A finiteness theorem for the Brauer group of abelian varieties and $K3$ surfaces

AbstractWe give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over {{\mathbf{Z}}}. This gives an analogue for K3 surfaces of Deligne’s description of the category of ordinary abelian varieties over a finite field, and refines earlier work by N.O. Nygaard and J.-D. Yu. Our main result is conditional on a conjecture on potential semi-stable reduction of K3 surfaces over p-adic fields. We give unconditional versions for K3 surfaces of large Picard rank and for K3 surfaces of small degree.

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Isogenies Between K3 Surfaces Over

International Mathematics Research Notices ◽

10.1093/imrn/rnaa176 ◽

2020 ◽

Author(s):

Ziquan Yang

Keyword(s):

Abelian Varieties ◽

K3 Surfaces ◽

Main Step ◽

Shimura Varieties ◽

K3 Surface ◽

Perfect Field ◽

Finite Height ◽

Mild Assumption

Abstract We generalize Mukai and Shafarevich’s definitions of isogenies between K3 surfaces over ${\mathbb{C}}$ to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over $\bar{{\mathbb{F}}}_p$ by prescribing linear algebraic data when $p$ is large. The main step is to show that isogenies between Kuga–Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on $p$.

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On the Frequency of Algebraic Brauer Classes on Certain Log K3 Surfaces

Canadian Mathematical Bulletin ◽

10.4153/s0008439518000590 ◽

2018 ◽

Vol 62 (3) ◽

pp. 551-563

Author(s):

Jörg Jahnel ◽

Damaris Schindler

Keyword(s):

K3 Surfaces ◽

Upper Bounds ◽

Hasse Principle ◽

Brauer Group ◽

Integral Points ◽

Quadratic Equations

AbstractGiven systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer–Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer–Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauer-group is trivial. We use resolvent constructions to give quantitative upper bounds on the number of exceptions.

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Kuga-Satake abelian varieties of K3 surfaces in mixed characteristic

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle.2010.078 ◽

2010 ◽

Vol 2010 (648) ◽

Cited By ~ 3

Author(s):

Jordan Rizov

Keyword(s):

Abelian Varieties ◽

K3 Surfaces ◽

Mixed Characteristic

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On the Supersingular Reduction of K3 Surfaces with Complex Multiplication

International Mathematics Research Notices ◽

10.1093/imrn/rny210 ◽

2018 ◽

Vol 2020 (20) ◽

pp. 7306-7346

Author(s):

Kazuhiro Ito

Keyword(s):

Comparison Theorem ◽

Complex Multiplication ◽

K3 Surfaces ◽

Explicit Description ◽

Brauer Group ◽

K3 Surface ◽

Good Reduction ◽

Picard Number ◽

Reduction Modulo ◽

Modulo P

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.

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