$p$-adic Periods and Selmer Scheme Images

The Chabauty–Kim method was developed with the aim of approaching effective Faltings’, the problem of explicitly determining the finite set of rational points on a hyperbolic curve. This method has seen success with the more particular Quadratic Chabauty method, but this method still applies only to certain curves. Previous applications of Chabauty–Kim beyond the quadratic level, as pursued by the authors, by S. Wewers, and by others, use mixed Tate motives and the $p$-adic period map of Chatzistamatiou-Ünver to approach the particular hyperbolic curve ${\mathbb{P}}^1\setminus \{0,1,\mathrm{\infty }\}$. The main purpose of this article is to lay foundations for extending the above approach to more general hyperbolic curves, in particular by defining an analogous $p$-adic period map for more general categories of motives and their non-conjectural cousins such as systems of realizations and $p$-adic Galois representations. We use this to describe a general setup for non-abelian Chabauty for an arbitrary hyperbolic curve. Our period map also connects the study of $p$-adic iterated integrals with Goncharov’s theory of motivic iterated integrals, and allows us to investigate Goncharov’s conjectures from a $p$-adic point of view. In particular, it suggests the possibility of evaluating syntomic regulators by writing elements of $K$-theory in terms of motivic iterated integrals. Lastly, it forms the basis for a certain generalization of the $p$-adic period conjecture of Yamashita for mixed Tate motives well-suited to applications in Chabauty–Kim theory.