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: - 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: 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. 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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