CourseNotes (J.S. Milne)

This directory contains the full course notes in dvi form for some of the advanced courses I've taught.

Group Theory

This is the first half of one of our core graduate algebra courses. Math 594g (Last revised August 21, 1996; v2.01; 57pp.)

Contents

  1. Basic Definitions
  2. Free Groups and Presentations
  3. Isomorphism Theorems; Extensions
  4. Groups Acting on Sets
  5. The Sylow Theorems; Applications
  6. Normal Series; Solvable and Nilpotent Groups

Fields and Galois Theory

This is the second half of one of our core graduate algebra courses. Math 594f (Last revised August 21, 1996; v2.01; 58pp)

Contents

  1. Extensions of Fields.
  2. Splitting Fields; Algebraic Closures.
  3. The Fundamental Theorem of Galois Theory.
  4. Computing Galois Groups.
  5. Applications of Galois Theory.
  6. Transcendental Extensions.

Algebraic Geometry

An introductory course concerned with algebraic varieties over algebraically closed fields. Math 631 (Last revised August 24, 1996; v2.01, 140pp)

Algebraic Number Theory

A fairly standard graduate course on algebraic number theory. Math 676 (Last revised August 14, 1996; v2.01; 144p).

Contents

  1. Preliminaries From Commutative Algebra
  2. Rings of Integers
  3. Dedekind Domains; Factorization
  4. The Finiteness of the Class Number
  5. The Unit Theorem
  6. Cyclotomic Extensions; Fermat's Last Theorem
  7. Valuations; Local Fields
  8. Global Fields

Modular Functions and Modular Forms

This course may serve as an introduction to Shimura's book (Introduction to the Arithmetic Theory of Automorphic Functions). Math 678 (dvi file) Math 678 (ps file) (Last revised May 22, 1997; v1.1, 128 pages)

Contents

  1. Preliminaries
  2. Elliptic modular curves as Riemann surfaces
  3. Elliptic functions
  4. Modular functions and modular forms
  5. Hecke operators
  6. The modular equation for Gamma_0(N)
  7. The canonical model of X_0(N) over Q
  8. Modular curves as moduli varieties
  9. Modular forms, Dirichlet series, and functional equations
  10. Correspondences on curves; the theorem of Eichler and Shimura
  11. Curves and their zeta functions
  12. Complex multiplication for elliptic curves

Elliptic Curves

This course was an introductory overview of the topic. Math 679 (Last revised August 21, 1996; v1.01; 158pp)

Contents

  1. Review of Plane Curves
  2. Rational Points on Plane Curves
  3. The Group Law on a Cubic Curve
  4. Functions on Algebraic Curves and the Riemann-Roch Theorem
  5. Definition of an Elliptic Curve
  6. Reduction of an Elliptic Curve Modulo p
  7. Elliptic Curves over Qp
  8. Torsion Points
  9. Neron Models
  10. Elliptic Curves over the Complex Numbers
  11. The Mordell-Weil Theorem: Statement and Strategy
  12. Group Cohomology
  13. The Selmer and Tate-Shafarevich Groups
  14. The Finiteness of the Selmer Group
  15. Heights
  16. Completion of the Proof of the Mordell-Weil Theorem, and Further Remarks
  17. Geometric Interpretation of the Cohomology Groups; Jacobians
  18. The Tate-Shafarevich Group; Failure of the Hasse Principle
  19. Elliptic Curves over Finite Fields
  20. The Conjecture of Birch and Swinnerton-Dyer
  21. Elliptic Curves and Sphere Packings
  22. Algorithms for Elliptic Curves
  23. The Riemann Surfaces X0(N)
  24. X0 as an Algebraic Curve over Q
  25. Modular Forms
  26. Modular Forms and the L-series of Elliptic Curves
  27. Statements of the Main Theorems
  28. How to get an Elliptic Curve from a Cusp Form
  29. Why the L-Series of E Agrees with the L-series of f
  30. Wiles's Proof
  31. Fermat, At Last

Class Field Theory

This is a course on Class Field Theory, roughly along the lines of the articles of Serre and Tate in Cassels-Frohlich. The new version (1997) is heavily revised and expanded. Math 776 (Last revised May 6, 1997; v3.1; 222 pages)

Contents

  1. Local Class Field Theory: Lubin-Tate Extensions
  2. Cohomology of Groups.
  3. Local Class Field Theory Continued.
  4. Brauer Groups
  5. Global Class Field Theory: Statements
  6. L-series and the Density of Primes
  7. Global Class Field Theory: Proofs
  8. Complements (Power reciprocity laws; quadratic forms; etc.)