The Tate conjecture for ordinary K 3 surfaces over finite fields
Let \(X_ 0\) be an ordinary K 3 surface over a finite field \(k={\mathbb{F}}_ q\), \(q=p^ a\). The Galois group acts on the étale cohomology groups \(H^ 2_{et}(X_ 0\otimes \bar k,{\mathbb{Q}}_{\ell}(1)),\) \(\ell \neq p.\) According to the Tate conjecture on algebraic cycles, the subspace fixed under the Galois action is spanned by cohomology classes of algebraic cycles on \(X_ 0\). This conjecture is proved in the paper. Half of the paper is devoted to the study of a canonical lifting \(X_{can}\) of \(X_ 0\) over the Witt ring \(W(k)\). \(X_ 0\) is ordinary implies that via the Serre-Tate isomorphism one can get a structure of a formal group on the universal formal deformation space S of \(X_ 0\) over the Witt ring \(W(k)\). Let \({\mathcal X}_{can}\) be the formal lifting obtained by pulling back the universal formal family \({\mathcal X}/S\) along the 0-section \(Spf W(k)\to S.\) \(X_{can}\) is the algebraization of \({\mathcal X}_{can}\). It is proved that \(X_{can}\) is the same as the canonical lifting of Deligne and Illusie ({\S} 1). - If \(A_{{\mathbb{C}}}\) is the abelian variety associated to \(X_{{\mathbb{C}}}=X_{can}\otimes {\mathbb{C}}\) by Kuga-Satake-Deligne, then theorem 2.7 says that the Galois representation \(H^ 1_{et}(A_{{\mathbb{C}}},{\mathbb{Q}}_ p)\) is semi-simple. This leads to the proof that the reduction \(A_ 0\) of \(A_{{\mathbb{C}}}\) is an ordinary abelian variety when \(X_ 0\) is ordinary ({\S} 2.8, 2.5). The geometric Frobenius of \(\bar A_ 0\) lifts to an element \(\sigma_ A\in End(A_{{\mathbb{C}}})\times {\mathbb{Q}}\) such that the action of \(\sigma_ A\) on \(H^ 1_{et}(A_{{\mathbb{C}}},{\mathbb{Q}}_{\ell})\) coincides with the action of geometric Frobenius and \(\sigma_ A\) induces an automorphism of the rational Hodge structure \(H^ 1(A_{{\mathbb{C}}},{\mathbb{Q}})\). Also it is proved ({\S} 3.1, 3.2) that the geometric Frobenius on \(P^ 2_{et}(X_{{\mathbb{C}}},{\mathbb{Q}}_{\ell}(1))\) descends to an autmorphism \(\sigma_ X\) of the rational Hodge structure \(P^ 2(X_{{\mathbb{C}}},{\mathbb{Q}}(1))\). Then there is a commutative diagram \(P^ 2(X_{{\mathbb{C}}},{\mathbb{Q}}(1))\to^{\alpha}End(H^ 1(A_{{\mathbb{C}}},{\mathbb{Q}}))\to^{conj(\sigma_ A)}End(H^ 1(A_{{\mathbb{C}}},{\mathbb{Q}})),\quad P^ 2(X_{{\mathbb{C}}},{\mathbb{Q}}(1))\to^{\sigma_ X}P^ 2(X_{{\mathbb{C}}},{\mathbb{Q}}(1))\to^{\beta}End(H^ 1(A\quad_{{\mathbb{C}}},{\mathbb{Q}}));\quad conj(\sigma_ A)\circ \alpha =\beta \circ \sigma_ X,\) \(\alpha\) and \(\beta\) being injections of Hodge structures. From this and the Tate conjecture for abelian varieties it is easy to deduce that the subspace of \(P^ 2(X_{{\mathbb{C}}},{\mathbb{Q}}(1))\) fixed by the Frobenius \(\sigma\) consists of algebraic classes (theorem 3.3). Tate's conjecture follows as an immediate corollary (3.4), as \(P^ 2_{et}(X_ 0\otimes \bar k,{\mathbb{Q}}_{\ell}(1))^{\sigma}=P^ 2(X_{{\mathbb{C}}},{\mathbb{Q}}(1))^{\sigma}\quad \otimes {\mathbb{Q}}_{\ell}\).
ordinary K 3 surface over a finite field
scientific article; zbMATH DE number 3887831
scientific article