It was a compulsion for Artin to present each argument in its purest form, to replace computation by conceptual arguments, to strip the theory of unnecessary ballast. What was the decisive point for him was to show the beauty of the subject to the reader. He himself has said: " We all believe that mathematics is an art. The author of a book, the lecturer in a classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this attempt, he must always fail."
Richard Brauer, BAMS 73 (1967), p38.
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.
At last count, the notes included about 1700 pages.
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).
| Course | Year | Required | Useful | Version | ||
|---|---|---|---|---|---|---|
| GT | Group Theory | First | 01.09.07; v3.00; 121 pages | |||
| FT | Fields and Galois Theory | First | GT | 11.02.08; v4.20; 111 pages | ||
| AG | Algebraic Geometry | Second | FT | 19.03.08; v5.10; 241 pages | ||
| ANT | Algebraic Number Theory | Second | GT, FT | 11.02.08; v3.00; 163 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 | |
| AAG | Algebraic groups and Arithmetic Groups | Second | GT, FT | AG | 04.06.06; v1.01; 219 pages | |
| AV | Abelian varieties | Third | AG, ANT | CFT | 16.03.08; v2.00; 172 pages | |
| LEC | Lectures on Etale Cohomology | Third | AG | ANT | 09.08.98; v2.01; 190 pages | |
| CFT | Class Field Theory | Third | ANT | 02.03.08; v4.00; 287 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.
Algebraic groups and arithmetic groups.
These notes provide an introductory overview of the theory
of algebraic groups, Lie algebras, Lie groups, and arithmetic groups.
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 new version (1997) is heavily revised and expanded.
