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