Motivic zeta function

In algebraic geometry, the motivic zeta function of a smooth algebraic variety $$X$$ is the formal power series:
 * $$Z(X,t)=\sum_{n=0}^\infty [X^{(n)}]t^n$$

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 $$1/(1-t)^{\chi(X)}$$.

Motivic measures
A motivic measure is a map $$\mu$$ from the set of finite type schemes over a field $$k$$ to a commutative ring $$A$$, satisfying the three properties
 * $$\mu(X)\,$$ depends only on the isomorphism class of $$X$$,
 * $$\mu(X)=\mu(Z)+\mu(X\setminus Z)$$ if $$Z$$ is a closed subscheme of $$X$$,
 * $$\mu(X_1\times X_2)=\mu(X_1)\mu(X_2)$$.

For example if $$k$$ is a finite field and $$A={\mathbb Z}$$ is the ring of integers, then $$\mu(X)=\#(X(k))$$ 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 $$\mu$$ is the formal power series in $$At$$ given by
 * $$Z_\mu(X,t)=\sum_{n=0}^\infty\mu(X^{(n)})t^n$$.

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,
 * $$[X]=[Z]+[X\setminus Z]$$ if $$Z$$ is a closed subvariety of $$X$$,
 * $$[X_1\times X_2]=[X_1]\cdot[X_2]$$.

The universal motivic measure gives rise to the motivic zeta function.

Examples
Let $$\mathbb L=[{\mathbb A}^1]$$ denote the class of the affine line.


 * $$Z({\mathbb A},t)=\frac{1}{1-{\mathbb L} t}$$


 * $$Z({\mathbb A}^n,t)=\frac{1}{1-{\mathbb L}^n t}$$


 * $$Z({\mathbb P}^n,t)=\prod_{i=0}^n\frac{1}{1-{\mathbb L}^i t}$$

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 $${\mathbb L}$$ is invertible, then
 * $$Z(X,t)=\frac{P(t)}{(1-t)(1-{\mathbb L}t)}\,,$$

where $$P(t)$$ is a polynomial of degree $$2g$$. 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


 * $$\sum_{n=0}^\infty[S^{[n]}]t^n=\prod_{m=1}^\infty Z(S,{\mathbb L}^{m-1}t^m)$$

Here $$S^{[n]}$$ is the Hilbert scheme of length $$n$$ subschemes of $$S$$. For the affine plane this formula gives


 * $$\sum_{n=0}^\infty[({\mathbb A}^2)^{[n]}]t^n=\prod_{m=1}^\infty \frac{1}{1-{\mathbb L}^{m+1}t^m}$$

This is essentially the partition function.