Time: 9h-11h, Thursday, 19/9/2019

Venue: Seminar room 302, Building A5, Institute of Mathematics Hanoi.

Abstract: In this report, I am going to talk about the topic ``Brauer--Manin obstruction for curves and surfaces'', in which I focus on the Brauer--Manin obstruction for textbf{Markoff surfaces}, a type of cubic surfaces as studied in my Master thesis.

Before coming to the main topic, I will recall briefly some basic results from ``Brauer groups of fields'' and ``Brauer groups of varieties'', including Azuyama algebras, cyclic algebras and their cohomological intepretations using Galois cohomology and Étale cohomology. Then I will introduce the basic theory of Brauer--Manin obstruction as a specific case of functorial obstructions to the Hasse principal. Afterward, I will give some examples for curves and surfaces, which places emphasis on the Brauer--Manin obstruction for Markoff cubic surfaces, whose Brauer groups are explicitly computed. During the talk, I will also mention the essential tools for computing Brauer groups such as Hochschild--Serre spectral sequences and textit{residue maps}, as well as the (local) Hasse invariant maps and the Hilbert reciprocity law for determining Brauer--Manin obtructions.

If the time allows, I will introduce some computational results of the frequency of Brauer--Manin obstructions for Markoff surfaces as discussed in my Master thesis. In the end, we may look at some open questions about cohomological obstructions to rational points on varieties.