Institut Henri Poincaré

À la redécouverte des points rationnels

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The arithmetic of K3 surfaces with complex multiplication.

Domenico Valloni

In the realm of K3 surfaces, the ones with complex multiplications (e.g. K3 surfaces with maximal Picard rank or Kummer surfaces associated to an Abelian surface with CM) play a very special role, due to their explicit arithmetic and Hodge-theoretic properties. For example, if X is a complex K3 surface with CM, then it is automatically defined over a number field. This was originally proved by Pyatetskii-Shapiro and Shafarevich, and it follows from the interpretation of the moduli spaces of K3 surfaces as Shimura varieties. In this talk, we will show how the CM theory for K3 surfaces is a natural tool to perform concrete computations on their Brauer groups and fields of definition. In particular, we will present a general algorithm to classify all Brauer groups that can appear as \(\text{Br}(\overline{X})^{G_K}\), where \(X/K\) is a K3 with CM and \(G_K\) is the absolute Galois group of K. Moreover, we will show how many complex K3 surface with CM admit a “canonical model” over an explicit abelian extension of the CM field. This will allow us to extend the classification provided by Schütt and Elkies of singular K3 surfaces that can be defined over \(\mathbb{Q}\), and to prove a finiteness theorem due to Shafarevich and later generalised by Orr and Skorobogatov.