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

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

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 ${𝐅}_q$ and the "arithmetic" Frobenius $f:X\to X$ (i.e., the inverse of the Frobenius in Grothendieck's formula),

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

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

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