## Zeta functions, Grothendieck groups, and the Witt ring.(English)Zbl 1337.11024

This is a well-written paper on zeta-functions, Grothendieck groups, motivic measures and the Witt ring. For any commutative ring $$A$$ with identity denote the big Witt ring with addition and with the multiplication $$\ast$$ by $$W(A)$$. For any natural $$n$$ there are a Frobenius ring homomorphsm $$F_n:W(A)\to W(A)$$ and an additive Verschiebung ring homomorphism $$V_n:W(A) \to W(A)$$. Let $$k$$ be a field. By $$GK_k$$ the author of the paper under review denotes the Grothendieck ring $$K_0(\mathrm{Var}_k)$$ of schemes of finite type over $$k$$.
The author proves the following. Theorem 2.1: Let $$X$$ and $$Y$$ be schemes of finite type over $$\mathbb F_q$$.
(i) The zeta function of the product $$X\times Y$$ is the Witt product of the zeta functions of $$X$$ and $$Y$$. In particular, $$Z(X^n,t)=Z(X,t)\ast\cdots\ast Z(X,t)$$.
(ii) The map $$\kappa:GK_{\mathbb F_q}\to W(\mathbb Z)$$ is a ring homomorphism. Hence $$X\mapsto Z(X,t)$$ is a motivic measure.
(iii) If $$X\to B$$ is a (Zariski locally trivial) fiber bundle with fibre $$F$$, namely, there is a covering of $$B$$ by Zariski opens $$U$$ with $$X\times_BU$$ isomorphic to $$U\times_{\mathrm{Spec }\mathbb F_q}F$$, then $$Z(X, t)=Z(B,t)\ast Z(F,t)$$.
(iv) For any $$m\in\mathbb N$$, let $$X_m$$ be the variety over $$\mathbb F_{q^m}$$ obtained by base change along $$b:\mathbb F_q\to\mathbb F_{q^m}$$. One has $$Z(X_m/\mathbb F_{q^m},t)=F_mZ(X/\mathbb F_q,t)$$.
(v) One has a commutative diagram of ring homomorphisms with upper $$b:GK_{\mathbb F_q}\to GK_{\mathbb F_q^m}$$ and low $$F_m:W(\mathbb Z)\to W(\mathbb Z)$$ horizontal arrows, and with left $$\kappa:GK_{\mathbb F_q}\to W(\mathbb Z)$$ and right $$\kappa:GK_{\mathbb F_q^m}\to W(\mathbb Z)$$ vertical arrows.
Analogical results were established by N. Naumann [Trans. Am. Math. Soc. 359, No. 4, 1653–1683 (2007; Zbl 1115.14004)] in connection with the Grothendieck ring of varieties.
Let $$A$$ be an abelian variety over $$\mathbb F_q$$ and $$A'$$ be an abelian variety over $$\mathbb F_q^m$$.
Theorem 2.6: Let notations be as above.
(a) Let $$A'$$ be an abelian variety over $$\mathbb F_q^m$$. Let $$P_1(A',t)=\prod_j(1-\alpha_jt)$$ and $$P_1(A,t)=\prod_r(1-\beta_rt)$$. One has $$P_1(A,t) =V_mP_1(A',t)=P_1(A',t^m)=\prod_j(1-\alpha_jt^m)$$. The set $$\{\beta_1^m,\cdots\}$$ coincides with the set $$\{\alpha_1,\cdots\}$$.
(b) For any smooth projective variety $$X'$$, one has $$P_1(X,t)=V_mP_1( X',t)$$.
(c) Let $$X'$$ be a smooth proper geometrically connected variety over $$\mathbb F_q^m$$. For each integer $$0<i\leq 2\dim X'$$, the polynomial $$P_i(X,t)$$ is divisible by $$V_mP_i( X',t)$$. In general, $$Z(X,t)\neq V_mZ(X',t)$$, $$Z(X\times_{\mathbb F_q}\mathbb F_q^m,t)=Z((X')^m,t)=Z(X,t)\ast\cdots\ast Z(X,t)$$.
The relation between $$a_r=\sharp X'(\mathbb F_q^r)$$ and $$b_r = \sharp X(\mathbb F_q^{mr})$$ can be described explicitly (using $$d=gcd(m,r)$$ and $$r=sd):b_r = a^d_s$$.
The paper is an excellent survey of the results that connecting zeta functions of varieties (over finite fields) with motivic measures on the category of schemes of finite type (over a field) and with the big Witt ring over $$\mathbb Z$$. For the sake of completeness the reader should also refer to the paper by S. Lichtenbaum [Fields Inst. Commun. 56, 249–255 (2009; Zbl 1245.14023)], to the book dealing with zeta elements by K. Kato et al. [Number theory 1. Fermat’s dream. Providence, RI: AMS (2000; Zbl 0953.11003)] and also to the preprint dealing with motivic zeta function with coefficients in the Grothendieck ring of varieties by M. Kapranov [“The elliptic curve in the $$S$$-duality theory and Eisenstein series for Kac-Moody groups”, Preprint, arXiv:math/0001005].

### MSC:

 11E81 Algebraic theory of quadratic forms; Witt groups and rings 11G25 Varieties over finite and local fields 13D15 Grothendieck groups, $$K$$-theory and commutative rings 14C15 (Equivariant) Chow groups and rings; motives 11M41 Other Dirichlet series and zeta functions

### Citations:

Zbl 1115.14004; Zbl 1245.14023; Zbl 0953.11003
Full Text:

### References:

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