I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form
$
\left(
\begin{array}{cc}
a+b & -a \\ 
a & b%
\end{array}%
\right),
$
where a and b are rational integers. By calculate I mean calculation of the coordinate ring of the quotient or any other insight into the situation. 



Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)
The matrix in the question is the representing matrix of multiplication with $b-a\zeta\_3$ on $\mathbb{Q}(\zeta\_3)$, for the basis $\{\zeta\_3,1\}$.
Therefore, the group of matrices in the question is $R\_{\mathbb{Q}(\zeta\_3)/\mathbb{Q}}(\mathbb{G}\_m/\mathbb{Q}(\zeta\_3))$, the Weil restriction of the multiplicative group from $\mathbb{Q}(\zeta\_3)$. 


The action of the matrix group in the question on $\mathbb{P}^1(\mathbb{Q})$ is transitive, via direct computation. After base change to $\mathbb{Q}(\zeta\_3)$, the action is conjugate to the standard action of $\mathbb{G}\_m^2/\mathbb{Q}(\zeta\_3)\cong R\_{\mathbb{Q}(\zeta\_3)/\mathbb{Q}}(\mathbb{G}\_m/\mathbb{Q}(\zeta\_3))\times\_{\mathbb{Q}}\mathbb{Q}(\zeta\_3)$ on $\mathbb{P}^1/\mathbb{Q}(\zeta\_3)$. So the quotient does not exist as a variety. 


The same will be true for all other algebraic groups of the form $R\_{K/\mathbb{Q}}(\mathbb{G}\_m)$ with $[K:\mathbb{Q}]=2$.


