Transcendental lattices and supersingular reduction lattices of a singular K3 surface

2006/11/08 by Ichiro Shimada, Shimada, Ichiro
Mathematics · #14J28 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14J28

paper · pdf · doi:10.48550/arxiv.math/0611208

40 pages, revised version, to appear in Transactions of the American Mathematical Society

arxiv created 2007/06/27 · arxiv updated 2009/12/01

Abstract

A (smooth) K3 surface X defined over a field k of characteristic 0 is called singular if the Néron-Severi lattice NS (X) of X over the algebraic closure of k is of rank 20. Let X be a singular K3 surface defined over a number field F. For each embedding σof F into the complex number field, we denote by T(X^σ) the transcendental lattice of the complex K3 surface X^σobtained from X by σ. For each prime ideal P of F at which X has a supersingular reduction X_P, we define L(X, P) to be the orthogonal complement of NS(X) in NS(X_P). We investigate the relation between these lattices T(X^σ) and L(X, P). As an application, we give a lower bound of the degree of a number field over which a singular K3 surface with a given transcendental lattice can be defined.

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