I was describing Manish Kumar's work a few weeks ago to a fellow graduate student, and she stumped me with a big-picture question I couldn't answer.

Manish Kumar proved that the commutator subgroup of $\pi\_1(\mathbb{A}^1\_K)$, where $K$ is a characteristic $p$, algebraically closed field, is pro-finite free. He proved this, in fact, for any smooth affine curve over $K$.

(He proved this for algebraically closed fields of char $p$ which are uncountable in his thesis: <http://www.math.msu.edu/~mkumar/Publication/thesis.pdf>; and without the cardinality restriction in: <http://arxiv.org/PS_cache/arxiv/pdf/0903/0903.4472v2.pdf>)

As I explained to my colleague, this is a geometric analogue of Shafarevich's conjecture, that $Gal(\mathbb{Q}^{ab})$ is pro-finite free. Indeed $Gal(\mathbb{Q}^{ab})=\pi\_1^c(Spec(\mathbb{Q}))$ ($c$ stands for the commutator subgroup). But why is $\mathbb{A}^1\_K$, for $K$ an algebraically closed characteristic $p$ field (or indeed, *any* smooth affine curve over $K$), an analogue of $Spec(\mathbb{Q})$? Usually $\mathbb{A}^1\_K$ (for $K$ an algebraically closed field) is an analogue of $Spec(\mathbb{Z})$. I came up with some partial explanations, but no full heuristic. Can you think of one?