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,
- the category of principal G-bundles over the relative curve .
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 (which is not a topological space) by a homotopy quotient (which is a topological space) 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 (smooth) cohomology and homology of .
It is known that is a smooth stack of dimension where is the genus of X. It is not of finite type but of locally of finite type; one thus usually uses a stratification by open substacks of finite type (cf. the Harder–Narasimhan stratification.) If G is a split reductive group, then the set of connected components is in a natural bijection with the fundamental group .
The Atiyah–Bott formula
Behrend's trace formula
This is a (conjectural) 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, then
where (see also Behrend's trace formula for the details)
- 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.
A priori, neither left nor right side in the formula converges. Thus, the formula states that the two sides converge to finite numbers and that those numbers coincide.
- Heinloth 2010, Proposition 2.1.2
- Lurie 2014, Conjecture 1.3.4.
- J. Heinloth, Lectures on the moduli stack of vector bundles on a curve, 2009 preliminary version
- J. Heinloth, A.H.W. Schmitt, The Cohomology Ring of Moduli Stacks of Principal Bundles over Curves, 2010 preprint, available at http://www.uni-essen.de/~hm0002/.
- Gaitsgory, D; Lurie, J.; Weil's Conjecture for Function Fields. 2014,