# Moduli stack of principal bundles

In algebraic geometry, given a smooth projective curve X 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 $\operatorname {Bun} _{G}(X)$ , is an algebraic stack given by: for any $\mathbf {F} _{q}$ -algebra R,

$\operatorname {Bun} _{G}(X)(R)=$ 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 $\operatorname {Bun} _{G}(X)$ , that is, $\operatorname {Bun} _{G}(X)(\mathbf {F} _{q})$ , is the category of G-bundles over X.

Similarly, $\operatorname {Bun} _{G}(X)$ can also be defined when the curve X is over the field of complex numbers. Roughly, in the complex case, one can define $\operatorname {Bun} _{G}(X)$ 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 $\operatorname {Bun} _{G}(X)$ .

In the finite field case, it is not common to define the homotopy type of $\operatorname {Bun} _{G}(X)$ . But one can still define a (smooth) cohomology and homology of $\operatorname {Bun} _{G}(X)$ .

## Basic properties

It is known that $\operatorname {Bun} _{G}(X)$ is a smooth stack of dimension $(g(X)-1)\dim G$ where $g(X)$ 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 $\pi _{0}(\operatorname {Bun} _{G}(X))$ is in a natural bijection with the fundamental group $\pi _{1}(G)$ .

## Behrend's trace formula

This is a (conjectural) version of the Lefschetz trace formula for $\operatorname {Bun} _{G}(X)$ 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

$\#\operatorname {Bun} _{G}(X)(\mathbf {F} _{q})=q^{\dim \operatorname {Bun} _{G}(X)}\operatorname {tr} (\phi ^{-1}|H^{*}(\operatorname {Bun} _{G}(X);\mathbb {Z} _{l}))$ • l is a prime number that is not p and the ring $\mathbb {Z} _{l}$ of l-adic integers is viewed as a subring of $\mathbb {C}$ .
• $\phi$ is the geometric Frobenius.
• $\#\operatorname {Bun} _{G}(X)(\mathbf {F} _{q})=\sum _{P}{1 \over \#\operatorname {Aut} (P)}$ , the sum running over all isomorphism classes of G-bundles on X and convergent.
• $\operatorname {tr} (\phi ^{-1}|V_{*})=\sum _{i=0}^{\infty }(-1)^{i}\operatorname {tr} (\phi ^{-1}|V_{i})$ for a graded vector space $V_{*}$ , provided the series on the right absolutely converges.