# zbMATH — the first resource for mathematics

$$K3$$ surfaces with symplectic group actions. (English) Zbl 1055.14043
Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 167-182 (2003).
Let $$k$$ be a field of positive characteristic, and let $$X$$ be a $$K3$$ surface over $$k$$. A finite group action $$G$$ on $$X$$ is called symplectic if every element of $$G$$ fixes the nowhere vanishing $$2$$-form on $$X$$. The paper under review looks at symplectic group actions on $$K3$$ surfaces defined over a field $$k$$ of positive characteristic, and discuss their influences on the quantities that exist exclusively in positive characteristic, e.g., the height of formal Brauer groups, Artin invariants of $$K3$$ surfaces. The main results established here are that the height of the formal Brauer group and the Artin invariant of a $$K3$$ surface are invariant under symplectic group actions. Special attention is given to the $$K3$$ surfaces, which are one-parameter deformations of $$K3$$ surfaces in weighted projective $$3$$-spaces. A typical example of such a surface is a deformation of a weighted diagonal $$K3$$ surface, and is defined by the equation of the form $$x_0^{d_0}+x_1^{d_1}+x_2^{d_2}+x_3^{d_3}-\lambda x_0x_1x_2x_3=0$$ with a parameter $$\lambda$$ in a weighted projective $$3$$-space. Picard numbers, the height of formal Brauer groups, and Artin invariants are computed for these $$K3$$ surfaces. Moreover, the Tate conjecture is proved for (some) of these $$K3$$ surfaces over a finite field $$k$$. Also a formula counting the number of rational points over a finite field of $$q$$-elements on these $$K3$$ surfaces is given.
For the entire collection see [Zbl 1022.00014].

##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces 11G25 Varieties over finite and local fields 14F22 Brauer groups of schemes 14L30 Group actions on varieties or schemes (quotients) 14C22 Picard groups
##### Keywords:
formal Brauer groups; Artin invariants; mirror symmetry