Group cohomology, Galois cohomology and arithmetic 1000-1M09KGA
In the last 50 years Galois cohomology has provided indispensable tools in arithemtic algebraic geometry for solving a number of fascinating problems, such as Fermat's Last Theorem. The main goal of the lecture is to present fundamental properties and results concerning group cohomology, Galois theory and Galois cohomology. I would also like to present some of the applications of Galois cohomology in arithmetic algebraic geometry, e.g., Kummer theory, Hilbert Theorem 90, Weak Mordell-Weil Theorem, Fermat's Last Theorem etc.
Course topics:
1. Field extensions. Galois theory.
2. Modules, complexes, cohomology.
3. Group cohomology.
4. Profinite groups. The group $G(\overline{F} /F)$.
5. Cohomology of profinite groups. Galois cohomology.
6. Theorems of Dedekind and Kummer. Hilbet 90 Theorem.
7. l-adic representations and Kummer theory.
8. Selmer and Tate-Shafarevich groups.
Type of course
Bibliography
1. M. Atiyach, K. Hall Cohomology of groups article in Algebraic number theory J.W.S Cassels, A. Frohlich ed. Academic Press 1967
2. K.S. Brown Cohomology of groups Springer 1982
3. K. Greenberg Profinite groups article in Algebraic number theory J.W.S Cassels, A. Frohlich ed. Academic Press 1967
4. Y. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number fields, Springer 2000
5. J-P Serre, Galois cohomology, Springer 1994
6. L, Washington Galois cohomology article in Modular forms and Fermat's last theorem G. Cornell, J.H. Silverman, G. Stevens ed., Springer 1997
7. C. Weibel, An introduction to homological algebra, Cambridge Univ. Press, 1994.
Additional information
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