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 $\operatorname {Bun} _{G}$, 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 $X\times _{\mathbf {F} _{q}}\operatorname {Spec} R$.

In particular, the category of $\mathbf {F} _{q}$-points of $\operatorname {Bun} _{G}$, that is, $\operatorname {Bun} _{G}(\mathbf {F} _{q})$, 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 $\operatorname {Bun} _{G}$ 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