\(K3\) surfaces, cyclotomic polynomials and orthogonal groups (Q6564867)

Let \(X\) be a projective \(K3\) surface over \(\mathbf{C}\). Let \(T_X\) (resp. \(S_X\)) denote its transcendental (resp. Picard) lattice. It is known that the characteristic polynomials of isometries of \(T_X\) induced by automorphisms of \(X\) are powers of cyclotomic polynomials.\N\NThis article answers the question of which powers of cyclotomic polynomials occur.\N\NLet \(m, r\) be integers with \(m\geq 3\) and \(r\geq 1\), and let \(C=\Phi_m^r\) where \(\Phi_m\) is the \(m\)-th cyclotomic polynommial.\N\NProposition 1. Assume that \(\mbox{deg}(C)\leq 20\). Then there exists an automorphism \(a: X\to X\) of a projective \(K3\) surface \(X\) such that the characteristic polynomial of the restriction of \(a^*\) to \(T_X\) is equal to \(C\).\N\NDenote by \(\mbox{Aut}(X)\) the group of automorphisms of \(X\), and by \(\mbox{Aut}_s(X)\) the subgroup of \(\mbox{Aut}(X)\) acting trivially on \(T_X\). Then there is the exact sequence \N\[\N1\to\mbox{Aut}_s(X)\to \mbox{Aut}(X)\to M_X\to 1\N\]\Nwhere\(M_X\) is a finite cyclic group of order \(m_X\).\N\NCorollary: Let \(m\geq 4\) be even such that \(\varphi(m)\leq 20\). Then there exists a projective \(K3\) surface \(X\) with \(m_X=m\).\N\NA natrual question is what if \(m\) is odd. An answer to this question is given by the following\N\NPropostion 2: There exists an automorphism \(a: X\to X\) of projective \(K3\) surface \(X\) such that \(a^*\) is the identity on \(S_X\) and the characteristic polynomial of the restruction of \(a^*\) to \(T_X\) is equal to \(C\) if and only if the following condtions hold:\N\N(i) \(C(-1)\) is a square,\N\N(ii) If \(C(1)=1\), then \(\mbox{deg}(C)\equiv 4\pmod 8\).\N\NThese results are proved using some arithmetic properties for cyclotomic polynomials, lattice theory as well as some geometric properties of \(K3\) surfaces. For instance, Proposition 2 (resp. Proposition 1) is established stuying Proerty P2 (resp. Property P1) for even unimodular lattices of signature \((R,S)\). The Property P2 holds if there exists an even, unimodular lattice \(L\) of signature \((R,S)\) and an isometry \(t:L\to L\) such that (a) The characteristic polynomial of \(t\) is \(C(X)(X-1)^{R+S-\mbox{deg}(C)}\) (b) The signature of the sublattice \(\mbox{Ker}(C(t))\) is \((c, \mbox{deg}(C)-c)\). The Property P2 holds if and only if the following conditions hold: (i) \(C(-1)\) is a square, (ii) If \(C(1)=1\), then \(\mbox{deg}(C)\equiv 2c\pmod 8\).\N\NLet \(N_X\) be the kernel of \(\mbox{Aut}(X)\to O(S_X)\). Then \(N_X\) is a cyclic subgroup of \(M_X\). Let \(n_X\) be its order.\N\NAlso a question: what are possible values for the pairs \((m_X, n_X)\) is discussed.