Let $K_0(Var_k)$ be the Grothendieck ring of $k$-varieties for a field $k$. Let $\mathbb{L}$ denote the class of the affine line over $k$. Let $S$ be a $k$-variety and $[S] \in \mathbb{Z}[\mathbb{L}]$, say $[S] = P(\mathbb{L})$ with $P(\mathbb{L})$ a polynomial in $\mathbb{L}$. Then if $k = \mathbb{F}_q$ is a finite field, one has that $\vert S \vert_q = P(q)$. Call elements of $\mathbb{Z}[\mathbb{L}]$ ''virtual Tate motives." I have read many times that ''after the Tate conjectures,'' if $\Delta$ is a $\mathbb{Z}$-variety such that there exists a polynomial $P_{\Delta}(X) \in \mathbb{Z}[X]$ for which $\vert \Delta \otimes_{\mathrm{Spec}(\mathbb{Z})} \mathrm{Spec}(\mathbb{F}_q) \vert_q = P_{\Delta}(q)$ for every finite field $\mathbb{F}_q$, then for each finite field $k$, the class $[\Delta_k]$ is given by $P_{\Delta}(\mathbb{L})$. (Here, $\Delta_k := \Delta \otimes_{\mathrm{Spec}(\mathbb{Z})} \mathrm{Spec}(k)$.)

I have two questions on this subject.

  • Can anyone explain which Tate conjectures are used here, and why this property follows ?

  • Suppose the above property is true for some set of prime powers $\Omega$. When can one conclude that if $q \in \Omega$, then $[\Delta_{\mathbb{F}_q}]$ is a virtual mixed Tate motive in $K_0(Var_{\mathbb{F}_q})$ ?



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