Course Notes - J.S. Milne Top
These are full notes for all the advanced (graduate-level) courses I have taught since 1986. Some of the notes give complete proofs (Group Theory, Fields and Galois Theory, Algebraic Number Theory, Class Field Theory, Algebraic Geometry), while others are more in the nature of introductory overviews to a topic. They have all been heavily revised from the originals. One day I may publish some of them as books, but until I do they are living documents, so please send me corrections (especially significant mathematical corrections) and suggestions for improvements.

Errata:This is a list of errors and additional comments not yet incorporated into the files on the web, mainly contributed by readers.

The following table indicates how advanced a course is (first, second, or third year graduate course in North American universities), and which courses are prerequisites for it (or would be useful).

Link Course Year Required Useful Version pdf crop eReader
GT Group Theory First

17.03.17; v3.14; 135 pages pdf crop pdf v3.11
FT Fields and Galois Theory First GT
27.05.17; v4.53; 138 pages pdf pdf v4.30
AG Algebraic Geometry Second FT
19.03.17; v6.02; 221 pages pdf

ANT Algebraic Number Theory Second GT, FT
18.03.17; v3.07; 165 pages pdf crop pdf v3.03
MF Modular Functions and Modular Forms Second GT, FT ANT 22.03.17; v1.31; 134 pages pdf crop
EC Elliptic Curves Second GT, FT ANT See books
AV Abelian varieties Third AG, ANT CFT 16.03.08; v2.00; 172 pages pdf crop
LEC Lectures on Etale Cohomology Third AG CFT 22.03.13; v2.21; 202 pages pdf crop
CFT Class Field Theory Third ANT
23.03.13; v4.02; 289 pages pdf crop
CM Complex Multiplication Third ANT, AV
07.04.06; v0.00; 113 pages pdf
iAG Algebraic Groups:
algebraic group schemes over fields
Third AG 20.12.15, v2.00; 528 pages pdf
AGS Basic Theory of Affine Group Schemes Third GT, FT AG 11.03.12, v1.00; 275 pages pdf
LAG Lie Algebras, Algebraic Groups, and Lie Groups Third GT, FT AG 05.05.13, v2.00; 186 pages pdf
RG Reductive Groups Third GT, FT AG, AGS 11.03.12, v1.00; 77 pages pdf

If the pdf files are placed in the same directory, some links will work between files.
The pdf files are formatted for printing on a4/letter paper.
The cropped files have had their margins cropped --- may be better for viewing on gadgets.
The eReader files are formatted for viewing on eReaders (they have double the number of pages).
At last count, the notes included about 2016 pages.

Group Theory
A concise introduction to the theory of groups, including the representation theory of finite groups.

Fields and Galois Theory
A concise treatment of Galois theory and the theory of fields, including transcendence degrees and infinite Galois extensions.

Algebraic Geometry
An introductory course. In contrast to most such accounts it studies abstract algebraic varieties, and not just subvarieties of affine and projective space. This approach leads more naturally into scheme theory.

Algebraic Number Theory
A fairly standard graduate course on algebraic number theory.

Modular Functions and Modular Forms
This is an introduction to the arithmetic theory of modular functions and modular forms, with a greater emphasis on the geometry than most accounts.

Elliptic Curves
This course is an introductory overview of the topic including some of the work leading up to Wiles's proof of the Taniyama conjecture for most elliptic curves and Fermat's Last Theorem.
These notes have been rewritten and published as a book.

Abelian Varieties
An introduction to both the geometry and the arithmetic of abelian varieties. It includes a discussion of the theorems of Honda and Tate concerning abelian varieties over finite fields and the paper of Faltings in which he proves Mordell's Conjecture. Warning: These notes are less polished than the others.

Lectures on Etale Cohomology
An introductory overview. In comparison with my book, the emphasis is on heuristics rather than formal proofs and on varieties rather than schemes, and it includes the proof of the Weil conjectures.

Class Field Theory
This is a course on Class Field Theory, roughly along the lines of the articles of Serre and Tate in Cassels-Fröhlich, except that the notes are more detailed and cover more. The have been heavily revised and expanded from earlier versions.

Algebraic groups, Lie groups, and their arithmetic groups.
Eventually, these notes will provide a modern exposition of the theory of algebraic group schemes, Lie algebras, Lie groups, and their arithmetic subgroups.

Complex Multiplication
These are preliminary notes for a modern exposition of the theory of complex multiplication.