Math 596/720 – Class Field Theory

This course will be a course on class field theory. Class field theory describes Galois groups of abelian Galois extensions of local and global fields in terms of the internal arithmetic of the fields.

We aim to cover both local and global class field theory and to develop a little bit of the theory of formal groups and Galois cohomology along the way.

Technical details

Instructor: Patrick Allen

Email:

Time and location: MW, 2:05 – 3:25 in 1205 BURN.

Prerequisites: Graduate level algebra and a first course in algebraic number theory.

Evaluation: Parts of this course will be conducted in a seminar style and students will be required to give some presentations. There may also be some homework.

References

We will closely follow Milne's notes:

• J.S. Milne. Class field theory.

There are many other references for class field theory. For the purely local story, I suggest:

• Serre, Jean-Pierre. Local fields. Translated from the French by Marvin Jay Greenberg. Graduate Texts in Mathematics, 67. Springer-Verlag, New York-Berlin, 1979.
• Iwasawa, Kenkichi Local class field theory. Oxford Science Publications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1986.

For the global (and local) theory, I suggest:

• Childress, Nancy. Class field theory. Universitext. Springer, New York, 2009.
• Algebraic number theory. Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich Academic Press, London; Thompson Book Co., Inc., Washington, D.C. 1967.
• Lang, Serge Algebraic number theory. Second edition. Graduate Texts in Mathematics, 110. Springer-Verlag, New York, 1994.

For some historical perspective, I recommend:

• Wyman, B. F. What is a reciprocity law? Amer. Math. Monthly 79 (1972), 571–586; correction, ibid. 80 (1973), 281.

And finally, for a mix of historical perspective and recent developments in nonabelian reciprocity, I recommend:

• Weinstein, Jared. Reciprocity laws and Galois representations: recent breakthroughs. Bull. Amer. Math. Soc. (N.S.) 53 (2016), no. 1, 1–39.

Course Log

Jan 5: Introduction.

Jan 9: The statements of local class field theory.

Jan 11: Lubin–Tate formal group laws.

Jan 16: No class.

Jan 18: No class.

Jan 23: Totally ramified extensions cut out by Lubin–Tate formal groups.