This is the page for a reading course on abelian varieties I took in Fall 2024 at NYU. The course initially followed Brian Conrad's lecture notes here while using Mumford's text, Milne's text, etc., as references. After that, we used this to learn about Galois representations and a little more about Tate Modules. I have uploaded my personal notes and solutions to problems (many handwritten, not in LaTeX because of time constraints).
Advisor: Fedor Bogomolov
Meetings: Once per week for 1 hour
Note: This is a slightly ambitious semester-long 3-credit course. If you want to copy this course, I recommend sticking with only Conrad's text (maybe spend a week reviewing scheme theory and representable functors).
Note: We do the problems in Conrad's notes and those on his website here.
| Week | Topic | Reading | Notes/Problems |
|---|---|---|---|
| 1 | Basic Theory | 1.1-1.4 in Conrad | PDF removed |
| 2 | Basic Theory | 1.4-1.7 in Conrad | PDF removed |
| 3 | The Picard Functor | 2.1-2.4 in Conrad | PDF removed |
| 4 | Line Bundles on Abelian Varieties | 3.1-3.4 in Conrad | PDF removed |
| 5 | Torsion | 4.1-4.2 in Conrad | PDF removed |
| 6 | The Dual Abelian Variety | 5.1-5.3 in Conrad | PDF removed |
| 7 | Descent | 6.1-6.2 in Conrad | PDF removed |
| 8 | More on Dual Abelian Variety | 7.1-7.4 in Conrad | PDF removed |
| 9 | More on Dual Abelian Variety | 7.4-7.6 in Conrad | PDF removed |
| 10 | The Weil Pairing | 8.1-8.2 in Conrad | PDF removed |
| 11 | The Mordell-Weil Theorem | 9.1-9.3 in Conrad | PDF removed |
| 12 | Heights | 10.1-10.3 in Conrad | PDF removed |
| 13 | Galois Representations | 2.1-2.4 in Lombardo | PDF removed |
| 14 | Galois Representations | 2.5-2.10 in Lombardo | PDF removed |
| 15 | p-adic Galois Representations | Read Lombardo's notes of Szamuely's talk | PDF removed |