An explicit theory of π₁^\un,\crys(ℙ¹ - \0,μ_N,∞\) - V-1 : The Frobenius extended to π₁^\un,\DR(ℙ¹ - \0,μ_p^αN,∞\)

2017/08/26 by David Jarossay, Jarossay, David
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1708.08009

openalex publication_date 2017/08/26 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

Let p a prime number. For all N ∈ ℕ^∗ prime to p, let k_N be a finite field of characteristic p containing a primitive N-th root of unity. Let X_k_N,N= ℙ¹ - (\0,∞\ ∪ μ_N) / k_N. This work is an explicit theory of the crystalline pro-unipotent fundamental groupoid (π₁^\un,\crys) of X_k_N,N. In the parts I to IV, we have considered each possible value of N separately. The purpose of part V is to study the role of the morphisms relating π₁^\un(ℙ¹ - \0,μ_N₁,∞\) and π₁^\un(ℙ¹ - \0,μ_N₂,∞\) when N₁ divides N₂. In V-1, we specify this question to the theme of part I, the computation of the Frobenius. For any N ∈ ℕ^∗, let K_N=ℚ_p(ξ_N) where ξ_N∈ ℚ_p is a primitive N-th root of unity, and X_K_N,N = ℙ¹ - (\0,∞\ ∪ μ_N) / K_N. For N prime to p, we are used to view the Frobenius of π₁^\un,\crys(X_k_N,N) as a structure on π₁^\un,\DR(X_K_N,N). In V-1, we show that the Frobenius of π₁^\un,\DR(X_K_N,N), iterated α∈ ℕ^∗ times, can be extended canonically as a structure of π₁^\un,\DR(X_K_p^αN,p^αN). This allows to define generalizations of adjoint p-adic multiple zeta values associated with roots of unity of order p^αN, and several related objects. This also gives a canonical framework to relate to each other the direct method of computation of the Frobenius of I-1 and the indirect methods of computation of the Frobenius of I-2 and I-3.

Related

• An explicit theory of π₁^un,crys(ℙ¹ - \0,μ_N,∞\) - II-3 : Sequences of multiple harmonic sums viewed as periods
• An explicit theory of π₁^\un,\crys(ℙ¹ - \0,μ_N,∞\) - II-2 : From standard algebraic relations of weighted multiple harmonic sums to those of cyclotomic p-adic multiple zeta values
• p-adic multiple zeta values and p-adic pro-unipotent harmonic\n actions : summary of parts I and II
• p-adic multiple zeta values at roots of unity and p-adic pro-unipotent harmonic actions - IV-1 : p-adic multiple zeta values at roots of unity extended to sequences of integers of any sign
• Arithmetic properties of the Frobenius traces defined by a rational\n abelian variety (with two appendices by J-P. Serre)