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This course is an introduction to arithmetic geometry, a subject that lies at the intersection of algebraic geometry and number theory.
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English
English [CC]
FREE
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Description
Its primary motivation is the study of classical Diophantine problems from the modern perspective of algebraic geometry.
Course content
- Introduction to Arithmetic Geometry Unlimited
- Rational Points on Conics Unlimited
- Finite Fields Unlimited
- The Ring of p-adic Integers Unlimited
- The Field of p-adic Numbers, Absolute Values, Ostrowski’s Theorem for Q Unlimited
- Ostrowski’s Theorem for Number Fields Unlimited
- Product Formula for Number Fields, Completions Unlimited
- Hensel’s Lemma Unlimited
- Quadratic Forms Unlimited
- Hilbert Symbols Unlimited
- Weak and Strong Approximation, Hasse-Minkowski Theorem for Q Unlimited
- Field Extensions, Algebraic Sets Unlimited
- Affine and Projective Varieties Unlimited
- Zariski Topology, Morphisms of Affine Varieties and Affine Algebras Unlimited
- Rational Maps and Function Fields Unlimited
- Products of Varieties and Chevalley’s criterion for Completeness Unlimited
- Tangent Spaces, Singular Points, Hypersurfaces Unlimited
- Smooth Projective Curves Unlimited
- Divisors, The Picard Group Unlimited
- Degree Theorem for Morphisms of Curves Unlimited
- Riemann-Roch Spaces Unlimited
- Proof of the Riemann-Roch Theorem for Curves Unlimited
- Elliptic Curves and Abelian Varieties Unlimited
- Isogenies and Torsion Points, The Nagell-Lutz Theorem Unlimited
- The Mordell-Weil Theorem Unlimited
- Jacobians of Genus One Curves, The Weil-Chatelet and Tate-Shafarevich Groups Unlimited
Instructor
Massachusetts Institute of Technology
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1916
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