Transcendental lattices and supersingular reduction lattices of a singular $K3$ surface

Details Transcendental lattices and supersingular reduction lattices of a singular $K3$ surface Transcendental lattices and supersingular reduction lattices of a singular $K3$ surface

2006-11-08

arxiv.org/abs/math/0611208v2

Mathematics

Algebraic Geometry

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

Scientific paper

A (smooth) K3 surface X defined over a field k of characteristic 0 is called singular if the N\'eron-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 \sigma of F into the complex number field, we denote by T(X^\sigma) the transcendental lattice of the complex K3 surface X^\sigma obtained from X by \sigma. 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^\sigma) 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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