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. I am (slowly) in the process of producing final versions of them and publishing them. Please continue to 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.
I occasionally search math.stackexchange for questions on my notes, but I no longer respond on that site.
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 | June 2021; v4.00; 139p | pdf v3.11 | ||||
| FT | Fields and Galois Theory | First | GT | Sept. 2022; v5.10; 144p | pdf v4.30 | |||
| AG | Algebraic Geometry | Second | FT | Nov. 11, 2024; v6.10; 231p | ||||
| ANT | Algebraic Number Theory | Second | GT, FT | July 2020; v3.08; 166p | crop | pdf v3.03 | ||
| MF | Modular Functions and Modular Forms | Second | GT, FT | ANT | March 2017; v1.31; 134p | crop | ||
| EC | Elliptic Curves | Second | GT, FT | ANT | See books | |||
| AV | Abelian varieties | Third | AG, ANT | CFT | March 2008; v2.00; 172p | crop | ||
| LEC | Lectures on Etale Cohomology | Third | AG | CFT | March 2013; v2.21; 202p | crop | ||
| CFT | Class Field Theory | Third | ANT | August 2020; v4.03; 296p | crop | |||
| CM | Complex Multiplication | Third | ANT, AV | July 2020; v0.10; 108p | ||||
| iAG | Algebraic Groups | Third | AG | See books | ||||
| LAG | Lie Algebras, Algebraic Groups, and Lie Groups | Third | GT, FT | AG | May 2013, v2.00; 186p | |||
| RG | Reductive Groups | Third | GT, FT | AG, AGS | March 2018, v2.00; 139p |
If the pdf files are placed in the same directory,
some links will work between files (you may have to get the correct version
and rename it, e.g., get AG510.pdf and rename it AG.pdf).
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 over 2022 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
This is a basic first course. In contrast to most such accounts the notes study 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.
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.
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.
Complex Multiplication
These are preliminary notes for a modern exposition of the theory of complex multiplication.
Algebraic groups, Lie algebras, Lie groups; reductive groups.
The notes provide a modern exposition of these topics.
