Motivic zeta function

In algebraic geometry, the motivic zeta function of a smooth algebraic variety $X$ is the formal power series

${\displaystyle Z(X,t)=\sum _{n=0}^{\infty }[X^{(n)}]t^{n}}$

Here ${\displaystyle X^{(n)}}$ 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 ${\displaystyle [X^{(n)}]}$ is the class of ${\displaystyle X^{(n)}}$ in the ring of motives (see below).

If the ground field is finite, and one applies the counting measure to ${\displaystyle Z(X,t)}$, 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 ${\displaystyle Z(X,t)}$, one obtains ${\displaystyle 1/(1-t)^{\chi (X)}}$.

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

${\displaystyle \mu (X)\,}$ depends only on the isomorphism class of $X$,

${\displaystyle \mu (X)=\mu (Z)+\mu (X\setminus Z)}$ if $Z$ is a closed subscheme of $X$,

${\displaystyle \mu (X_{1}\times X_{2})=\mu (X_{1})\mu (X_{2})}$.

For example if $k$ is a finite field and ${\displaystyle A={\mathbb {Z}}}$ is the ring of integers, then ${\displaystyle \mu (X)=\#(X(k))}$ 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 ${\displaystyle A[[t]]}$ given by

${\displaystyle Z_{\mu }(X,t)=\sum _{n=0}^{\infty }\mu (X^{(n)})t^{n}}$.

There is a universal motivic measure. It takes values in the K-ring of varieties, ${\displaystyle A=K(V)}$, which is the ring generated by the symbols $[X]$, for all varieties $X$, subject to the relations

${\displaystyle [X']=[X]\,}$ if $X'$ and $X$ are isomorphic,

${\displaystyle [X]=[Z]+[X\setminus Z]}$ if $Z$ is a closed subvariety of $X$,

${\displaystyle [X_{1}\times X_{2}]=[X_{1}]\cdot [X_{2}]}$.

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

Examples

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

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

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

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 ${\displaystyle {\mathbb {L}}}$ is invertible, then

${\displaystyle Z(X,t)={\frac {P(t)}{(1-t)(1-{\mathbb {L}}t)}}\,,}$

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 ${\displaystyle 0}$, 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

${\displaystyle \sum _{n=0}^{\infty }[({\mathbb {A}}^{2})^{[n]}]t^{n}=\prod _{m=1}^{\infty }{\frac {1}{1-{\mathbb {L}}^{m+1}t^{m}}}}$

This is essentially the partition function.