Let \(X\) be a smooth proper variety defined over a finite field \(k\) of characteristic \(p\), and let \(X_a\) be the corresponding variety defined over an algebraic closure of \(k\). For any positive integer \(m\) and each rational prime \(\ell\) different from \(p\), let \(H^{2m} (X_a, \mathbb{Q}_\ell)(m)\) denote the \(2m\)-th twisted \(\ell\)-adic cohomology group. The Galois group \(G(k)\) acts on \(H^{2m} (X_a, \mathbb{Q}_\ell)(m)\) in a natural way. The Tate conjecture claims that the subspace \((H^{2m} (X_a, \mathbb{Q}_\ell) (m))^{G(k)}\) fixed under the Galois action is spanned by cohomology classes of codimension \(m\) algebraic cycles on \(X\) over \(k\). More precisely, write \(\text{Al}^m(X)\) for the \(\mathbb{Q}_\ell\)-vector subspace of \(H^{2m} (X_a, \mathbb{Q}_\ell)(m)\) spanned by cohomology classes of all algebraic cycles of codimension \(m\) on \(X\) defined over \(k\). It is well-known that \(\text{Al}^m(X) \subseteq (H^{2m} (X_a,\mathbb{Q}_\ell) (m))^{G(k)}\). Elements of \(H^{2m} (X_a,\mathbb{Q}_\ell) (m)^{G(k)}\) are called Tate classes on \(X\). The conjecture is that Tate classes exhaust all the algebraic cycles of codimension \(m\) on \(X\) over \(k\). The Tate conjecture has been proved in the affirmative for a number of cases by several mathematicians, e.g., Fermat varieties satisfying certain conditions, abelian varieties for \(m=1\), certain classes of abelian varieties for arbitrary \(m\), ordinary K3 surfaces, and K3 surfaces of finite height.
In this paper, the author establishes the validity of Tate’s conjecture for powers \(X=Y^r\) of an ordinary K3 surface, \(Y\). For an ordinary K3 surface \(Y\), the Tate conjecture is true so that \(H^2(Y_a,\mathbb{Q}_\ell) (1)^{G (k)} = \text{NS}(Y_a)_\ell\) where \(\text{NS}(Y_a)\) is the Néron-Severi group of \(Y_a\), and \(\text{NS}(Y_a)_\ell: =\text{NS}(Y_a) \otimes \mathbb{Q}_\ell\). Let \(V_\ell(Y)\) denote the orthogonal complement of \(\text{NS} (Y_a)_\ell\) in \(H^2(X_a, \mathbb{Q}_\ell)(1)\). Then there is a canonical splitting \(H^2(X_a, \mathbb{Q}_\ell) (1)= \text{NS}(Y_a)_\ell \oplus V_\ell(Y)\), which is Galois invariant. Let \(\varphi_k\) denote the geometric Frobenius automorphism, and let \(P_{Y,\text{tr}}(t): =\text{det(id}-\varphi_kt \mid V_\ell(Y))\) be the characteristic polynomial. In the earlier paper [Duke J. Math. 72, No. 1, 65-83 (1993; Zbl 0819.14005)], the author obtained two results (which are relevant to this paper):
(A) \(P_{Y,\text{tr}} (t)\) is irreducible over \(\mathbb{Q}\) (and hence all roots are simple), and
(B) if \(R_{Y,\text{tr}}\) denotes the set of roots of \(P_{Y,\text{tr}}\), then there is a subset \(B \subset R_{Y,\text{tr}}\) such that \(B\) consists of multiplicatively independent elements, and that \(R_{Y,\text{tr}}\) is the disjoint union of \(B\) and \(B^{-1}: =\{\alpha^{-1} \mid\alpha \in B\}\).
The proof of the Tate conjecture for powers \(X=Y^r\) of an ordinary K3 surface \(Y\) over \(k\) is streamlined as follows. First the Tate conjecture is established for the product \(Y^2\) using (A), and then for an arbitrary power \(Y^r\) using (B). Indeed, it is shown that the pullbacks of Tate classes on \(Y^2\) with respect to all projection maps \(Y^r\to Y\) and \(Y^r\to Y^2\) generate the \(\mathbb{Q}_\ell\)-subalgebra of Tate classes on \(Y^r\). More precisely, for \(X=Y^2\), \(H^4(X_a, \mathbb{Q}_\ell) (2)^{G(k)}\) is generated as a \(\mathbb{Q}_\ell\)-vector subspace by cohomology classes of the following three types: (1) products of divisor classes, (2) the classes of graphs of Frobenius and its iterates, and (3) the classes of \(Y\times\{\text{point}\}\) and \(\{\text{point}\} \times Y\). In particular, the Tate conjecture holds. Now let \(r\) be a natural number \(>1\), and let \(X=Y^r\). Then each cohomology class in \((H^{2m} (X_a,\mathbb{Q}_\ell) (m))^{G(k)}\) can be presented as a linear combination of products of pullbacks of Tate classes on \(Y\) and \(Y^2\) with respect to the projection maps \(X\to Y\) and \(X\to Y^2\). Therefore, the Tate conjecture is true for \(X=Y^r\).