Many arithmetic questions such as the existence of rational points or the classification of forms over a base field can be naturally formulated in terms of "Galois cohomology", the group cohomology associated with the action of a Galois group on an abelian or non-abelian group. This lecture intends to convey a working knowledge of this method and to apply it in particular to the classification of simple linear algebraic groups over number fields. The course will start with a recap of necessary tools from algebraic number theory. Afterwards, we will consider Galois cohomology of finite abelian modules and discuss the corresponding local-global principle called Poitou-Tate duality. In the final part of the lecture, we will use this result to obtain a corresponding local-global principle for simple groups over number fields.
Begin: Thu., 10/10/2024
Time/place: Thu., 10:30 a.m. - 12:00 p.m., seminar room 25.22.U1.72, HHU
Prerequisites: The lecture is designed for the PhD students in our joint Research Training Group 2240 with the University of Wuppertal. It is however also open for algebraically inclined Master students and is suitable to prepare oneself for writing a Master thesis in the field. For further information, please contact me via email.
Begin: Fr.,10/25/2024
Time/place: Fr., 04:30 p.m. - 6:00 p.m., seminar room 25.22.02.81, every fortnight
Master students who want to earn credit points for this course will have to take part in the exercise class. Homework problems will be provided in class.
• Jean-Pierre Serre Galois cohomology, Springer Monographs in Mathematics, available when connected to the campus network on SpringerLink.
• Jean-Pierre Serre, Local Fields, Springer GTM 67, available when connected to the campus network on SpringerLink.
• David Harari, Galois Cohomology and Class Field Theory, Springer Universitext, available when connected to the campus network on SpringerLink.
