Quadratic forms over function fields of p-adic curves

A classical theorem of Hasse-Minkowski asserts that a quadratic form over a number field is isotropic, i.e., it represents zero nontrivially, if it is isotropic over completions at all its places. As a consequence, every quadratic form in at least five variables over a totally imaginary number field is isotropic. A natural question is to extend such a Hasse principle to function fields of curves over p-adic fields and number fields. We discuss some recent results in this direction as well as similar Hasse principle for rational points on homogeneous spaces under connected linear algebraic groups.