A non-abelian conjecture of Birch and Swinnerton-Dyer type for
hyperbolic curves

We state a conjectural criterion for identifying global integral points on a
hyperbolic curve over $\Z$ in terms of Selmer schemes inside non-abelian
cohomology functors with coefficients in $\Q_p$-unipotent fundamental groups.
For $\P^1\setminus \{0,1,\infty\}$ and the complement of the origin in
semi-stable elliptic curves of rank 0, we compute the local image of global
Selmer schemes, which then allows us to numerically confirm our conjecture in a
wide range of cases.