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# Local-Global principles for tori over arithmetic surfaces

## Julia Hartmann

Given a field $$F$$ and a collection of overfields $$F_i$$ ($$i \in I$$), we say that the local global principle holds for an $$F$$-variety $$Z$$ if the existence of a rational point over each $$F_i$$ 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. Parimala and V. Suresh.