Class Field Theory
This is a website for the Part III (non-examinable, graduate) course Class Field Theory (2012, L24); here are some material for this course.
(from the official website of the department)
Meetings: Tue / Thu / Sat 11am, at MR 12.
First lecture: Th., 19th January, 2012.
Last lecture: Tu., 13th March, 2012.
p.37 of the [pdf] Mathematical Tripos Part III Lecture Courses.
I may post some typed/scanned lecture notes here. There will be some overlap with my notes from the Postech Summer School:
[pdf] Notes for Postech Summer School 2010.
The goal would be to cover the proofs of local/global class field theory for number fields, with no essential part left out. We use a "modern" language, namely adeles/ideles and Galois cohomology. This means that we may need to skip some straightforward exercises in commutative/homological algebra, but I will not assume them as prerequisites (i.e. will give all the necessary definitions), and try not to omit the number theoretic arguments.
I don't intend to make much changes from the same course given last year.
Lecture 1 (Th. 19/1/12) [scan]: 1.1 Preliminaries (tensor products, limits)
Handout on Categories and Functors (updated after the lecture)
Lecture 2 (Sa. 21/1/12) [scan]: 1.2 Galois theory (infinite extensions, cyclotomic characters)
Lecture 3 (Tu. 24/1/12) [scan]: 1.3 Dedekind domains
Lecture 4 (Th. 26/1/12) [scan]: 1.4 Cyclotomic extensions, CFT of Q
Lecture 5 (Sa. 28/1/12) [scan]: 1.5 CFT of Q via adeles
Lecture 6 (Tu. 30/1/12) [scan]: 1.6 Local fields, Weil groups
Lecture 7 (Th. 2/2/12) [scan]: 2.1 A statement of local CFT
Lecture 8 (Sa. 4/2/12) [scan]: 2.2 Lubin-Tate groups
Lecture 9 (Tu. 7/2/12) [scan]: 2.3 Lubin-Tate extensions
Lecture 10 (Th. 9/2/12) [scan]: 2.3 (continued: Lubin-Tate spaces)
Lecture 11 (Sa. 11/2/12) [scan] : 2.4 The Artin map
Lecture 12 (Tu. 14/2/12) [scan]: 2.5 Norm groups
Handout: lemmas on Coleman operators / Handout: a lemma on infinite norms
Handout on Categories and Functors + Homological Algebra
Lecture 13 (Th. 16/2/12) : 3.1 Group Cohomology
Lecture 14 (Sa. 18/2/12) : 3.2 Res-Inf sequence / 3.3 Standard complex / 3.4 Tate Cohomology
Handout: the Res-Inf exact sequence
Lecture 15 (Tu. 21/2/12) : 3.5 Shift morphisms
Handout: the shift morphisms
Lecture 16 (Th. 23/2/12) : 3.6 Tate's theorem
Lecture 17 (Sa. 25/2/12) : 4.1 Galois cohomology / 4.2 Class formations
Lecture 18 (Tu. 28/2/12) : 4.3 Local CFT
Handout: an alternative proof of existence theorem
Lecture 19 (Th. 1/3/12) : 5.1 Number fields
Lecture 20 (Sa. 3/3/12) : 5.2 Cohomology of idele groups
Handout: functoriality of Shapiro's lemma
Lecture 21 (Tu. 6/3/12) : 5.3 Global theorems / 1st inequality
Handout: weak approximation for ideles
Lecture 22 (Th. 8/3/12) : 5.4 2nd inequality
Lecture 23 (Sa. 10/3/12) : 5.4 (continued) / 5.5 Invariant maps
Handout: local power index
Lecture 24 (Tu. 13/3/12) : 5.5 (continued) / 5.6 Global CFT
Lecture 25 (Th. 15/3/12) [additional] : 5.6 (continued)
Handout: global existence theorem
Here are some references - they are not essential at all.
M. Reid, Undergraduate Commutative Algebra, CUP (London Mathematical Society Student Texts), 1996.
M.F. Atiyah, I.G. MacDonald, Introduction to Commutative Algebra, Westview Press, 1994.
N. Bourbaki, Commutative Algebra: Chapters 1-7, Springer, 1998.
C.A. Weibel, An Introduction to Homological Algebra, CUP (Cambridge Studies in Advanced Mathematics), 1995.
A. Grothendieck, Sur quelques points d'alg'ebre homologique (translation), Tohoku Mathematics Journal 9-2 (1957), 119-221.
S. Maclane, Categories for the Working Mathematician, Springer (Graduate Texts in Mathematics), 1998.
H. Cartan, S. Eilenberg, Homological Algebra, Princeton UP, 1956.
J.-P. Serre, Local Fields, Springer (Graduate Texts in Mathematics), 1980.
K. Iwasawa, Local Class Field Theory, Oxford UP, 1986.
T. Yoshida, Local class field theory via Lubin-Tate theory, Annales de la Faculte des Sciences de Toulouse, Ser. 6, 17-2 (2008), 411-438.
E. Artin, J. Tate, Class Field Theory, Benjamin, 1967.
J.W.S. Cassels, A. Fr"ohlich, Algebraic Number Theory, Academic Press, 1967.
A. Weil, Basic Number Theory, Springer (Classics in Mathematics), 1995.
J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, Springer (Grundlehren der Mathematischen Wissenschaften 323), 2000.
2011 course website