Appears in collection : Rational Points on Fano and Similar Varieties

Given a field F and a collection of overfields Fi (i ∈ I), we say that the local global principle holds for an F-variety Z if the existence of a rational point over each Fi implies the existence of an F-rational point.In this talk, we study this question when F is a semi-global field, i.e., the function field of a curve X over a complete discretely valued field, and Z is a principal homogeneous space under a torus. It is known that a local-global principle need not hold in general. We give a formula which often leads to an explicit description of the obstruction set in the case when the torus is defined over X. This is joint work with J.L. Colliot-Thélène, D. Harbater, D. Krashen, R. Pari- mala and V. Suresh.