Summary: We conjecture that if \(C\) is a curve of genus \(>1\) over a number field \(k\) such that \(C(k)=\emptyset\), then a method of Victor Scharaschkin [Local-global problems and the Brauer-Manin obstruction. Ph. D. thesis, University of Michigam (1999)] (essentially equivalent to the Brauer-Manin obstruction in the context of curves) supplies a proof that \(C(k)=\emptyset\). As evidence, we prove a corresponding statement in which \(C(\mathbb F_v)\) is replaced by a random subset of the same size in \(J(\mathbb F_v)\) for each residue field \(\mathbb F_v\) at a place \(v\) of good reduction for \(C\), and the orders of Jacobians over finite fields are assumed to be smooth (in the sense of having only small prime divisors) as often as random integers of the same size. If our conjecture holds, and if Tate-Shafarevich groups are finite, then there exists an algorithm to decide whether a curve over \(k\) has a \(k\)-point, and the Brauer-Manin obstruction to the Hasse principle for curves over the number fields is the only one.