(math.stackexchange.com 3079835).

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 pdf crop eReader GT Group Theory First



July 2020; v3.16; 137p pdf crop pdf v3.11 FT Fields and Galois Theory First GT

April 2020; v4.61; 138p pdf crop pdf v4.30 AG Algebraic Geometry Second FT

March 2017; v6.02; 221p pdf



ANT Algebraic Number Theory Second GT, FT

July 2020; v3.08; 166p pdf crop pdf v3.03 MF Modular Functions and Modular Forms Second GT, FT ANT March 2017; v1.31; 134p pdf crop

EC Elliptic Curves Second GT, FT ANT See books

AV Abelian varieties Third AG, ANT CFT March 2008; v2.00; 172p pdf crop LEC Lectures on Etale Cohomology Third AG CFT March 2013; v2.21; 202p pdf crop CFT Class Field Theory Third ANT

August 2020; v4.03; 296p pdf crop CM Complex Multiplication Third ANT, AV

July 2020; v0.10; 108p pdf iAG Algebraic Groups Third AG See books LAG Lie Algebras, Algebraic Groups, and Lie Groups Third GT, FT AG May 2013, v2.00; 186p pdf RG Reductive Groups Third GT, FT AG, AGS March 2018, v2.00; 139p pdf

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 2020 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.