18.782 | Fall 2013 | Undergraduate
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
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Fall 2013
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