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Oberseminar SoSe 2008 Abstract to the talk by M. Schütt (Harvard)

K3 surfaces of Picard rank 20

In this talk, we will consider complex K3 surfaces of Picard number 20. These surfaces, often called singular, behave in many ways like elliptic curves with complex multiplication. For instance, they can be defined over some class fields, and their zeta-functions are expressed in terms of Hecke characters.

Using the Artin-Tate conjecture, modularity and class group theory, we will find the minimal field such that the Neron-Severi group of a given singular K3 surface is generated by divisors over this field. In particular, we will determine all K3 surfaces with Picard rank 20 over the rationals. This last result has been used by Elkies to show that an elliptic K3 surface cannot have Mordell-Weil rank 18 over the rationals.