Zeta function of algebraic variety over $\mathbb{F}_q$, formula for product of Witt vectors.

Let $X$ be an algebraic variety over $\mathbb{F}_q$. We have the definition of the zeta function of $X$ as follows:$$Z(X, t) = \prod_{x \, = \,\text{Fr}_q\text{-orbit in }X(\overline{\mathbb{F}}_q)}(1 - t^{\deg x})^{-1},$$where we write $\deg x = \text{number of elements in }x$.

Question. What is $Z(X_1 \times X_2, t)$ in terms of $Z(X_1, t)$ and $Z(X_2, t)$? The result supposedly be a formula for the product of Witt vectors...

Solution 1:

The zeta function can be considered as a map $$ Var_k \to 1 + t \mathbb{Z}[[t]] =: W $$ from isomorphism classes of varieties over $k = \mathbb{F}_q$ to the multiplicative group of power series with coefficient $1$. More precisely we should be working with the Grothendieck group of varieties $K_0(Var_k)$.

$W$ is an abelian group under multiplication of power series which I will denote by $\times$. However $W$ also has a product $*$ that makes $(W,\times, *)$ a ring with addition given by $\times$ and product given by $*$. The product $*$ is characterized by the identity $$ \frac{1}{1 - at} * \frac{1}{1 - bt} = \frac{1}{1 - abt} $$ for $a,b \in \mathbb{Z}$. This is the (big) Witt vector ring.

Then the following is true: $$ Z(X \times Y, t) = Z(X,t) * Z(Y,t) $$ This result is Theorem 2.1(i) in the following paper:

Ramachandran, N. (2015), "Zeta functions, Grothendieck groups, and the Witt ring." Bull. Sci. Math. 139 (2015), no. 6, 599–627.

This paper is also relevant:

Ramachandran, N., Tabuada, G., "Exponentiable motivic measures." J. Ramanujan Math. Soc. 30 (2015), no. 4, 349–360.

PDFs of both are available on Ramachandran's website.