On the Supersingular Reduction of K3 Surfaces with Complex Multiplication. (4th September 2018)

• Record Type: Journal Article
• Authors: Ito, Kazuhiro
• Is Part Of: International mathematics research notices. Volume 2020:Number 20(2020)
• Page Start: 7306
• Page End: 7346
• Publication Date: 2018-09-04
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rny210
• Languages: English
• ISSNs: 1073-7928

Abstract

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.