This coincise and carefully written paper studies the non-symplectic index of supersingular \(K3\) surfaces over algebraically closed fields of odd characteristic.
Given a \(K3\) surface \(X\) over an algebraically closed field \(k\), the image of the representation of its automorphism group \[ \mathrm{Aut}(X)\to GL(H^{0}(X,\Omega_{X/k}^{2})) \] is a finite cyclic group. Its order \(N_{X}\) is called the non-symplectic order of \(X\).
If \(k=\mathbb C\) is the field of complex numbers, it is known that the non-symplectic index can be any integer \(N\) with \(1\leq N\leq 66\) and \(N\not= 60\).
Recall that if \(k\) is of positive characteristic, a \(K3\) surface over \(k\) is said to be supersingular when it has maximal Picard number, i.e. when its Neron-Severi lattice \(NS(X)\) has rank 22. Recall also that the Artin invariant of \(X\) is half the dimension of the discriminant group of \(NS(X)\) as an \(\mathbb{F}_{p}\)-vector space.
Let \(k\) be of characteristic \(p>2\). The result in the paper under revision shows that for a supersingular \(K3\) surface \(X\) over \(k\) of Artin invariant \(\sigma\), the non-symplectic index is \(p^{m}+1\) where \(m=0\) or \(\sigma/m\) is an odd integer. Dimensions of the families they form are also computed in terms of \(m\). The main ingredient of the proof is the crystalline Torelli theorem. Furthermore, a supersingular \(K3\) surface with maximal non-symplectic index, i.e. \(N_{X}=p^{\sigma}+1\), is unique up to isomorphism. Examples of these are provided.