$p$-adic Periods and Selmer Scheme Images

TL;DR

This paper develops a foundational $p$-adic period map for motives, aiming to extend non-abelian Chabauty methods to more general hyperbolic curves and connect $p$-adic iterated integrals with motivic theory.

Contribution

It introduces a $p$-adic period map for broader categories of motives, facilitating non-abelian Chabauty for general hyperbolic curves and linking $p$-adic integrals with motivic conjectures.

Findings

Defines a $p$-adic period map for general motives.

Connects $p$-adic iterated integrals with Goncharov's motivic integrals.

Suggests evaluating syntomic regulators via motivic iterated integrals.

Abstract

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-\"Unver to approach the particular hyperbolic curve $\mathbb{P}^1\setminus\{0,1,\infty\}$. 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.