In algebraic geometry, the motivic zeta function of a smooth algebraic variety
X
is the formal power series
Z
(
X
,
t
)
=
∑
n
=
0
∞
[
X
(
n
)
]
t
n
Here
X
(
n
)
is the
n
-th symmetric power of
X
, i.e., the quotient of
X
n
by the action of the symmetric group
S
n
, and
[
X
(
n
)
]
is the class of
X
(
n
)
in the ring of motives (see below).
If the ground field is finite, and one applies the counting measure to
Z
(
X
,
t
)
, one obtains the local zeta function of
X
.
If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to
Z
(
X
,
t
)
, one obtains
1
/
(
1
−
t
)
χ
(
X
)
.
A motivic measure is a map
μ
from the set of finite type schemes over a field
k
to a commutative ring
A
, satisfying the three properties
μ
(
X
)
depends only on the isomorphism class of
X
,
μ
(
X
)
=
μ
(
Z
)
+
μ
(
X
∖
Z
)
if
Z
is a closed subscheme of
X
,
μ
(
X
1
×
X
2
)
=
μ
(
X
1
)
μ
(
X
2
)
.
For example if
k
is a finite field and
A
=
Z
is the ring of integers, then
μ
(
X
)
=
#
(
X
(
k
)
)
defines a motivic measure, the counting measure.
If the ground field is the complex numbers, then Euler characteristic with compact supports defines a motivic measure with values in the integers.
The zeta function with respect to a motivic measure
μ
is the formal power series in
A
[
[
t
]
]
given by
Z
μ
(
X
,
t
)
=
∑
n
=
0
∞
μ
(
X
(
n
)
)
t
n
.
There is a universal motivic measure. It takes values in the K-ring of varieties,
A
=
K
(
V
)
, which is the ring generated by the symbols
[
X
]
, for all varieties
X
, subject to the relations
[
X
′
]
=
[
X
]
if
X
′
and
X
are isomorphic,
[
X
]
=
[
Z
]
+
[
X
∖
Z
]
if
Z
is a closed subvariety of
X
,
[
X
1
×
X
2
]
=
[
X
1
]
⋅
[
X
2
]
.
The universal motivic measure gives rise to the motivic zeta function.
Let
L
=
[
A
1
]
denote the class of the affine line.
Z
(
A
n
,
t
)
=
1
1
−
L
n
t
Z
(
P
n
,
t
)
=
∏
i
=
0
n
1
1
−
L
i
t
If
X
is a smooth projective irreducible curve of genus
g
admitting a line bundle of degree 1, and the motivic measure takes values in a field in which
L
is invertible, then
Z
(
X
,
t
)
=
P
(
t
)
(
1
−
t
)
(
1
−
L
t
)
,
where
P
(
t
)
is a polynomial of degree
2
g
. Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.
If
S
is a smooth surface over an algebraically closed field of characteristic
0
, then the generating function for the motives of the Hilbert schemes of
S
can be expressed in terms of the motivic zeta function by Göttsche's Formula
∑
n
=
0
∞
[
S
[
n
]
]
t
n
=
∏
m
=
1
∞
Z
(
S
,
L
m
−
1
t
m
)
Here
S
[
n
]
is the Hilbert scheme of length
n
subschemes of
S
. For the affine plane this formula gives
∑
n
=
0
∞
[
(
A
2
)
[
n
]
]
t
n
=
∏
m
=
1
∞
1
1
−
L
m
+
1
t
m
This is essentially the partition function.
