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
- Basic Definitions
- Free Groups and Presentations
- Isomorphism Theorems; Extensions
- Groups Acting on Sets
- The Sylow Theorems; Applications
- 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
- Extensions of Fields.
- Splitting Fields; Algebraic Closures.
- The Fundamental Theorem of Galois Theory.
- Computing Galois Groups.
- Applications of Galois Theory.
- 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
- Preliminaries From Commutative Algebra
- Rings of Integers
- Dedekind Domains; Factorization
- The Finiteness of the Class Number
- The Unit Theorem
- Cyclotomic Extensions; Fermat's Last Theorem
- Valuations; Local Fields
- 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
- Preliminaries
- Elliptic modular curves as Riemann surfaces
- Elliptic functions
- Modular functions and modular forms
- Hecke operators
- The modular equation for Gamma_0(N)
- The canonical model of X_0(N) over Q
- Modular curves as moduli varieties
- Modular forms, Dirichlet series, and functional equations
- Correspondences on curves; the theorem of Eichler and Shimura
- Curves and their zeta functions
- 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
- Review of Plane Curves
- Rational Points on Plane Curves
- The Group Law on a Cubic Curve
- Functions on Algebraic Curves and the Riemann-Roch Theorem
- Definition of an Elliptic Curve
- Reduction of an Elliptic Curve Modulo p
- Elliptic Curves over Qp
- Torsion Points
- Neron Models
- Elliptic Curves over the Complex Numbers
- The Mordell-Weil Theorem: Statement and Strategy
- Group Cohomology
- The Selmer and Tate-Shafarevich Groups
- The Finiteness of the Selmer Group
- Heights
- Completion of the Proof of the Mordell-Weil Theorem, and Further
Remarks
- Geometric Interpretation of the Cohomology Groups; Jacobians
- The Tate-Shafarevich Group; Failure of the Hasse Principle
- Elliptic Curves over Finite Fields
- The Conjecture of Birch and Swinnerton-Dyer
- Elliptic Curves and Sphere Packings
- Algorithms for Elliptic Curves
- The Riemann Surfaces X0(N)
- X0 as an Algebraic Curve over Q
- Modular Forms
- Modular Forms and the L-series of Elliptic Curves
- Statements of the Main Theorems
- How to get an Elliptic Curve from a Cusp Form
- Why the L-Series of E Agrees with the
L-series of f
- Wiles's Proof
- 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
- Local Class Field Theory: Lubin-Tate Extensions
- Cohomology of Groups.
- Local Class Field Theory Continued.
- Brauer Groups
- Global Class Field Theory: Statements
- L-series and the Density of Primes
- Global Class Field Theory: Proofs
- Complements (Power reciprocity laws; quadratic forms; etc.)