Fields of definition of K3 surfaces with complex multiplication

DOI10.1016/J.JNT.2022.04.013arXiv1907.01336WikidataQ114156493 ScholiaQ114156493MaRDI QIDQ6321406

Domenico Valloni

Publication date: 2 July 2019

Abstract: Let $X/mathbbC$ be a K3 surface with complex multiplication by the ring of integers of a CM field $E$. We show that $X$ can always be defined over an Abelian extension $K/E$ explicitly determined by the discriminant form of the lattice $mathrmNS(X)$. We then construct a model of $X$ over $K$ via Galois-descent and we study some of its basic properties, in particular we determine its Galois representation explicitly. Finally, we apply our results to give upper and lower bounds for a minimal field of definition for $X$ in terms of the class number of $E$ and the discriminant of $mathrmNS(X)$.

Mathematics Subject Classification ID

(K3) surfaces and Enriques surfaces (14J28) Complex multiplication and moduli of abelian varieties (11G15) Brauer groups of schemes (14F22) Special surfaces (14J25) Automorphisms of surfaces and higher-dimensional varieties (14J50)

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