In algebraic geometry, the motivic zeta function of a smooth algebraic variety
is the formal power series:[1]
![{\displaystyle Z(X,t)=\sum _{n=0}^{\infty }[X^{(n)}]t^{n}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Here
is the
-th symmetric power of
, i.e., the quotient of
by the action of the symmetric group
, and
is the class of
in the ring of motives (see below).
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
, one obtains
.
Motivic measures
A motivic measure is a map
from the set of finite type schemes over a field
to a commutative ring
, satisfying the three properties
depends only on the isomorphism class of
,
if
is a closed subscheme of
,
.
For example if
is a finite field and
is the ring of integers, then
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
is the formal power series in
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
, for all varieties
, subject to the relations
if
and
are isomorphic,
if
is a closed subvariety of
,
.
The universal motivic measure gives rise to the motivic zeta function.
Examples
Let
denote the class of the affine line.
![{\displaystyle Z({\mathbb {A} },t)={\frac {1}{1-{\mathbb {L} }t}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle Z({\mathbb {A} }^{n},t)={\frac {1}{1-{\mathbb {L} }^{n}t}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle Z({\mathbb {P} }^{n},t)=\prod _{i=0}^{n}{\frac {1}{1-{\mathbb {L} }^{i}t}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
If
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
is invertible, then
![{\displaystyle Z(X,t)={\frac {P(t)}{(1-t)(1-{\mathbb {L} }t)}}\,,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
where
is a polynomial of degree
. Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.
If
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
![{\displaystyle \sum _{n=0}^{\infty }[S^{[n]}]t^{n}=\prod _{m=1}^{\infty }Z(S,{\mathbb {L} }^{m-1}t^{m})}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Here
is the Hilbert scheme of length
subschemes of
. For the affine plane this formula gives
![{\displaystyle \sum _{n=0}^{\infty }[({\mathbb {A} }^{2})^{[n]}]t^{n}=\prod _{m=1}^{\infty }{\frac {1}{1-{\mathbb {L} }^{m+1}t^{m}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
This is essentially the partition function.
References
- ^ Marcolli, Matilde (2010). Feynman Motives. World Scientific. p. 115. ISBN 9789814304481. Retrieved 26 April 2023.