An explicit theory of $π_{1}^{\un,\crys}(\mathbb{P}^{1} - \{0,μ_{N},\infty\})$ - V-1 : The Frobenius extended to $π_{1}^{\un,\DR}(\mathbb{P}^{1} - \{0,μ_{p^αN},\infty\})$

Abstract

Let $p$ a prime number. For all $N\in {\mathbb{N}}^{\ast }$ prime to $p$, let $k_N$ be a finite field of characteristic $p$ containing a primitive $N$-th root of unity. Let $X_{k_N,N}={\mathbb{P}}^1-(\{0,\infty \}\cup {\mu }_N)/k_N$. This work is an explicit theory of the crystalline pro-unipotent fundamental groupoid $(π_{1}^{\un,\crys})$ 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 $π_{1}^{\un}(\mathbb{P}^{1} - \{0,μ_{N_{1}},\infty\})$ and $π_{1}^{\un}(\mathbb{P}^{1} - \{0,μ_{N_{2}},\infty\})$ when $N_1$ divides $N_2$. In V-1, we specify this question to the theme of part I, the computation of the Frobenius. For any $N\in {\mathbb{N}}^{\ast }$, let $K_N={\mathbb{Q}}_p({\xi }_N)$ where ${\xi }_N\in \overline{{\mathbb{Q}}_p}$ is a primitive $N$-th root of unity, and $X_{K_N,N}={\mathbb{P}}^1-(\{0,\infty \}\cup {\mu }_N)/K_N$. For $N$ prime to $p$, we are used to view the Frobenius of $π_{1}^{\un,\crys}(X_{k_{N},N})$ as a structure on $π_{1}^{\un,\DR}(X_{K_{N},N})$. In V-1, we show that the Frobenius of $π_{1}^{\un,\DR}(X_{K_{N},N})$, iterated $\alpha \in {\mathbb{N}}^{\ast }$ times, can be extended canonically as a structure of $π_{1}^{\un,\DR}(X_{K_{p^αN},p^αN})$. This allows to define generalizations of adjoint $p$-adic multiple zeta values associated with roots of unity of order $p^{\alpha }N$, 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.

32 pages