When deciding the existence of integer solutions of a system of polynomial equations with integer coefficients (defining a variety X), one could first reduce the problem modulo every integer N, which is equivalent to considering solutions in every Z_p (the p-adic integers). Similarly, we can consider rational solutions, by first looking at the candidate solutions “locally” in every p-adic field Q_p. Does the existence of "local" solutions in every Q_p give a "global" solution over Q (known as "local-global principle")? Sometimes yes (e.g Hasse-Minkowski theorem for quadratic forms), but not always. In fact, when local points ∏_p X(Q_p) which potentially come from global points X(Q) are paired with elements of the Brauer group Br(X) or other cohomology groups, they should satisfy certain restrictions, e.g. the exact sequence from Class Field Theory relating Br(Q) to Brauer groups of all the local Q_p, and this defines the Brauer-Manin obstruction. This obstruction is enough to detect the existence of rational points when X satisfies certain properties, e.g. being homogeneous spaces of some nice algebraic groups like tori.
When there do exist rational points, sometimes we can simultaneously approximate Q_p-points for finitely many places p by a ration point, i.e. X(Q) is dense in ∏_pX(Q_p), known as weak approximation. Similarly, we look for obstructions when this doesn't hold: we hope that the closure of X(Q) in ∏_p X(Q_p) should be in some (closed) subset, e.g. the one defined by the Brauer-Manin obstruction.
Now, instead of working over number fields, we want to generalize these results to function fields of complex curves or surfaces, where we can define the counterpart of local p-adic completions using valuations given by codimension 1 points.
