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#### Seminar der Forschergruppe 'Symmetrie, Geometrie und Arithmetik'

„Complex Multiplication and K3 surfaces over finite fields“
Prof. Dr. Lenny Taelman, Universität Amsterdam

K3 surfaces form a widely studied class of algebraic surfaces. Having Kodaira dimension 0 (just like abelian surfaces), they exhibit more interesting geometry than say rational surfaces, yet at the same time they are more tractable than the relatively unstructured surfaces of general type. Over the complex numbers K3 surfaces are classified in terms of Hodge structures and quadratic forms, by what are commonly called the `Torelli theorems for K3 surfaces'. This is similar to the classification of complex elliptic curves or abelian varieties in terms of lattices. Although this classification is a purely transcendental affair, one can show that complex K3 surfaces whose Hodge structures have Complex Multiplication are defined over number fields. Reducing such CM K3 surfaces at primes of good reduction gives a powerful method of producing K3 surfaces over finite fields. In this lecture, I'll explain how this method can be used to attack questions about K3 surfaces over finite fields. In particular, I'll show how one can (almost) classify the zeta functions of K3 surfaces over a given finite field $\mathbf{F}_q$. No prior knowledge of K3 surfaces will be assumed.

Freitag, den 4. November 2016 um 13.30 Uhr, in INF205, SR C Freitag, den 4. November 2016 at 13.30, in INF205, SR C

Der Vortrag folgt der Einladung von The lecture takes place at invitation by Dr. Giulia Battiston