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18.782

Introduction to Arithmetic Geometry

3-0-9 units · Undergraduate

Exposes students to arithmetic geometry, motivated by the problem of finding rational points on curves. Includes an introduction to p-adic numbers and some fundamental results from number theory and algebraic geometry, such as the Hasse-Minkowski theorem and the Riemann-Roch theorem for curves. Additional topics may include Mordell's theorem, the Weil conjectures, and Jacobian varieties.

Fall 2013

Topics

  • Introduction to Arithmetic Geometry
  • Rational Points on Conics
  • Finite Fields
  • The Ring of p -adic Integers
  • The Field of p -adic Numbers, Absolute Values, Ostrowski’s Theorem for Q
  • Ostrowski’s Theorem for Number Fields
  • Product Formula for Number Fields, Completions
  • Hensel’s Lemma
  • Quadratic Forms
  • Hilbert Symbols
  • Weak and Strong Approximation, Hasse-Minkowski Theorem for Q
  • Field Extensions, Algebraic Sets
  • Affine and Projective Varieties
  • Zariski Topology, Morphisms of Affine Varieties and Affine Algebras
  • Rational Maps and Function Fields
  • Products of Varieties and Chevalley’s criterion for Completeness
  • Tangent Spaces, Singular Points, Hypersurfaces
  • Smooth Projective Curves
  • Divisors, The Picard Group
  • Degree Theorem for Morphisms of Curves
  • Riemann-Roch Spaces
  • Proof of the Riemann-Roch Theorem for Curves
  • Elliptic Curves and Abelian Varieties
  • Isogenies and Torsion Points, The Nagell-Lutz Theorem
  • The Mordell-Weil Theorem
  • Jacobians of Genus One Curves, The Weil-Chatelet and Tate-Shafarevich Groups

Readings

Prerequisites & course information

Prereq: 18.702 Acad Year 2027-2028: U (Spring) Acad Year 2028-2029: Not offered 3-0-9 units

Sources