TITLE: Why is $(\mathbb{Z}/3\mathbb{Z})^3$ not a class group of an imaginary quadratic number field ?
QUESTION [22 upvotes]: In general, it seems not known which finite abelian groups are class groups of quadratic number fields.
For imaginary quadratic number fileds, I  read that $(\mathbb{Z}/3\mathbb{Z})^3$ is the smallest 
abelian group which does not occur. It does occur as the class group of $\mathbb{Q}(\sqrt{188184253})$.
What other groups are known not to be the class groups of imaginary quadratic number fields ? 
How does one prove this  ? 
What is the situation for class groups of real quadratic number fields ?

REPLY [3 votes]: Hello MO, the system will not let me make a comment only an answer, but I just wanted to point out that, contrary to what "v08ltu" commented, the case of $(Z/3)^k$ is not clearly easier for the methods of Arno and Watkins, but in fact in one aspect is close to the worst case (aside from genera at 2).
The point is, you can have heaps of $(a,b,c)$ forms with $a\sim d^{1/4}$, the multiplicative structure is all gone. In the notation of these papers, the $A(s)=\sum_a 1/a^s$ is only bounded by $|A(1/2+it)|\ge 1-(h-1)/d^{1/8}$, and you want to keep this away from zero (the trick of Watkins is to consider smaller $a$ multiplicatively, and larger ones additively in a variant of $A$, but here all is additive). It seems that Watkins has just enough to get $h=81$ to work, as $d\sim 80^8\sim 2^{52}$, which was his sieveing limit. I sincerely doubt that $(Z/3)^5$ is feasible the same way, as the bound jumps by $3^8$. Watkins barely mentions this at the end of his work, saying that workload is at least fourth-power growth in sieving limit from $h/d^{1/4}$ "but also the lower bounds on $|A(s)|$ become worse", and I think they particularly jam you up here, being 8th power not 4th.
On the other hand, once you actually get to the sieving problem, even with a higher limit, it is then easier as you can demand that all small primes (up to $d^{1/4}$ which is at least almost 10000) are inert, whereas Arno and Watkins split up into a myriad of cases to handle alternative behavior vis-a-vis accounting of smallish split primes and their effect. Sieving up to $728^8\sim 10^{23}$ could be feasible, as Sorenson has done $10^{25}$ for pseudosquares (using doubly-focussed methods).