Much progress in understanding $\Bbb Z$ is made from analogies between $\Bbb F_q[x]$ and $\Bbb Z$.

Can there be analogies between arithmetic in $\Bbb F_q[x_1,x_2]$ and a suitable object related to $\Bbb Z$ as there are analogies between $\Bbb F_q[x]$ and $\Bbb Z$?