An explicit theroy of $pi_{1}^{mathrm{un,crys}}(mathbb{P}^{1}-{0,1,infty})$ : summary of parts I and II (Various Aspects of Multiple Zeta Value)

Abstract

This is a review of our study of the crystalline pro-umipotent fundamental groupoid of mathrm{I}mathrm{P}^{1}-({0, infty}cup$mu$_{N}) ([Jn], nin{1, ldots , 12 This review is restricted to the parts I and II of this study ([Jn], nin{1, ldots , 6 and to N=1. The part I ([JI], [J2], [J3]) is an explicit computation of the Frobenius. The part II ([J4], [J5], [J6]) is an explicit algebraic study of p-adic multiple zeta values, which are associated with the Frobenius. The theory is centered around the notion of harmonic Ihara action, defined in [J2], [J3]. The harmonic Ihara action is connected to the usual Ihara action, which is a byproduct of a motivic Galois action ; it has three different incarnations related to each other by "comparison " maps ; it enables to express our explicit formulas for the Frobenius, which relate p-adic multiple zeta values and certain sequences of multiple harmonic sums to each other ; it then also enables to understand jointly the algebraic properties of these two objects. We try to emphasize the ideas and the philosophy of this work. We review its context and its motivations (§1) and, after a few technical preliminaries (§2), our general strategy (S3). We state most of the main results of part I (§4) and of part II (§5), and we also summarize and motivate the methods of the proofs. We conclude on the main messages of the theory (§6).