In algebraic geometry, given a smooth projective curve X over a finite field
F
q
and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by
Bun
G
(
X
)
, is an algebraic stack given by: for any
F
q
-algebra R,
Bun
G
(
X
)
(
R
)
=
the category of principal
G-bundles over the relative curve
X
×
F
q
Spec
R
.
In particular, the category of
F
q
-points of
Bun
G
(
X
)
, that is,
Bun
G
(
X
)
(
F
q
)
, is the category of G-bundles over X.
Similarly,
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
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
Bun
G
(
X
)
.
In the finite field case, it is not common to define the homotopy type of
Bun
G
(
X
)
. But one can still define a (smooth) cohomology and homology of
Bun
G
(
X
)
.
It is known that
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
π
0
(
Bun
G
(
X
)
)
is in a natural bijection with the fundamental group
π
1
(
G
)
.
This is a (conjectural) version of the Lefschetz trace formula for
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
#
Bun
G
(
X
)
(
F
q
)
=
q
dim
Bun
G
(
X
)
tr
(
ϕ
−
1
|
H
∗
(
Bun
G
(
X
)
;
Z
l
)
)
where (see also Behrend's trace formula for the details)
l is a prime number that is not p and the ring
Z
l
of l-adic integers is viewed as a subring of
C
.
ϕ
is the geometric Frobenius.
#
Bun
G
(
X
)
(
F
q
)
=
∑
P
1
#
Aut
(
P
)
, the sum running over all isomorphism classes of G-bundles on X and convergent.
tr
(
ϕ
−
1
|
V
∗
)
=
∑
i
=
0
∞
(
−
1
)
i
tr
(
ϕ
−
1
|
V
i
)
for a graded vector space
V
∗
, 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.
