Arithmetic of linear algebraic groups over function fields of p-adic curves and beyond

Hasse principle analogous to number fields has been studied in the context of function fields of p-adic curves.
An analogue of Hasse-Minkowski’s theorem holds in this context implying that 9-dimensional quadratic forms over such fields represent zero nontrivially. We discuss some questions and conjectures concerning homogeneous spaces under reductive groups over function fields of p-adic curves and more generally over function fields of curves over complete discrete valued fields.