An explicit theory of $π_{1}^\mathrm{un,crys}(\mathbb{P}^{1} - \{0,μ_{N},\infty\})$ - II-3 : Sequences of multiple harmonic sums viewed as periods

Abstract

Let $X=\text{ }\mathbb{P}^{1} - (\{0,\infty\} \cup μ_{N})\text{ }/\text{ }W(\mathbb{F}_{q})$, with $N \in \mathbb{N}^{\ast}$ and $\mathbb{F}_{q}$ of characteristic $p$ prime to $N$ and containing a primitive $N$-th root of unity. We establish an explicit theory of the crystalline Frobenius of the pro-unipotent fundamental groupoid of $X$. In part I, we have computed explicitly the Frobenius action. In part II, we use this computation to understand explicitly the algebraic relations of cyclotomic $p$-adic multiple zeta values. We have used the ideas and the vocabulary of the Galois theory of periods, and in our framework, certain sequences of prime weighted multiple harmonic sums have been dealt with as if they were periods. In this II-3, we define three notions which essentialize our three types of computations, respectively : a "continuous" groupoid $π_{1}^{\un,\DR}(X_{K})^{\hat{\text{cont}}}$, a "localization" $π_{1}^{\un,\DR}(X_{K})^{\loc}$ of $π_{1}^{\un,\DR}(X_{K})$, and a "rational counterpart at zero" $π_{1}^{\un,\RT,0}(X_{K})$ of $π_{1}^{\un,\DR}(X_{K})$. As an application, and as a conlcusion of this part II, we justify and clarify our Galois-theoretic point of view, in particular, we construct period maps and state period conjectures for sequences of prime weighted multiple harmonic sums.