Values of zeta functions of varieties over finite fields. (English) Zbl 0611.14020

Milne, J. S.

Am. J. Math. 108, 297-360 (1986); addendum ibid. 137, No. 6, 1703-1712 (2015).

Let X be a smooth projective variety of dimension d over a finite field k of q elements. The zeta function \(\zeta\) (X,s) of X satisfies \(\zeta (X,s)\sim C_ X(r)/(1-q^{r-s})^{\rho_ r}\) as \(s\to r\), for some integer \(\rho_ r\) and rational number \(C_ X(r)\). Let \(\bar k\) be the algebraic closure of k, let \(\gamma\) be the canonical generator of \(\Gamma =Gal(\bar k/k)\) and let \(\bar X=X\otimes_ k\bar k\). It has been conjectured SS(X,r,\(\ell):\) The minimal polynomial of \(\gamma\) acting on \(H^{2r}(\bar X,{\mathbb{Q}}_{\ell}(r))\) does not have 1 as multiple root. Note that the case \(\ell =p\), the characteristic of k, it not excluded. A rational number \(\chi\) (X,\({\hat {\mathbb{Z}}}(r))\), related to the cohomology groups \(H^ i(X,{\hat {\mathbb{Z}}}(r))=\prod_{\ell}H^ i(X,{\mathbb{Z}}_{\ell}(r))\quad can\) be defined if and only if conjecture SS(X,r,\(\ell)\) holds for all \(\ell\), in which case the author proves that \[ \zeta (X,s)\sim \pm \chi (X,{\hat {\mathbb{Z}}}(r))q^{\chi (X,{\mathcal O}_ X,r)}(1-q^{r-s})^{-\rho_ r}\quad as\quad r\to s. \] The groups \(H^ i(X,{\mathbb{Z}}_ p(r))\) play a role similar to the étale cohomology groups \(H^ i(X,{\mathbb{Z}}_{\ell}(r))\) for \(\ell \neq p\). They are closely related to the crystalline cohomology groups, and their behaviour is studied throughout the paper. There are cycle maps \(c^ r_{\ell}: CH^ r(X)\to H^{2r}(\bar X,{\mathbb{Q}}_{\ell}(r)),\) compatible with intersection and cup products, defined for all \(\ell\). Let \(B^ r(X)=c^ r_{\ell}(CH^ r(X)){\mathbb{Q}}_{\ell}\); it is conjectured that the dimension of \(B^ r_{\ell}(X)\) is the multiplicity of \(q^ r\) as an inverse root of \(P_{2r}(X,t)=\det (1-\gamma t)| H^{2r}(\bar X,{\mathbb{Q}}_{\ell}))\). This statement is equivalent to SS(X,r,\(\ell)\), together with Tate’s conjecture: \(B^ r_{\ell}(X)=H^{2r}(\bar X,{\mathbb{Q}}_{\ell}(r))^{\Gamma}\).

Reviewer: P.Bayer

MSC:

14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14C15	(Equivariant) Chow groups and rings; motives
14F30	\(p\)-adic cohomology, crystalline cohomology
14G99	Arithmetic problems in algebraic geometry; Diophantine geometry
14G15	Finite ground fields in algebraic geometry

Keywords:

Quillen K-groups; Chow groups; zeta function of varieties over finite field; étale cohomology groups; crystalline cohomology groups; Tate’s conjecture