TITLE: Strict Class Numbers of Totally Real Fields
QUESTION [9 upvotes]: In their paper Computing Systems of Hecke Eigenvalues Associated to Hilbert Modular Forms, Greenberg and Voight remark that
...it is a folklore conjecture that if one orders totally real fields by their discriminant, then a (substantial) positive proportion of fields will have strict class number 1.
I've tried searching for more details about this, but haven't found anything. 
Is this conjecture based solely on calculations, or are there heuristics which explain why this should be true?

REPLY [9 votes]: Maybe it's worth a word about why Cohen-Lenstra predicts this behavior.  Suppose K is a field with r archimedean places.  Then Spec O_K can be thought of as analogous to a curve over a finite field k with r punctures, which is an affine scheme Spec R.  Write C for the (unpunctured) curve.  Then the class group of R is the quotient of Pic(C)(k) by the subgroup generated by the classes of the punctures -- or, what is the same, the quotient of Jac(C)(k) by the subgroup generated by degree-0 divisors supported on the punctures.  (This last subgroup is just the image of a natural homomorphim from Z^{r-1} to Jac(C)(k).)
The Cohen-Lenstra philosophy is that these groups and the puncture data are "random" -- that is, you should expect that the p-part of the class group of R looks just like what you would get if you chose a random finite abelian p-group (where a group A is weighted by 1/|Aut(A)|) and mod out by the image of a random homomorphism from Z^{r-1}.  (There are various ways in which this description is slightly off the mark but this gives the general point.)
It turns out that when r > 1 the chance is quite good that a random homomorphism from Z^{r-1} to A is surjective.  In fact, the probability is close enough to 1 that when you take a product over all p you still get a positive number.  In other words, when r > 1 Cohen-Lenstra predicts a positive probability that the class group will have trivial p-part for all p; in other words, it is trivial.  (In fact, it predicts a precise probability, which fits experimental data quite well.)
When r = 1, on the other hand, the class group is just A itself, and the probability its p-part is trivial is on order 1-1/p.  Now the product over all p is 0, so one does NOT expect to see a positive proportion of trivial class groups.  And in fact, when there is just one archimedean place -- i.e. when K is imaginary quadratic -- this is just what happens!