everyone, I want to ask is there any result in the literature
 similar to the following: 


Let $ X=\mathbb{P}^1\backslash \{0,1,\infty\}$, then $X$ is defined over $\mathbb{Z}$. Let $X\_{\mathbb{Q}}$ denote the its generic fiber, and for any prime number $p$, $X\_{\mathbb{Q}\_p}:= X\_{\mathbb{Q}}\times Spec(\mathbb{Q}\_p)$ be the base change to $\mathbb{Q}\_p$.


Consider the global section of log differntial form $\Omega\_{X\_{\mathbb{Q}}}(log(0,1,\infty))$. It has a basis $\{\frac{dz}{z},\frac{dz}{z-1}\}$ , where $z$ is the standard affine coordinate. 


The result is : 


For any global section of log 1-form $\omega=a\frac{dz}{z} + b\frac{dz}{z-1}$, with $a,b\in \mathbb{Z}$,
one can associate a system of representation of algebraic fundamental group. More explicitly, for any prime number $p$, such that $p\nmid a$, or $p\nmid b$, then one has a nontrivial rank two $\mathbb{Z}\_p$-representation of $\pi\_1^{alg}(X\_{\mathbb{Q}\_p})$, where $\pi\_1^{alg}$ denote the algebraic fundamental group, that is 


$$
\rho\_{\omega,p}: \pi\_1^{alg}(X\_{\mathbb{Q}\_p})\to GL\_2(\mathbb{Z}\_p).
$$
It has the following property:


(i): the restriction of $\rho\_{\omega,p}$ to the geometry $\pi\_1$ is an nontrivial extension of two trivial representation, so when restricted to geometry $\pi\_1$, it will never be trivial.


(ii): If one has a $\mathbb{Q}\_p$-point of $X\_{\mathbb{Q}\_p}$, then the induced Galois represention of $Gal(\bar{\mathbb{Q}}\_p/\mathbb{Q}\_p)$ is an extension of trivial representation and Tate twist $\mathbb{Z}\_p(1)$.


I really want to know wether there is a trivial way to realize the above association. Thank you!



representation of algebraic fundamental group of projective line minus three point
Yes, I think so. The geometric $\pi\_1$ is a normal subgroup of $\pi\_1$, and by (1) consists of upper triangular unipotent matrices. So the whole representation must live in the normalizer, which is all upper-triangular matrices. In other words, the representation is an extension of two one-dimensional representations, $\chi\_1$ and $\chi\_2$. $\pi\_1^{geom}$ acts trivially on these, so are just Galois representations. By (2), we see these are the trivial representation and the Tate twist $\mathbb Z\_p(1)$.


In $\pi\_1$-representations, extensions of the $\mathbb Z\_p$ by $\mathbb Z\_p(1)$ are classified by $Hom(\pi\_1^{geom}, \mathbb Z\_p(1)) = H^1\_{et} ( X\_{\overline {\mathbb Q}}, \mathbb Z\_p(1))$. This is a rank-two lattice over $\mathbb Z\_p$. The most natural way to view it is as triples of numbers $a,b,c$ satisfying $a+b+c=0$, where $a$ is the local contribution at $0$, $b$ is the local contribution at $1$, and $c$ is the local contribution at $\infty$. (In De Rham cohomology, we would view $a,b,c$ as measuring the integral of the $1$-form over a small loops around the respective point.)


Presumably this is your associated representation?


