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].


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
Full Text: DOI arXiv


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