Motivic zeta function

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In algebraic geometry, the motivic zeta function of a smooth algebraic variety ${\displaystyle X}$ is the formal power series:^[1]

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

Here ${\displaystyle X^{(n)}}$ is the ${\displaystyle n}$-th symmetric power of ${\displaystyle X}$, i.e., the quotient of ${\displaystyle X^{n}}$ by the action of the symmetric group ${\displaystyle S_{n}}$, 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 ${\displaystyle X}$.

If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to ${\displaystyle Z(X,t)}$, one obtains ${\displaystyle 1/(1-t)^{\chi (X)}}$.

A motivic measure is a map ${\displaystyle \mu }$ from the set of finite type schemes over a field ${\displaystyle k}$ to a commutative ring ${\displaystyle A}$, satisfying the three properties

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

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

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

For example if ${\displaystyle k}$ 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 ${\displaystyle \mu }$ 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 ${\displaystyle [X]}$, for all varieties ${\displaystyle X}$, subject to the relations

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

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

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

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

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

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

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

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

If ${\displaystyle X}$ is a smooth projective irreducible curve of genus ${\displaystyle g}$ 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 ${\displaystyle P(t)}$ is a polynomial of degree ${\displaystyle 2g}$. Thus, in this case, the motivic zeta function is rational. In higher dimension, the motivic zeta function is not always rational.

If ${\displaystyle S}$ 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 ${\displaystyle S}$ can be expressed in terms of the motivic zeta function by Göttsche's Formula

${\displaystyle \sum _{n=0}^{\infty }[S^{[n]}]t^{n}=\prod _{m=1}^{\infty }Z(S,{\mathbb {L} }^{m-1}t^{m})}$

Here ${\displaystyle S^{[n]}}$ is the Hilbert scheme of length ${\displaystyle n}$ subschemes of ${\displaystyle S}$. 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.