# 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...

## 1 Answer

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.

• I would be surprised if this result didn't go back to one of Weil, Dwork, or Grothendieck. Certainly the papers you linked to prove a stronger result than what I'm looking for? – user361298 Sep 18 '16 at 14:17
• It is likely that it was known earlier, this is just the first place I've seen the result. This formula is exactly the content of 2.1(i) of the first paper above. Of course the rest of the paper, and the second reference, do prove more. – Dori Bejleri Sep 18 '16 at 22:28