Galois cohomology is the study of group cohomology for modules of Galois groups of field extensions. It is the natural continuation of previous classical results such as Kummer theory, and plays a crucial role in theories such as étale cohomology, class field theory (and its generalization as part of the Langlands program) and in the study of abelian varieties, such as elliptic curves. One of the main results of this theory is the Poitou-Tate duality, which consists of a series of duality statements captured in a 9-term exact sequence (Thm. 4.10 of [1]), and will be the main focus of this seminar.

We will begin with the basics of group cohomology and Galois cohomology, where the focus of these preliminaries will be tailored to the needs of the participants. Here we will follow mostly the references by Serre, supplementing with the book by Neukirch et. al. when needed.

We will then proceed with sections 1,2 and 4 of chapter 1 of our main reference [1], covering the duality theorems in Galois cohomology for class formations, local fields and global fields, respectively. Some theorems of class field theory will be stated and used (probably) without proof.

Depending on the number of participants and their interests, we may see applications to the computation of Euler-Poincaré characteristics and to the theory of abelian varieties, if time permits.

It is highly recommended to the participants of this seminar to follow the lecture "Introduction to étale cohomology". Likewise, participants of that lecture will profit from taking part in this seminar, as well as participants of the previous seminar on adelic number theory.

The preliminary meeting will be online via zoom on Friday, July 29th at 16:00 via the following link:

https://uni-regensburg.zoom.us/j/61742166352?pwd=ZTVuV0ZCVk5hQnk1N2lYUXpVVTJPUT09

Meeting-ID: 617 4216 6352

password: galois

If you could not join the preliminary meeting, but still would like to participate in the seminar, please send me an email: Julio.de-Mello-Bezerra"at"mathematik.uni-regensburg.de

J. S. Milne - Arithmetic Duality Theorems [1];

J. Neukirch, A. Smchmidt, K. Wingberg - Cohomology of Number Fields ;

J. P. Serre - Local Fields ;

J. P. Serre - Galois Cohomology .

• Organisational meeting/distribution of topics: Friday, July 29th at 16:00 via the following
  link: https://uni-regensburg.zoom.us/j/61742166352?pwd=ZTVuV0ZCVk5hQnk1N2lYUXpVVTJPUT09
  Meeting-ID: 617 4216 6352 password: galois
• If you would like to take part in this seminar, please write me an
  email: Julio.de-Mello-Bezerra"at"mathematik.uni-regensburg.de
• Registration for course work/examination/ECTS: FlexNow

• Presentation: Giving a seminar talk of roughly 90 minutes

• Detailed written report of the seminar talk