Let S be an open subscheme of Spec Z and let X be an S-model of a hyperbolic curve. In the last decade, Minhyong Kim has developed a new approach to the study of integral points which uses Deligne's theory of the unipotent fundamental group to construct certain subsets X(Zp)_n of the set of Z_p-points which contain X(S) and are conjectured to converge to X(S) as n grows. In the special case of the punctured line, the unipotent fundamental group is known to be motivic, opening the door to motivic methods. Our main goal in this project is to use Goncharov's theory of motivic iterated integrals, as well as methods developed by Francis Brown, to construct an algorithm for computing the sets X(Z_p)_n for the thrice punctured line over Q, as well as for more general curves over more general bases.
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