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

$\#C=\sum _{p}{1 \over \#\operatorname {Aut} (p)},$

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),

$\#X(\mathbf {F} _{q})=q^{\operatorname {dim} X}\sum _{i\geq 0}(-1)^{i}\operatorname {tr} (f;H^{i}(X(\mathbf {F} _{q}),\mathbb {Q} _{l})),$

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

$\#X(\mathbf {F} _{q})=\tau (G)\prod _{x}{1 \over \operatorname {vol} (G({\mathcal {O}}_{x}))}$