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