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 | crop | eReader | |
---|---|---|---|---|---|---|---|---|

GT | Group Theory | First | 15.03.13; v3.13; 135 pages | crop | pdf v3.11 | |||

FT | Fields and Galois Theory | First | GT | 31.08.15; v4.51; 138 pages | crop | pdf v4.30 | ||

AG | Algebraic Geometry | Second | FT | 22.08.15; v6.01; 226 pages | crop | |||

ANT | Algebraic Number Theory | Second | GT, FT | 28.05.14; v3.06; 164 pages | crop | pdf v3.03 | ||

MF | Modular Functions and Modular Forms | Second | GT, FT | ANT | 26.04.12; v1.30; 138 pages | crop | ||

EC | Elliptic Curves | Second | GT, FT | ANT | See books | |||

AV | Abelian varieties | Third | AG, ANT | CFT | 16.03.08; v2.00; 172 pages | crop | ||

LEC | Lectures on Etale Cohomology | Third | AG | CFT | 22.03.13; v2.21; 202 pages | crop | ||

CFT | Class Field Theory | Third | ANT | 23.03.13; v4.02; 289 pages | crop | |||

CM | Complex Multiplication | Third | ANT, AV | 07.04.06; v0.00; 113 pages | ||||

iAG | Algebraic Groups: algebraic group schemes over fields |
Third | AG | 20.12.15, v2.00; 528 pages | ||||

AGS | Basic Theory of Affine Group Schemes | Third | GT, FT | AG | 11.03.12, v1.00; 275 pages | |||

LAG | Lie Algebras, Algebraic Groups, and Lie Groups | Third | GT, FT | AG | 05.05.13, v2.00; 186 pages | |||

RG | Reductive Groups | Third | GT, FT | AG, AGS | 11.03.12, v1.00; 77 pages |

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.