Monthly Archives: July 2013

SNU Topology of number fields, week of 15 July

I was looking over the notes and saw a few items omitted by Milne.

First, in the proof of Hilbert’s theorem 90, there is no proof of the linear independence of distinct characters on a group. You can try to figure it out yourself (quite easy) or look at the notes with that title Keith Conrad’s webpage. (For some strange reason, I can’t create a direct link to the paper.)

Secondly, Milne has a cryptic remark without definition or proof about a ‘Verlagerung’ (transfer map) in Proposition 3.2. This is reference to the fact that the restriction map for Tate cohomology in degree minus 2 corresponds to a classical formula for a map between abelianizations. This proof is not so straightforward, and can be found in Prop.11.12 of these nice notes by Holden Lee. It seems they’re from a reading course he took with Professor Shin Sug Woo.

If some of you would like to read a more leisurely exposition of group cohomology, I found online these notes by Kenneth Brown.

Finally, this old paper of Oort contains a discussion of extensions and cup products.

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SNU Topology of Number Fields

Here are some documents to help you with the course.

1. General Motivation:

Lecture at the Cambridge workshop on non-abelian fundamental groups in arithmetic geometry (2009)

Lecture at Bordeaux conference in honor of Martin Taylor’s 60th birthday (2012)

Lecture at British Mathematical Colloquium (2011)

Lecture at AMC, Busan (2013)

2. James Milne’s lecture notes:

Class Field Theory

Algebraic Number Theory

Arithmetic Duality Theorems

3. Langlands programme:

Langlands’s lecture at Helsinki ICM

Report on work of Bao Chau Ngo (with Sugwoo Shin)

4. Topology of number fields:

Mazur’s paper on etale cohomology of number fields

Morishita’s paper on knots and primes

Furusho’s paper on Galois action on knots