# Motivic zeta function

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

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

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

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

## Examples

Let denote the class of the affine line.

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

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*

Here is the Hilbert scheme of length subschemes of . For the affine plane this formula gives

This is essentially the partition function.