Four-spiral semigroup

Formulation

By definition, if C is a category in which morphisms are isomorphisms, the number of points in $C$ is denoted by

$\#C=\sum _p\frac{1}{\#\mathrm{Aut}(p)},$

with the sum running over objects p. (The series may diverge in general.) The formula states: for a smooth algebraic stack X over the finite field ${\mathbf{F}}_q$ and the "arithmetic" Frobenius $f:X\to X$ (i.e., the inverse of the Frobenius in Grothendieck's formula),

$\#X({\mathbf{F}}_q)=q^{\dim X}\sum _{i\ge 0}(-1{)}^i\mathrm{tr}(f;H^i(X({\mathbf{F}}_q),{\mathbb{\mathbb{Q}}}_l)),$

The Siegel mass formula computes the left-hand side: if G is a semisimple group with tamagawa number $\tau (G)$,

$\#X({\mathbf{F}}_q)=\tau (G)\prod _x\frac{1}{\mathrm{vol}(G({\mathcal{𝒪}}_x))}$