsion points
Associated Reading:
- The part of Silverman's elliptic curves book covers of curves
Video of class (downloadable)
|
Feb. 20
In Lecture: (Meeting at 9:00 AM in 4N30)
- The torsion point times root of unity pairing, continued
Associated Reading:
- The part of Silverman's elliptic curves book covers of curves
There was no video for this class.
| Feb. 22
In Lecture: (Meeting at 9:00 AM in 4N30)
- The inverse Weil pairing
- Elliptic curves with complex multiplication
- Constructing elliptic curves with all \ell-torsion defined over a given finite field
Associated Reading:
- The part of Silverman's elliptic curves book covers of curves
Video of class (downloadable)
|
Feb. 27
In Lecture: (Meeting at 9:00 AM in 4N30)
- Comparing the inverse Weil pairing to the the torsion point times root of unity pairing.
- Constructing elliptic curves with all \ell-torsion defined over a given finite field
Associated Reading:
- The part of Silverman's elliptic curves book covers of curves
Video of class (downloadable)
| Mar. 1
No class
|
Mar. 6
No class, spring break
| Mar. 8
No class, spring break
|
March 13
In Lecture: (Meeting at 9:00 AM in 4N30)
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| March 15
In Lecture: (Meeting at 9:00 AM in 4N30)
- The Zariski, stale and flat sites
- Etale presheaves
- The case of the spectrum of a field
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
|
March 20
In Lecture: (Meeting at 9:00 AM in 4N30)
- Galois morphisms
- The stale sheaf associated to a coherent sheaf
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| March 22
No class due to Penn's closure in the blizzard
|
March 27
In Lecture: (Meeting at 9:00 AM in 4N30)
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| March 29
In Lecture: (Meeting at 9:00 AM in 4N30)
- Constant group schemes
- Cartier duals of group schemes
- Cech cohomology in the etale topology
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
|
March 27
In Lecture: (Meeting at 9:00 AM in 4N30)
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| March 29
In Lecture: (Meeting at 9:00 AM in 4N30)
- Constant group schemes
- Cartier duals of group schemes
- Cech cohomology in the etale topology
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| April 3
In Lecture: (Meeting at 9:00 AM in 4N30)
- Cech cohomology and group cohomology
- Derived functors
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| April 5
In Lecture: (Meeting at 9:00 AM in 4N30)
- Homological algebra behind the existence of derived functors
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| April 10
In Lecture: (Meeting at 9:00 AM in 4N30)
- Exact sequences of pre sheaves and of sheave
- Push forward and pull backs
- Examples
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| April 12
No class
| April 17
In Lecture: (Meeting at 9:00 AM in 4N30)
- Stalks of sheaves in the etale topology
- Construction of injective resolutions
- Mapping cylinder constructions of stale sheaves
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| April 19
In Lecture: (Meeting at 9:00 AM in 4N30)
- Examples of mapping cylinder constructions
- The six functors associated to mapping cylinders
- Torsors
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| April 23
In Lecture: (Meeting at 9:00 AM in 4N30)
- Torsors, sheaf torsors and cohomology
- The cohomology of G_m
- A formula for triple products on curves
Associated Reading:
- Milne's "Etale cohomology"
Video of class (downloadable)
| April 25
No class!
|