Lecture Schedule
# | Date | Topic | Reading | Extra resources |
---|---|---|---|---|
1 | 9/2 | Introduction to arithmetic geometry | slides | Ellenberg, Poonen, Drew's slides, Amit (fruit math) |
2 | 9/4 | Rational points on conics | notes | Cremona-Ruskin |
9/7 | No class (Labor day) | |||
3 | 9/9 | Finite fields | notes | Rabin |
4 | 9/11 | Finite fields, the ring of p-adic integers | notes | Roberts|
5 | 9/14 | The field of p-adic numbers, absolute values, Ostrowski's theorem | notes | Conrad |
6 | 9/16 | Product formula for number fields | notes (7.1-7.3) | |
7 | 9/18 | Completions | notes (7.4) notes (8.1) | |
8 | 9/21 | Hensel's lemma and Newton polygons | notes (8.2-8.3) | Conrad, Stoll (§3) |
9 | 9/23 | Quadratic forms | notes | Conway |
10 | 9/25 | Hilbert symbols | notes | |
11 | 9/28 | Weak and strong approximation, Hasse-Minkowski theorem | notes | |
12 | 9/30 | Genera of quadratic forms, class groups of binary quadratic forms | Martin (4.1-2,4.4) | Conway-Sloane (§7) |
13 | 10/2 | 15 and 290 theorems (Add date) | Bhargava (pp 27-38) | Henke |
14 | 10/5 | Field extensions, algebraic sets | notes | |
15 | 10/7 | Affine varieties | notes (13.1) | |
16 | 10/9 | Projective varieties | notes (13.2-5) | |
10/12 | No class (Indigenous peoples' day); class on Tuesday instead | |||
17 | 10/13 | Zariski topology, morphisms of affine varieties | notes | |
18 | 10/14 | Affine algebras, rational maps | notes | |
19 | 10/16 | Function fields | notes | |
20 | 10/19 | Tangent spaces, singular points, smoothness, hypersurfaces | notes | |
21 | 10/21 | Smooth projective curves | notes (18.1-2) | |
22 | 10/23 | Function fields and curves | notes (18.3) notes (19.1) | |
23 | 10/26 | Divisors | notes (19.2) | |
24 | 10/28 | The Picard group | none | |
25 | 10/30 | Degree theorem for morphisms of curves | notes | |
26 | 11/2 | Riemann-Roch spaces | notes | |
27 | 11/4 | Genus of curves | notes (22.1-2) | |
28 | 11/6 | Canonical divisors and Riemann-Roch theorem | notes (22.3-22.4) | |
29 | 11/9 | Elliptic curves and abelian varieties | notes | |
11/11 | No class (Veteran's day) | |||
30 | 11/13 | Isogenies and torsion points, the Nagell-Lutz theorem | notes | |
31 | 11/16 | The Mordell-Weil theorem 1 | notes (25.1-4) | |
32 | 11/18 | The Mordell-Weil theorem 2 (Drop date) | notes (25.5-7) | |
33 | 11/20 | Falting's theorem | notes | |
11/23 | No class (Thanksgiving) | |||
11/25 | No class (Thanksgiving) | |||
11/27 | No class (Thanksgiving) | |||
34 | 11/30 | Weil conjectures: zeta functions | notes | |
35 | 12/2 | Weil conjectures: rationality and functional equation | notes | |
36 | 12/4 | Weil conjectures: etale cohomology | notes | |
37 | 12/7 | Weil conjectures: the Riemann hypothesis | notes | |
38 | 12/9 | The Honda-Tate theorem | notes |