Obstructions to weak approximation for reductive groups over p-adic function fields

Poitou–Tate sequence for complex of tori over $p$-adic function fields

“…So far, the theorem was known when i = 1 by [Tia21,Theorem 1.18]. Thus the above result provides a generalization and complement to loc.…”

“…This paper is some sort of complement to [Tia21] where the author established some arithmetic duality results and obtained obstructions to weak approximation for connected reductive algebraic groups over K. In the present paper, we give a full picture of arithmetic duality results for a short complex of tori and deduce a 12-term PoitouśTate style exact sequence (in particular, we obtain such an exact sequence for groups of multiplicative type over K).…”

A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation

“…Actually the question that whether the canonical image of a certain map is unramified was first raised by Colliot-Thélène (see also [3,Remarque 4.3 (b)]). Later [11,Appendix] obtained a partial answer when the torsor is trivial. Therefore it is an interesting question to describe the cohomological obstruction to the Hasse principle using unramified cohomology groups for general torsors under tori.…”

A Comparison of Cohomological Obstructions to the Hasse Principle and to Weak Approximation

“…Actually the question that whether the canonical image of a certain map is unramified was first raised by Colliot-Thélène (see also [CT15, Remarque 4.3(b)]). Later [Tia21,Appendix] obtained a partial answer when the torsor is trivial. Therefore it is an interesting question to describe the cohomological obstruction to the Hasse principle using unramified cohomology groups for general torsors under tori.…”