Lecture Schedule
| # | Date | Topic (references) | Materials |
|---|---|---|---|
| 1 | 02/06 | Overview of arithmetic geometry (Ellenberg, Poonen) | slides |
| 2 | 02/08 | Rational points on curves by genus | notes (2.1-2.2), Drew's slides |
| 3 | 02/10 | Reduction modulo primes; Legendre's theorem and descent algorithm | Drew's slides, notes (2.3) |
| 4 | 02/13 | Finite fields | notes (3.1-3.3) |
| 5 | 02/15 | Solving polynomials over finite fields; p-adic integers | notes (3.4), notes |
| 6 | 02/17 | p-adic numbers | notes (5.1, 5.2) |
| 7 | 02/21 | Ostrowski's theorem | notes (5.3), Conrad (Sections 1-3) |
| 8 | 02/22 | Product formula; Basics of Algebraic Number Theory | notes (7.2, 7.3), Milne (Chapter 2) |
| 9 | 02/24 | Product formula; Completions | notes (7.1, 7.3, 7.4) |
| 10 | 02/27 | Equivalence of definitions of p-adics; Root finding in p-adics | notes (8.1, 8.2) |
| 11 | 03/01 | Hensel's lemma; Quadratic forms | notes (8.3) notes |
| 12 | 03/03 | Hilbert symbol | notes |
| 13 | 03/06 | Hilbert symbol continued; Hasse-Minkowski theorem | notes notes (11.3) |
| 14 | 03/08 | Hilbert reciprocity; Proof of Hasse-Minkowski in dim 4 | notes (Theorem 10.11) notes |
| 15 | 03/10 | Proof of Hasse-Minkowski in dim ≥ 5 | notes |
| 16 | 03/13 | Field extensions | notes (12.1) |
| 17 | 03/15 | Affine algebraic sets and varieties | notes (12.2),notes (13.1) |
| 18 | 03/17 | Projective varieties | notes (13.2-13.5) |
| 19 | 03/20 | Zariski topology, Affine morphisms | notes |
| 20 | 03/22 | Rational maps of affine varieties | notes (15.1) |
| 21 | 03/24 | Products of varieties | notes (16.1) |
| 22 | 04/03 | Rational maps and morphisms of projective varieties; Complete varieties | notes (15.2) notes (16.2) |
| 23 | 04/05 | Valuative criterion of completeness | notes (16.3-16.5) |
| 24 | 04/07 | Projective varieties are complete; Tangent and cotangent spaces | notes (16.5) notes |
| 25 | 04/10 | Smooth points; algebraic characterisation of cotangent space; Regular local ring | notes notes (18.1) |
| 26 | 04/12 | Smooth projective curves | notes (Section 18.2, Definition 18.9, Statements of 18.10, 18.13, 18.14) |
| 27 | 04/14 | Divisors | notes |
| 28 | 04/19 | Degree theorem | notes |
| 29 | 04/21 | Riemann-Roch spaces | notes |
| 30 | 04/24 | Adele spaces; Canonical divisor | notes (22.1-22.3) |
| 31 | 04/26 | Riemann-Roch theorem; Curves of genus <= 1 with a rational point | notes (22.4) notes (23.1, 23.2) |
| 32 | 04/28 | Elliptic curves and abelian varieties | notes (23.3, 23.4) |
| 33 | 05/01 | Isogenies and torsion points on elliptic curves | notes (24.1, 24.2) |
| 34 | 05/03 | Torsion points on elliptic curves over Q | notes (24.3) |
| 35 | 05/05 | Mordell-Weil theorem - 1 (weak version) | notes (25.1-25.4) |
| 36 | 05/08 | Mordell-Weil theorem - 2 (heights); Principal homogeneous spaces and Weil-Chatelet group | notes (25.5-25.7) notes (26.4-26.6) |