The representations of the automorphism groups and the Frobenius invariants of \(K3\) surfaces
The aim of the article under review is to study a \(K3\) surface over an algebraically-closed field \(k\) of odd characteristic. Denoting by \(W\) the ring of Witt vectors of \(k\) and \(\mathrm{NS}(X)\) the Néron-Severi lattice of \(X\), there are embeddings of the \(\mathbb{Z}_l\)-module \(\mathrm{NS}(X)\otimes\mathbb{Z}_l\) into the second étale cohomogoly, and of the \(W\)-module \(\mathrm{NS}(X)\otimes W\) into the second crystelline cohomology, with the orthogonal complements of which are denoted by \(T_l(X)\) and \(T_{\mathrm{cris}}(X)\), respectively. There are representations of the automorphism group of \(X\): \(\chi_{l,X}\) into \(O(T_l(X))\),\, \(\chi_{\mathrm{cris},X}\) into \(O(T_{\mathrm{cris}}(X))\),\, \(\rho_X\) into \(\mathrm{GL}(H^0(X,\,\Omega_{x\slash k}^2))\), and \(\nu_X\) into \(O(\mathrm{NS}(X)^*\slash \mathrm{NS}(X))\).NEWLINENEWLINEMotivated by the properties of representations of the automorphism group for a complex \(K3\) surface, the first part is devoted to a comparison of the images of \(\rho_X\) and \(\nu_X\), and analogous results are obtained. Moreover, the eigenvalues of \(\chi_{l,X}(\alpha)\) and \(\alpha^*|_{H^2_{\text{ét}}(X,\mathbb{Q}_l)}\) for an automorphism \(\alpha\) are also given. There is also given a criterion for the rank of \(T_l(X)\) is divisible by the Euler function of the order of the image of \(\rho_X\).NEWLINENEWLINEThe second part gives a criterion for the \(K3\) surface \(X\) to be supersingular in terms of Frobenius invariant, and the order of the image of \(\rho_X\). Here, the Frobenius invariant is a pair of the Artin invariant, and the height of the formal Brauer group \(\widehat{Br}(X)\) of \(X\).
scientific article; zbMATH DE number 6572608