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.
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 | |
|---|---|---|---|---|---|---|
| GT | Group Theory | First | 17.05.08; v3.01; 124 pages | |||
| FT | Fields and Galois Theory | First | GT | 28.09.08; v4.21; 111 pages | ||
| AG | Algebraic Geometry | Second | FT | 19.03.08; v5.10; 241 pages | ||
| ANT | Algebraic Number Theory | Second | GT, FT | 30.04.09; v3.02; 164 pages | ||
| MF | Modular Functions and Modular Forms | Second | GT, FT | ANT | 22.05.97; v1.10; 128 pages | |
| EC | Elliptic Curves | Second | GT, FT | ANT | See books | |
| AV | Abelian varieties | Third | AG, ANT | CFT | 16.03.08; v2.00; 172 pages | |
| LEC | Lectures on Etale Cohomology | Third | AG | CFT | 20.05.08; v2.10; 196 pages | |
| CFT | Class Field Theory | Third | ANT | 02.03.08; v4.00; 287 pages | ||
| BAG | Algebraic groups, Lie groups, and their arithmetic subgroups | Third | GT, FT | AG | 10.05.09; v1.01; 192 pages | |
| CM | Complex Multiplication | Third | ANT, AV | 07.04.06; v0.00; 113 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 groups, Lie groups, and their
arithmetic subgroups (and they will supersede the notes Algebraic groups and arithmetic groups)
. At present, only the first two chapters (of six)
are available.
Complex Multiplication
These are preliminary notes for a modern exposition of the theory of complex multiplication.
At last count, the notes included about 1700 pages.
