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Abstract des Vortrages von Roberto Laface

Fields of definition of singular K3 surfaces and classification of their Q-models

After the groundbreaking work of Shioda-Mitani and Shioda-Inose, the theory of K3 surfaces has been enriched of an arithmetic flavour due to the study of singular K3 surfaces. In 1977, Shioda and Inose proved that every singular K3 surface can be defined over a number field; later, Shafarevich proved that for every positive integer n there is a finite number of singular K3 surfaces defined over a field of degree n over the rational numbers.

Arithmetically speaking, one wants to classify all K3 surfaces with equation over Q: by using class field theory, Schütt narrowed down the investigation to 101 cases (plus possibly one exception). With this motivation in mind, I will present some of the techniques and the results in this direction: I will discuss the problem of finding the minimal field of definition of a singular K3 surface, and picture the classification of Q-models in the case of K3 surfaces of low (generalized) class number.