Motivic iterated integrals and integral points

设S是Spec Z的开子概型，X是双曲曲线的S-模型.在过去的十年中，Minhyong Kim发展了一种研究整点的新方法，它利用Deligne的幂幺基本群理论构造了包含X（S）的Z_p-点集的某些子集X（Z_p）_n，并证明了这些子集X（Z_p）_n随着n的增长收敛到X（S）。在穿孔线的特殊情况下，幂幺基本群被称为motivic，为motivic方法打开了大门。我们的主要目标是利用Goncharov的motivic迭代积分理论，以及弗朗西斯Brown的方法，构造一个算法来计算Q上三次穿孔线的集合X（Z_p）_n，以及更一般基上更一般曲线的集合X（Z_p）_n。

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.