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}{# 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),

# X (𝐅_{q}) = q^{\dim X} \sum_{i \geq 0} (- 1)^{i} tr (f; H^{i} (X (𝐅_{q}), ℚ_{l})),

The Siegel mass formula computes the left-hand side:^[1] if G is a semisimple group with tamagawa number $τ (G)$ ,

# X (𝐅_{q}) = τ (G) \prod_{x} \frac{1}{vol (G (𝒪_{x}))}

↑ Template:Harvnb

[1] Template:Harvnb

[1]

Four-spiral semigroup

Formulation

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