Math 702 schedule

Tuesday Thursday
Jan. 9

In Lecture:
  • No Class!
Jan. 11

In Lecture:
  • Overview of the course
  • The cryptography of sharing secrets
  • The Weil pairing

Associated Reading:
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:
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:
Video of class (downloadable)
Jan. 30

In Lecture:
  • Miller's algorithm, continued

Associated Reading:
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:
Video of class (downloadable)
Feb. 6

No class!
Feb. 8

No class due to eagles parade!
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)
  • Etale morphisms

Associated Reading:
  • Milne's "Etale topology"

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 topology"

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 topology"

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)
  • Group schemes

Associated Reading:
  • Milne's "Etale topology"

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 stale topology

Associated Reading:
  • Milne's "Etale topology"

Video of class (downloadable)
March 27

In Lecture: (Meeting at 9:00 AM in 4N30)
  • Group schemes

Associated Reading:
  • Milne's "Etale topology"

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 topology"

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 topology"

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 topology"

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 topology"

Video of class (downloadable)
April 12

No class
April 17

In Lecture: (Meeting at 9:00 AM in 4N30)
  • Stalks of sheaves in the stale topology
  • Construction of injective resolutions
  • Mapping cylinder constructions of stale sheaves

Associated Reading:
  • Milne's "Etale topology"

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 topology"

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 topology"

Video of class (downloadable)
April 25

No class!

Last updated: 4/22/18
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