Orbits of action of $SL_2(\mathbb{Z})$ on lattice

Question

I'm interested in the action of $SL_2(\mathbb{Z})$ on $\mathbb{Z}^2$: if $A\in SL_2(\mathbb{Z})$ and $v\in\mathbb{Z}^2$, then $Av\in\mathbb{Z}^2$.

Specifically, what are the orbits of this action?

Answer 1

By Bezout's theorem, for all $v,w\in\Bbb Z$ there exist $x,y\in\Bbb Z$ such that $vx+wy=\gcd(v,w)$. Hence

$$\begin{pmatrix}v/\gcd(v,w) & -y \\ w/\gcd(v,w) & x\end{pmatrix}\begin{pmatrix} \gcd(v,w) \\ 0 \end{pmatrix}=\begin{pmatrix} v \\ w \end{pmatrix} $$

shows that $(\begin{smallmatrix} v\\w\end{smallmatrix})$ is in the same ${\rm SL}_2(\Bbb Z)$-orbit as $(\begin{smallmatrix}g \\ 0\end{smallmatrix})$. Since $\gcd:\Bbb Z^2\to\Bbb N$ is ${\rm SL}_2(\Bbb Z)$-invariant, the orbits admit the vectors $(\begin{smallmatrix}g\\0\end{smallmatrix})$ for $g=0,1,2,3,\cdots$ as a system of representatives, and the orbit of $(\begin{smallmatrix}g\\0\end{smallmatrix})$ is the set of all vectors $(\begin{smallmatrix} v\\w\end{smallmatrix})$ such that $\gcd(v,w)=g$.