Specific mechanical energy

In algebraic geometry, given a smooth projective curve over a finite field ${\mathbf{F}}_q$ and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by ${\mathrm{Bun}}_G$, is an algebraic stack over the category of G-bundles on X that is given by: for any ${\mathbf{F}}_q$-algebra R,

$R\mapsto$ the category of (principal) G-bundles over the relative curve $X{\times }_{{\mathbf{F}}_q}\mathrm{Spec}R$.

In particular, the category of ${\mathbf{F}}_q$-points of ${\mathrm{Bun}}_G$, that is, ${\mathrm{Bun}}_G({\mathbf{F}}_q)$, is the category of G-bundles over X. Similarly, the moduli stack of bundles is defined when the curve X is over the field of complex numbers.

It is known that ${\mathrm{Bun}}_G$ is a smooth stack.

Behrend's formula

This is a version of Lefschetz trace formula when X is over a finite field.

Notes

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References

Besides the obvious, there are

• http://www.math.ubc.ca/~behrend/thesis.pdf
• http://staff.science.uva.nl/~heinloth/StacksCourse_v2.pdf

• The Cohomology Ring of Moduli Stacks of Principal Bundles over Curves

Further reading

• http://mathoverflow.net/questions/87171/tamagawa-number-for-functional-fields