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_^\infty ^^n
Here $X^$ 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 ^/math> is the class of $X^$ 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 $1/(1-t)^$.

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 $A=$ 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 A t given by
:$Z_\mu(X,t)=\sum_^\infty\mu(X^)t^n$.

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 /math>, for all varieties $X$, subject to the relations
: ' , if $X'$ and $X$ are isomorphic,
: \setminus Z/math> if $Z$ is a closed subvariety of $X$,
: _1\times X_2 _1cdot _2/math>.
The universal motivic measure gives rise to the motivic zeta function.

Examples

Let \mathbb L= 1/math> denote the class of the affine line.

:$Z(,t)=\frac$

:$Z(^n,t)=\frac$

:$Z(^n,t)=\prod_^n\frac$

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 $$ is invertible, then
:$Z(X,t)=\frac\,,$
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_^\infty ^^n=\prod_^\infty Z(S,^t^m)

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

:\sum_^\infty ^2)^^n=\prod_^\infty \frac

This is essentially the partition function.

References

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Functions and mappings
Algebraic geometry