An Enriques involution of a supersingular K3 surface over odd characteristic

TL;DR

This paper characterizes when supersingular K3 surfaces over fields of odd characteristic admit Enriques involutions, extending complex case results using crystalline Torelli theorem and lattice calculations.

Contribution

It provides a criterion for Enriques involutions on supersingular K3 surfaces in odd characteristic, linking lattice embeddings to geometric properties.

Findings

Supersingular K3 surfaces have Enriques involutions if certain lattice conditions are met.

For p=19 or p>23, the surface is Enriques iff Artin invariant < 6.

The criterion extends complex case results to positive characteristic.

Abstract

In this paper, we prove, as the complex case, a supersingular K3 surface over a field of odd characteristic has an Enriques involution if and only if there exists a primitive embedding of the twice of the Enriques lattice into the Neron-Severi group such that the orthogonal complement of the embedding has no vector of self-intersection -2 using the Crystalline Torelli theorem. By this criterion and some lattice calculation, we prove that when the characteristic of the base field is p=19 or p>23, a superingular K3 surface is an Enriques K3 surface if and only if the Artin invariant is less than 6.