# Generalization of Gauss's class number one problem

Gauss's class number one problem for imaginary quadratic fields, now a theorem due to Heegner, Stark, and Baker (independently), asserts that the set of imaginary quadratic fields having class number equal to one correspond exactly to the discriminants $$\{-3,-4,-7,-8,-11,-19,-43,-67,-163\}$$. Of course, except for $$-4,-8$$ each of the discriminants which appear are prime: this is of course because genus theory tells us that otherwise the 2-part of the class group is non-trivial (and in particular the class number is even), and hence cannot equal one.

Since we have such concrete information coming from genus theory in for the 2-part of the class group, a natural question is the following: for which fundamental (negative) discriminants $$-d$$ is it the case that $$h_{-d} = h_{-d}[2]$$? Here $$h_{-d}$$ is the class number of the imaginary quadratic field $$\mathbb{Q}(\sqrt{-d})$$ and $$h_{-d}[2]$$ is the size of the 2-part of the class group.

If we let $$S^{-}$$ denote the set of fundamental discriminants $$-d$$ with $$h_{-d} = h_{-d}[2]$$, then clearly

$$\displaystyle \{-3,-4,-7,-8,-11,-19,-43,-67,-163\} \subseteq S^{-}.$$

Moreover, by the Brauer-Siegel theorem and the fact that imaginary quadratic fields have no units of infinite order (i.e., the regulator is bounded) it follows that $$h_{-d} \gg_\varepsilon d^{1/2 - \varepsilon}$$. Since genus theory tells us that the 2-part of the class group is essentially the number of divisors of $$d$$, which gives $$h_{-d}[2] = O(d^{1/\log \log d})$$, it follows that $$S^{-}$$ is surely finite. However the Brauer-Siegel theorem is ineffective, so it still requires some work to compute the set $$S^{-}$$.

Is it known exactly what $$S^{-}$$ is? If not, what progress has been made towards computing it?

• In terms of progress: presumably you know about the work on classifying $-d$ with $h(-d) = n$ for small $n$. This allows you to determine much more of $S^-$. Commented Aug 4, 2022 at 0:35
• PS You seem to be missing the class number criterion in your definition of $S^-$. Commented Aug 4, 2022 at 0:36
• @Kimball yes I noticed that; thank you for pointing it out! Commented Aug 4, 2022 at 1:17

This question is actually older than Gauss's class number question. These numbers a called Euler's idoneal numbers and they are of interest because this is exactly when the prime numbers of the form $$x^2 + d y^2$$ are determined by congruence conditions (equivalently when the Hilbert class field is abelian over $$\mathbb{Q}$$. They are determined under the assumption of GRH but not otherwise. See https://www.mast.queensu.ca/~kani/papers/idoneal-f.pdf or https://en.wikipedia.org/wiki/Idoneal_number for more references.