Moduli stack of principal bundles
In algebraic geometry, given a smooth projective curve X over a finite field and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by, is an algebraic stack given by: for any -algebra R,
In particular, the category of -points of, that is,, is the category of G-bundles over X.
Similarly, can also be defined when the curve X is over the field of complex numbers. Roughly, in the complex case, one can define as the quotient stack of the space of holomorphic connections on X by the gauge group. Replacing the quotient stack by a homotopy quotient gives the homotopy type of.
In the finite field case, it is not common to define the homotopy type of. But one can still define a cohomology and homology of.
Basic properties
It is known that is a smooth stack of dimension where is the genus of X. It is not of finite type but locally of finite type; one thus usually uses a stratification by open substacks of finite type, also for parahoric G over curve X see and for G only a flat group scheme of finite type over X see.If G is a split reductive group, then the set of connected components is in a natural bijection with the fundamental group.
Behrend's trace formula
This is a version of the Lefschetz trace formula for when X is over a finite field, introduced by Behrend in 1993. It states: if G is a smooth affine group scheme with semisimple connected generic fiber, thenwhere
- l is a prime number that is not p and the ring of l-adic integers is viewed as a subring of.
- is the geometric Frobenius.
- , the sum running over all isomorphism classes of G-bundles on X and convergent.
- for a graded vector space, provided the series on the right absolutely converges.