On the supersingular reduction of K3 surfaces with complex multiplication

2017/09/30 by Ito, Kazuhiro
#11G25 #14G15 #14J28 (Primary) #14K22 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1710.00225

Abstract

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. 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 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 comparison theorem recently established by Bhatt, Morrow, and Scholze.

Related

• Complex multiplication and Brauer groups of K3 surfaces
• Existence of supersingular reduction for families of K3 surfaces with large Picard number in positive characteristic
• Complex Multiplication for K3 Surfaces
• K3 surfaces with maximal complex multiplication
• Arithmetic of K3 surfaces