TITLE: Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: 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 $[K:\mathbb{Q}]=2$.