An explicit theory of π₁^un,crys(ℙ¹ - \0,μ_N,∞\) - II-3 : Sequences of multiple harmonic sums viewed as periods

2016/01/06 by David Jarossay, Jarossay, David · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1601.01159

openalex publication_date 2016/01/06 · openalex created_date 2016/09/16 · openalex updated_date 2026/07/28

Abstract

Let X= ℙ¹ - (\0,∞\ ∪ μ_N) / W(\mathbbF_q), with N ∈ ℕ^∗ and \mathbbF_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 π₁^\un,\DR(X_K)^cont, a "localization" π₁^\un,\DR(X_K)^\loc of π₁^\un,\DR(X_K), and a "rational counterpart at zero" π₁^\un,\RT,0(X_K) of π₁^\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.

Citations

• Multiple polylogarithms and mixed Tate motives

Cited by

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

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