website for
Class Field Theory
(2012, L24)

This is a website for the Part III (non-examinable, graduate) course Class Field Theory (2012, L24); here are some material for this course.

______time/syllabus
(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.

    ______notes/plan

    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

    ______literature

    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.

    ______old stuff

  • 2011 course website

    • Last modified: March 14, 2012.