Poincaré institute

Reinventing rational points

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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.