# Class number formula for binary hermitian forms

The class number of integral binary quadratic forms of discriminant $D$ correspond to the class number of $\mathbb Q(\sqrt{D})$. In this case the class number formula is the classical one of Dirichlet.

For the class number of binary hermitian forms of discriminant $D$, on the other hand, there is no such correspondence I am aware of. Still, there is a similar formula proved by G. Humbert around 1920, but it does not give $h(D)$ itself (see here). In a special case: $$\sum_{j=1}^{h(D)}\frac{D}{4}\sigma(F_j)=\frac{\pi}{4}\prod_{p|D,p\neq2}(1+\Big(\frac{-1}{p}\Big)\frac{1}{p})$$ where $\sigma(F_j)$ is a certain collection of fundamental domains.

I'd like to know if there is a more precise formula available.

The main theorem (Theorem 3) establishes a bijection between what he calls "regular classes of ideals" in the maximal order of a certain quaternion algebra, and the classes of binary hermitian forms of discriminant $\alpha$ (in his notation) with coefficients in $\mathbb{Z}$. Actually, he works in greater generality, over a number field $K$ and replacing $\mathbb{Z}$ with $\mathcal{O}_K$ (which he calls $G$). Latimer himself says in the Introduction that what he's doing is a generalization of the classical relation between quadratic forms and class groups of quadratic fields.
• Well, what do you mean by "closed formula"? In which sense is Dirichlet Class Number Formula a closed one? Or are you looking for an analytic formula, say expressing this quaternionic class group as special value of some $L$-function, for instance? – Filippo Alberto Edoardo Jan 18 '17 at 22:13