TITLE: What's the analogue of the Hilbert class field in the following analogy?
QUESTION [21 upvotes]: There's a wonderful analogy I've been trying to understand which asserts that field extensions are analogous to covering spaces, Galois groups are analogous to deck transformation groups, and algebraic closures are analogous to universal covering spaces, hence the absolute Galois group is analogous to the fundamental group.  (My vague understanding is that the machinery around etale cohomology makes this analogy precise.)  
Does the Hilbert class field (of a number field) fit anywhere into this analogy, and how?
Phrased another way, what does the Hilbert class field of the function field of a nonsinguar curve defined over $\mathbb{C}$ (say) look like geometrically?

REPLY [7 votes]: The absolute Galois group is $\pi_1( Spec K)$ and the Galois group of the maximal unramified extension of $K$ is $\pi_1( Spec R)$, where $R$ is the ring of integers of $K$ and the class group of $K$ is the abelianization of $\pi_1( Spec R)$ and the Hilbert class field is the field corresponding to this quotient.
A curve has a function field and therefore an absolute Galois group, which is much bigger than the fundamental group and describes all branched covers of the curve. The fundamental group corresponds of course to unramified covers. Working over C gives you infinite groups even for the abelianization. If you work over a finite field, then the analogy is more precise.