Lecture Notes

Some of the theorems presented in lecture will be demonstrated using the Sage computer algebra system, which is based on Python™. You can download a copy of Sage to run on your own machine if you wish, or create an account for free on the SageMathCloud™.

LEC # TOPICS
1 Introduction to Arithmetic Geometry (PDF), 18.782 Lecture 1 (SWS)
2 Rational Points on Conics (PDF)
3 Finite Fields (PDF), 18.782 Lecture 3 (SWS)
4 The Ring of p-adic Integers (PDF)
5 The Field of p-adic Numbers, Absolute Values, Ostrowski's Theorem for Q (PDF)
6 Ostrowski's Theorem for Number Fields (No lecture notes but see Ostrowski’s Theorem for Number Fields by Keith Conrad)
7 Product Formula for Number Fields, Completions (PDF)
8 Hensel's Lemma (PDF)
9 Quadratic Forms (PDF)
10 Hilbert Symbols (PDF), 18.782 Lecture 10 (SWS)
11 Weak and Strong Approximation, Hasse-Minkowski Theorem for Q (PDF)
12 Field Extensions, Algebraic Sets (PDF)
13 Affine and Projective Varieties (PDF)
14 Zariski Topology, Morphisms of Affine Varieties and Affine Algebras (PDF)
15 Rational Maps and Function Fields (PDF)
16 Products of Varieties and Chevalley's criterion for Completeness (PDF)
17 Tangent Spaces, Singular Points, Hypersurfaces (PDF)
18 Smooth Projective Curves (PDF)
19 Divisors, The Picard Group (PDF)
20 Degree Theorem for Morphisms of Curves (PDF)
21 Riemann-Roch Spaces (PDF)
22 Proof of the Riemann-Roch Theorem for Curves (PDF)
23 Elliptic Curves and Abelian Varieties (PDF)
24 Isogenies and Torsion Points, The Nagell-Lutz Theorem (PDF)
25 The Mordell-Weil Theorem (PDF)
26 Jacobians of Genus One Curves, The Weil-Chatelet and Tate-Shafarevich Groups (PDF)