# 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.