We show that the Brauer–Manin obstructions to the Hasse principle and weak approximation for homogeneous spaces under connected reductive groups over global function fields with connected reductive stabilizers are the only ones, extending some of Borovoi’s results (and thus also proving a partial case of a conjecture of Colliot-Thélène) in this regard. Along the way, we extend some perfect pairings and an important local-global exact sequence (an analog of a Cassels–Tate’s exact sequence) proved by Sansuc for connected linear algebraic groups defined over number fields, to the case of connected reductive groups over global function fields and beyond.

Let $E/K$ be an elliptic curve with $j$-invariant 1728 defined over a number field $K$. In this note, we give a simple condition on $K$ which determines whether all quartic twists of $E/K$ have the same root number or not. This completes a series of works on the same root number of twists begun in [DD1] and [BK].

This is the sequel to the author’s previous paper [1] with Matthias Franz. In the present paper, we introduce the notion of equivariant total Chern class of a GKM graph and show that the pair of graph equivariant cohomology and the equivariant total Chern class determines the GKM graph completely. We also show that for a torus graph in the sense of Maeda–Masuda–Panov, the pair of graph equivariant cohomology and the equivariant 1-st Chern class determines the torus graph completely.