nLab class number formula

Idea

Given a number field $K$, the Dedekind zeta function $\zeta_K$ of $K$ has a simple pole at $s = 1$. The class number formula says that its residue there is proportional the product of the regulator with the class number of $K$

$$\underset{s\to 1}{\lim} (s-1) \zeta_K(s) \propto ClassNumber_K \cdot Regulator_K \,.$$

Variants of this for arithmetic varieties over $\mathbb{Q}$ are the Birch and Swinnerton-Dyer conjecture and the Beilinson conjecture.

References

• Wikipedia, Class number formula