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Oberseminar SoSe 2011 Zusammenfassung des Vortrags von Frithjof Schulze

Singular K3 surfaces with class number 2

We will call a complex K3 surfaces with Picard number 20 a singular K3 surface. To every such surface we can associate a positive definite even binary quadratic form Q, which gives the discriminant and the class number of the surface.

It is known that singular K3 surfaces always have a model over some number field, but in general it it unclear what the minimal field of definition is. In particular it is not clear which singular K3 surfaces can be defined over the rationals.

I will talk about lower and upper bounds for the field of definition, the implications for surfaces with class number 2, and the construction of singular K3 surfaces over the rationals.