On the Monodromy of Motivic Fundamental Groups and Non-Abelian Chabauty for High Genus Curves

We investigate the arithmetic structure of the unipotent motivic fundamental group 1mot(X) associated with a smooth projective curve X of genus g 2 defined over a number field K. By analyzing the non-abelian Galois cohomology of the pro-unipotent completion of 1(X), we construct a tower of Selmer varieties which generalize the classical Selmer groups of Jacobians. We focus on the monodromy action of the absolute Galois group GK on the lower central series quotients of 1un(X), establishing a correspondence between the Zariski density of global points in the Chabauty-Kim locus and the non-triviality of certain extension classes in the category of mixed Tate motives. Furthermore, we provide a rigorous formulation of the non-abelian Chabauty method, utilizing p-adic iterated Coleman integrals to define analytic functions that vanish on the set of rational points X(K). We demonstrate that under suitable hypothesis on the crystalline realization of the motivic fundamental group, the set X(K) is contained within the zero set of a non-trivial p-adic analytic function derived from the unipotent Albanese map.