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).
|GT||Group Theory||First||15.03.13; v3.13; 135 pages||pdf v3.11|
|FT||Fields and Galois Theory||First||GT||31.08.15; v4.51; 138 pages||pdf v4.30|
|AG||Algebraic Geometry||Second||FT||22.08.15; v6.01; 226 pages|
|ANT||Algebraic Number Theory||Second||GT, FT||28.05.14; v3.06; 164 pages||pdf v3.03|
|MF||Modular Functions and Modular Forms||Second||GT, FT||ANT||26.04.12; v1.30; 138 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||22.03.13; v2.21; 202 pages|
|CFT||Class Field Theory||Third||ANT||23.03.13; v4.02; 289 pages|
|CM||Complex Multiplication||Third||ANT, AV||07.04.06; v0.00; 113 pages|
|iAG||Algebraic Groups: An introduction to the
theory of algebraic group schemes over fields
|Third||AG||20.09.14, v1.00; 331 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 eReader files are formatted for viewing on eReaders (they have double the number of pages).
At last count, the notes included about 2015 pages.
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.
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.
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.
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.