The field of moduli of singular K3 surfaces

Abstract

We study the field of moduli of singular K3 surfaces. We discuss both the field of moduli over the CM field and over Q. We also discuss non-finiteness with respect to the degree of the field of moduli. Finally, we provide an explicit approach to the computation of the field of moduli.

Decompositions of singular abelian surfaces

“…By observing that a form and its inverse lie in the same genus, we conclude that This suggests a connection between the field of moduli of a singular K3 surface and the genus of its transcendental lattice. In fact, the problem of characterizing the field of moduli of singular K3 surfaces is dealt with in a paper of the author [4].…”

“…(D) := m|f (D) C(D/m 2 ), (drop m in all factors)and consequently we define the extended ideal group4 asC(O) := O⊆O ′ ⊆O K C(O ′ ).The bijection C(D) ↔ C(O) yields an analogous bijection C(D) ↔ C(O); this allows us to work with ideal classes rather than forms.…”

Complex multiplication and Brauer groups of K3 surfaces

“…Finally, in Section 10, we determine the field of moduli of the tuple (T (X), B, ι), which is going to be one of the K3 class fields introduced before. In [Laf16], Laface computes the field of moduli of K3 surfaces with maximal Picard rank. If X/C is such a surface and E ⊂ C denotes its CM field, then the field of moduli of X (over E) corresponds to the fixed field of {τ ∈ Aut(C/E) : X τ ∼ = X}.…”