We discuss conjectures for the distribution of Galois groups
of the maximal unramified extension of quadratic fields and results
towards these conjectures in the function field case (as the size of
the finite field goes to infinity). We will give a construction of a
random pro-odd group whose moments match those we see in the function
field results, as a candidate random group for the distribution of the
maximal pro-odd unramified extension of quadratic fields. Venturing
into the even part, we will give a conjecture for the average number
of unramified G-extensions of an imaginary (or real) quadratic field
for any finite group G, motivated by further function field results.
The talk with include joint work with Boston and with Y. Liu.