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Transcendental lattices and supersingular reduction lattices of a singular $ K3$ surface

Author: Ichiro Shimada
Journal: Trans. Amer. Math. Soc. 361 (2009), 909-949
MSC (2000): Primary 14J28; Secondary 14J20, 14H52
Published electronically: July 30, 2008
MathSciNet review: 2452829
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Abstract: A $ K3$ surface $ X$ defined over a field $ k$ of characteristic 0 is called singular if the Néron-Severi lattice $ \mathrm{NS}(X)$ of $ X\otimes \overline{k}$ is of rank $ 20$. Let $ X$ be a singular $ K3$ surface defined over a number field $ F$. For each embedding $ \sigma: F\hookrightarrow \mathbb{C}$, we denote by $ T(X^\sigma)$ the transcendental lattice of the complex $ K3$ surface $ X^\sigma$ obtained from $ X$ by $ \sigma$. For each prime $ \mathfrak{p}$ of $ F$ at which $ X$ has a supersingular reduction $ X_{\mathfrak{p}}$, we define $ L(X, \mathfrak{p})$ to be the orthogonal complement of $ \mathrm{NS}(X)$ in $ \mathrm{NS}(X_{\mathfrak{p}})$. We investigate the relation between these lattices $ T(X\sp\sigma)$ and $ L(X,\mathfrak{p})$. As an application, we give a lower bound for the degree of a number field over which a singular $ K3$ surface with a given transcendental lattice can be defined.

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Additional Information

Ichiro Shimada
Affiliation: Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526 Japan

Received by editor(s): November 8, 2006
Received by editor(s) in revised form: April 16, 2007
Published electronically: July 30, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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