Algebraische Zahlentheorie 2

The course is a continuation of Gabor Wiese’s course. The central topic will be class field theory. We will start with local class field theory, following the approach of Lubin and Tate to explicitly construct the ‘‘reciprocity map’‘ using formal groups. In the second part of the course we will discuss global class field theory using cohomological methods.

Dates: Tue, 10-12, T03 R04 D10; Fri, 10-12, T03 R04 D10. The course starts on Tuesday, April 13.

Exercise sessions: There will be weekly exercise sessions (provisional date: Mon, 16-18, start: April 19).

Prerequisites: Algebraic Number Theory as covered in Gabor Wiese’s course (Global theory: finiteness of the ideal class group; Dirichlet’s unit theorem; ramification … Local theory: valuations, Hensel’s lemma)

Contact: Ulrich Görtz, ulrich.goertz@uni-due.de

Exercises		Due
Sheet 1	pdf	April 23
Sheet 2	pdf	April 30
Sheet 3	pdf	May 7
Sheet 4	pdf	May 14
Sheet 5	pdf	May 21
Sheet 6	pdf	May 28
Sheet 7	pdf	June 4
Sheet 8	pdf	June 11
Sheet 9	pdf	June 18
Sheet 10	pdf	June 25
Sheet 11	pdf	July 2
Sheet 12	pdf	July 9
Sheet 13	pdf	July 16

References

• Milne, Class field theory, see website
• Cassels, Fröhlich (eds.), Algebraic Number Theory
• Neukirch, Klassenkörpertheorie
• Neukirch, Algebraic Number theory
• Artin, Tate, Class field theory
• 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. math.NT/0606108
• Serre, Local fields / Corps locaux
• Fesenko, Vostokov, Local fields and their extensions, AMS Transl. of Math. Monographs 121

Plan of the course (provisional)

1. Introduction
2. Local class field theory
   1. The main theorems of local class field theory
   2. Remindes on local fields and discrete valuation rings
   3. Formal groups and Lubin-Tate theory
   4. The local Kronecker-Weber theorem
3. Cohomology of groups
4. Global class field theory
   1. Adeles, Ideles
   2. The main theorems of global class field theory