In algebraic geometry, the motivic zeta function of a smooth algebraic variety
is the formal power series:[1] Here
by the action of the symmetric group
If the ground field is finite, and one applies the counting measure to
, one obtains the local zeta function of
If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to
A motivic measure is a map
from the set of finite type schemes over a field
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
given by There is a universal motivic measure.
It takes values in the K-ring of varieties,
, which is the ring generated by the symbols
, subject to the relations The universal motivic measure gives rise to the motivic zeta function.
denote the class of the affine line.
is a smooth projective irreducible curve of genus
admitting a line bundle of degree 1, and the motivic measure takes values in a field in which
Thus, in this case, the motivic zeta function is rational.
In higher dimension, the motivic zeta function is not always rational.
is a smooth surface over an algebraically closed field of characteristic
, then the generating function for the motives of the Hilbert schemes of
can be expressed in terms of the motivic zeta function by Göttsche's Formula Here
is the Hilbert scheme of length
For the affine plane this formula gives This is essentially the partition function.