Local-global principle for groups of type An over semi global fields

TL;DR

This paper investigates the local-global principle for certain algebraic groups over semi-global fields, providing a description of obstructions and conditions under which they vanish, advancing understanding in algebraic geometry and number theory.

Contribution

It offers a new description of obstructions to the local-global principle for groups of type An over semi-global fields, and identifies conditions for their vanishing.

Findings

Obstructions are described via R-equivalence classes.

The obstruction vanishes under specific residue field conditions.

Results extend understanding of local-global principles in algebraic groups.

Abstract

Let F be the function field of a curve over a complete discretely valued field K. Let G be a semisimple simply connected linear algebraic group over F of type An. We give a description of the obstruction to local global principle for principal homogeneous spaces under G over F with respect to discrete valuations of F in terms of R-equivalence classes of G over some suitable over fields. Using this description we prove that this obstruction vanishes under some conditions on the residue field K.