McGill University

Department of Mathematics & Statistics

Class Field Theory

189-596A

Course Blog

Lecture 1. Wednesday, September 5.
Overview of the course. Statement of the Kronecker Weber theorem. Gauss's proof of quadratic reciprocity using Gauss sums.

Lecture 2. Friday, September 7.
End of the overview. A quick review of local fields. Unramified and totally ramified extensions. Statement of the local Kronecker-Weber theorem.

Lecture 3. Tuesday, September 11.
Reduction of the Kronecker-Weber theorem to its local variant. A brief digression through Kummer theory. Some basic facts about local fields.

Assignment 1 is now available!!

Lecture 4. Friday, September 14.
Further discussion of Kummer theory and of local fields. Beginning of the proof of the local Kronecker-Weber theorem.

Lecture 5. Monday, September 17.
End of proof of the local Kronecker-Weber theorem. Beginning of local class field theory. Statement of the local reciprocity law of class field theory.

Lecture 6. Tuesday, September 18.
Statement of Lubin-Tate theory. Formal groups.

Lecture 7. Monday, September 24.
Assignment 1 to be handed in today.
Formal groups.

Lecture 8. Tuesday, September 25.
Lubin-Tate theory, following chapter I.2 of Milne.

Lecture 9. Friday, September 28.
Abelian extensions arising from Lubin-Tate formal groups.
Assignment 2 is now available!!

Lecture 10. Tuesday, October 9.
The local reciprocity map in terms of Lubin Tate formal groups. Proof that it is independent of the choice of uniformiser.

Lecture 11. Friday, October 12.
Preliminaries on group cohomology. Derived functors.

Lecture 12. Monday, October 15.
Group cohomology, continued. Homology of groups.

Lecture 13. Tuesday, October 16.
Group cohomology, continued. The inflation-restriction sequence. Tate cohomology groups.
Assignment 3 is now available!!

Lecture 14. Monday, October 22.
More Galois cohomology. Cohomology of profinite groups.

Lecture 15. Tuesday, October 23.
Galois cohomology of local fields. The Galois cohomology of the multiplicative group of unramified extensions.

Lecture 16. Monday, October 29.
Galois cohomology of local fields. The Galois cohomology of the multiplicative group of unramified extensions.

Lecture 17. Tuesday, October 30.
Assignment 3 is due today.
Tate's theorem on class formation. Description of the local reciprocity map for unramified extensions.

Lecture 18. Monday, November 5.
The cohomology of ramified extensions. The local Artin map.

Assignment 4 is now available!!!

Lecture 19. Tuesday, November 6.
End of the proof of the main results of local class field theory.

Lecture 20. Monday, November 12.
Global class field theory. Statements of the main theorems.

Lecture 21. Tuesday, November 13.
Global class field theory. Statement in terms of Ideles. Cohomology of the Ideles.

Lecture 22. Monday, November 19.
Cohomology of the S-units, and of the Idele class group. The first inequality.

Lecture 23. Tuesday, November 20.
Cohomology of the Idele class group. The second inequality.

Assignment 5 is now available!!!

(With apologies for the tight deadline. I think it is shorter, and, hopefully, more manageable than some of the other ones.)

Lecture 24. Monday, November 26.
$L$-functions. Proof of the second inequality.

Lecture 25. Tuesday, November 27.
The second inequality, cont'd.

Lecture 26. Monday, December 3.
Proof of the global reciprocity law, final words. Some review.

Final Exam Friday, December 7, 10:00 AM to 1:00 PM.
To be written in Burnside Hall 920.