Arithmetic of Linear Algebraic Groups over Two-dimensional Fields

In 1962 Serre posed a conjecture, now referred to as Conjecture II, which states that principal homogeneous spaces under semisimple simply connected linear algebraic groups over perfect fields of cohomological dimension two have rational points. In this talk, after summarising the status of Conjecture II, we shall discuss progress concerning the study of principal homogeneous spaces under linear algebraic groups over function fields of two-dimensional schemes: surfaces over algebraically closed fields, strict Henselian two dimensional local domains and arithmetic surfaces that are relative curves over p-adic integers.