Arithmetic of algebraic groups over 2-dimensional fields

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For a connected linear algebraic group defined over field there has been considerable progress in recent years in the study of homogeneous spaces over function fields of curves over a p-adic field. It has been proved recently by Parimala and Suresh that every quadratic form in at least nine variables over the function field of a p-adic curve has a nontrivial zero. Further, patching techniques developed by Harbater-Hartmann-Krashen have provided tools to study certain local-global principle for existence of rational points on homogeneous spaces over such fields under the assumption that the algebraic group is rational. We propose to study homogeneous spaces under linear algebraic groups over function fields of curves over local and global fields. The study over function fields over global fields would have tremendous consequences and could lead to the fact that every quadratic form in large enough number of variables over function fields of curves over totally imaginary number fields has a nontrivial zero. We propose to study questions of Hasse principle for existence of rational points on homogeneous spaces under connected linear algebraic groups over such fields. In the context of simply connected groups, the Rost invariant for principal homogeneous spaces is a powerful tool to study these spaces. We propose to study obstruction to the injectivity of the Rost invariant for function fields of curves over local and global fields.The study of homogeneous spaces under linear algebraic groups includes study of several interesting algebraic objects like quadratic forms, central simple algebras, Octonian and Albert algebras. The study over number fields is enriched by class field theory techniques. A classical theorem of Hasse-Maass-Schiling for number fields is the following Hasse principle: an element in the number field is a reduced norm from a central simple algebra if and only if it is positive at all real completions where the division algebra is ramified. A similar criterion for reduced norms for 2-dimensional fields is a far-reaching extension of this classical result for number fields. We propose to replace class field theory by purely homological properties of these fields and the geometry of the associated arithmetic surfaces to study properties of homogeneous spaces in the general setting of function fields of curves over local and global fields.