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.