On the supersingular reduction of K3 surfaces with complex multiplication

TL;DR

This paper investigates the reduction behavior of K3 surfaces with complex multiplication, focusing on Picard number, formal Brauer group height, and Artin invariant in supersingular cases, using advanced algebraic and p-adic techniques.

Contribution

It extends previous results by explicitly calculating invariants of supersingular reductions of CM K3 surfaces, utilizing recent breakthroughs in p-adic Hodge theory and complex multiplication theory.

Findings

Calculated Picard number and Brauer group height for good reductions.

Determined Artin invariant for supersingular reductions under certain conditions.

Generalized results of Shimada for K3 surfaces with Picard number 20.

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.