If $K = \mathbb{Q}(\alpha)$ is a number field, where $\alpha$ is algebraic, and $\mathcal{O}_K$ the ring of integers in $K$, then the set of fractional ideals over $\mathcal{O}_K$ forms a group and if we mod out by the set of principal ideals, the resulting group is finite and we call its size the class number of $K$, which we denote $h(K)$.
I have two questions regarding the class number of imaginary quadratic fields:
If we consider the function $f(d) = h(\mathbb{Q}(\sqrt{d}))$ which maps the negative integers to the positive integers, do we know that this function is surjective? That is, can every positive integer be realized as the class number of an imaginary quadratic field extension of $\mathbb{Q}$?
We know that $h(\mathbb{Q}(\sqrt{d}))$ tends to infinity as $d$ tends to negative infinity, since there are at most finitely many imaginary quadratic fields with a given class number. However, do we have a rough estimate at how large class numbers can be relative to $|d|$? That is, if we consider the function $f(D) = \max_{|d| \leq D} h(\mathbb{Q}(\sqrt{d}))$ where $d < 0$, do we have any idea how large $f$ can be relative to $D$?
Thanks for any insights.
Best Answer
This is from Buell, Binary Quadratic Forms. From page 84, the class number for a negative discriminant $\Delta$ is about $$\frac{\sqrt{|\Delta|}}{\pi},$$ which comes from an $L$-function calculation on page 83.
Let's see, on page 101, he points out that for negative field discriminants, class group and narrow class group are identical. Then on page 103, the group of classes of binary quadratic forms is isomorphic to the narrow class group. So that works out.
I don't know about surjectivity of class numbers. I imagine so. See OEIS
I wrote a little program up to 1000, here it is up to 111. The first number that achieves a given class number tends to be squarefree, an exception being h=104. EDIT: I've run this up to 4000 so far. To get a class number $h,$ there was always some $k$ with $k < 4 h^2.$ The largest $h$ where $h^2$ did not suffice was $h=677,$ with smallest $k = 601247.$ For $678 \leq h \leq 4000,$ there was always some $k \leq 0.751517... h^2,$ equality at $h=857, k=551951.$