Some arithmetic properties of linear groups over $p$-adic function fields

2021-12-27 15:00-16:00 Zoom

Abstract: In this talk, we first recall several classical results about the arithmetic of linear groups over number fields. Namely, we talk briefly about the Hasse principal, weak approximation and the Borel--Serre theorem. Subsequently, we introduce some arithmetic dualities associated to reductive groups and obstructions to weak approximation for certain reductive groups over $p$-adic function fields. These works motivate us to study weak approximation for semi-simple simply connected groups and related Galois cohomology theoretic results on the finiteness (trivialness) of the Tate--Shafarevich set.  If we still have some time, we will also talk about the obstruction to the Hasse principal for torsors under tori and obstructions to weak approximation for homogenous spaces under reductive groups.

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