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April 13, 2012
Friday, April 13
Algebra Seminar
Time: 14:40
Speaker: Claudio Quadrelli (University of Milano-Bicocca)
Title: "Bloch-Kato groups and Galois groups?"
Room: MC 107

Abstract: Every profinite group is a Galois group, but which one is also an ${\textit{absolute}}$ Galois group? The cohomological implications of the Bloch-Kato conjecture -- positively solved by M.~Rost and V.~Voevodsky -- allows us to define ${\bf{Bloch-Kato}}$ ${\bf{pro-}}$$p$ ${\bf{groups}}$, which play a crucial role, since they arise naturally as maximal pro-$p$ quotients and Sylow pro-$p$ subgroups of absolute Galois groups. In this seminar I will present the state of the art of the research on Bloch-Kato groups, with a particular mention of the 'Elementary Type Conjecture' of maximal pro-$p$ Galois groups. Yet, there's still a lot of work to do: indeed every maximal pro-$p$ Galois group is equipped with an ${\textit{orientation}}$ $G_F(p)\rightarrow\mathbb{Z}_p^\times$, arising from the action on the group of the roots of unity of $p$-power order. The study of such orientation for Bloch-Kato groups will provide hopefully new results.