## $$K3$$ surfaces with an automorphism of order 11.(English)Zbl 1290.14028

Let $$k=\bar{k}$$ be a field of arbitrary characteristic. Consider $$K3$$ surface $$X$$ over $$k$$ admitting an automorphism $$\varphi$$ of order $$11$$. In particular in case of the characteristic of $$k$$ being $$11$$, such a surface is wild and $$X$$ has a structure of an elliptic fibration that is compatible with the automorphism $$\varphi$$.
The main theorem of the paper under review is that such $$K3$$ surfaces over $$k$$ with characteristic $$11$$ have generically Picard number $$2$$, which is related the Tate conjecture, and answers several questions asked in a paper by Dolgachev and Keum. As corollaries, one obtains that the height of such $$K3$$ is $$10$$, an example of supersingular $$K3$$ surface admitting an order-$$11$$ automorphism, and that if a complex $$K3$$ surface $$X$$ is reduced to $$K3$$ surfaces $$X_\varepsilon$$ or $$X_\gamma$$, then, the Picard number of $$X$$ is $$2$$. Here $$X_\varepsilon : y^2=x^3+\varepsilon x^2 + t^{11}-t, \, X_\gamma : y^2=x^3+\gamma x + t^{11}-t$$.
The main theorem is proved by firstly reducing to the case of elliptic $$K3$$ surfaces $$X_\varepsilon$$ or $$X_\gamma$$; and then, computing the characteristic polynomial of the induced Frobenius mappting on $$H^2_{\mathrm{\'et}}(\bar{X},\, \mathbb{Q}_l)$$. The second step is done with Lefschetz’s Fixed Point Formula.

### MSC:

 14J28 $$K3$$ surfaces and Enriques surfaces 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations 14J50 Automorphisms of surfaces and higher-dimensional varieties
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### References:

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