$p$-adic multiple zeta values and $p$-adic pro-unipotent harmonic actions : summary of parts I and II

Abstract: This is a review on the two first parts of our work on $p$-adic multiple zeta values at $N$-th roots of unity ($p$MZV$\mu_{N}$'s), the $p$-adic periods of the crystalline pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - {0,\mu_{N},\infty}$ (where $N$ and $p$ are coprime). We restrict for simplicity the review to the case of $N=1$, i.e. the case of $p$-adic multiple zeta values ($p$MZV's). The main tools are new objects which we call $p$-adic pro-unipotent harmonic actions. These are continuous group actions on a space containing the non-commutative generating series of weighted multiple harmonic sums, they are related to the motivic Galois action on $\pi_{1}^{\un}(\mathbb{P}^{1} - {0,1,\infty})$ and to the Poisson-Ihara bracket, and interrelated by some maps. They are defined in \cite{J2} and \cite{J3} ; the definition relies on a simplification of the differential equation of the Frobenius, proved as a preliminary technical fact by \cite{J1}. Part I (\cite{J1},\cite{J2},\cite{J3}) is an explicit computation of the Frobenius of $\pi_{1}^{\un,\crys}(\mathbb{P}^{1} - {0,1,\infty})$, and in particular of $p$MZV's. We give formulas which keep a track of the motivic Galois action. Part II (\cite{J4},\cite{J5},\cite{J6}) is a study of the algebraic properties of $p$MZV's brought together with the formulas of part I. We state an explicit elementary version of the Galois theory of $p$MZV's.