On the supersingular reduction of K3 surfaces with complex multiplication

Kazuhiro Ito

Published 2017-09-30Version 1

We study the good reduction modulo p of K3 surfaces with complex multiplication. If a K3 surface with complex multiplication has good reduction, we calculate the Picard number and the height of the formal Brauer group of the reduction. Moreover, if the reduction is supersingular, we calculate its Artin invariant under some assumptions on the discriminant of the Picard group and the endomorphism algebra of the Hodge structure. Our results generalize some results of Shimada for K3 surfaces with Picard number 20. Our methods rely on the main theorem of complex multiplication for K3 surfaces proved by Rizov, an explicit description of the Breuil-Kisin modules associated with Lubin-Tate characters due to Andreatta, Goren, Howard, and Madapusi Pera, and the integral p-adic Hodge theory recently established by Bhatt, Morrow, and Scholze.