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\(K3\) 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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