representation of algebraic fundamental group of projective line minus three point

Question

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!

Answer 1

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?