Introduction to Arithmetic Geometry

Calendar

LEC #	TOPICS	KEY DATES
1	Introduction to Arithmetic Geometry
2	Rational Points on Conics
3	Finite Fields
4	The Ring of p-adic Integers	Problem Set 1 Due
5	The Field of p-adic Numbers, Absolute Values, Ostrowski’s Theorem for Q
6	Ostrowski’s Theorem for Number Fields	Problem Set 2 Due
7	Product Formula for Number Fields, Completions
8	Hensel’s Lemma	Problem Set 3 Due
9	Quadratic Forms
10	Hilbert Symbols	Problem Set 4 Due
11	Weak and Strong Approximation, Hasse-Minkowski Theorem for Q
12	Field Extensions, Algebraic Sets
13	Affine and Projective Varieties	Problem Set 5 Due
14	Zariski Topology, Morphisms of Affine Varieties and Affine Algebras
15	Rational Maps and Function Fields	Problem Set 6 Due
16	Products of Varieties and Chevalley’s criterion for Completeness
17	Tangent Spaces, Singular Points, Hypersurfaces	Problem Set 7 Due
18	Smooth Projective Curves
19	Divisors, The Picard Group	Problem Set 8 Due
20	Degree Theorem for Morphisms of Curves
21	Riemann-Roch Spaces
22	Proof of the Riemann-Roch Theorem for Curves	Problem Set 9 Due
23	Elliptic Curves and Abelian Varieties
24	Isogenies and Torsion Points, The Nagell-Lutz Theorem	Problem Set 10 Due
25	The Mordell-Weil Theorem
26	Jacobians of Genus One Curves, The Weil-Chatelet and Tate-Shafarevich Groups	Problem Set 11 Due