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Last updated:

December 27, 2022


Unlimited Duration


This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration


This course is an introduction to arithmetic geometry, a subject that lies at the intersection of algebraic geometry and number theory.

Its primary motivation is the study of classical Diophantine problems from the modern perspective of algebraic geometry.

Course Curriculum

  • 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

About the instructor

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Massachusetts Institute of Technology
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